Adaptive Estimation and Equalization of Doubly-selective Fading
Channels using Basis Expansion Models
by
Hyosung Kim
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
December 13, 2010
Keywords: basis expansion model, adaptive channel estimation, doubly-selective channels
Copyright 2010 by Hyosung Kim
Approved by
Jitendra K. Tugnait, Chair, James B. Davis Professor of Electrical and Computer
Engineering
Stanley J. Reeves, Professor of Electrical and Computer Engineering
Shiwen Mao, Assistant Professor of Electrical and Computer Engineering
Abstract
Wireless channels, due to multipath propagation and Doppler spread, are characterized
by frequency- and time-selectivity, so-called doubly-selective wireless channels. In this disser-
tation, we concentrate on adaptive channel estimation and equalization for communications
systems over doubly selective channels, exploiting basis expansion models (BEM). Since the
time-varying nature of the channel is well captured in the complex exponential basis expan-
sion model (CE-BEM) by the known exponential basis functions, the time variations of the
(unknown) BEM coefficients are likely much slower than those of the channel and thus more
convenient to track.
First, a subblock-wise channel estimation based on CE-BEM is considered, where we
track the BEM coefficients using time-multiplexed (TM) periodic training symbols. Assum-
ing the BEM coefficients follow a first-order AR model, Kalman filtering is used to track
the BEM coefficients. This first-order AR assumption, however, is not necessarily true and
possibly incurs significant modeling errors in estimation. We then seek adaptive channel esti-
mation schemes with no a priori model for the BEM coefficients using recursive least-square
(RLS) algorithm with finite memory.
Next, taking the performance of BEM-based approach into account, we investigate an
adaptive soft-in soft-out turbo equalization for coded communication systems, exploiting CE-
BEM for the overall channel variations and AR model for the BEM coefficients. We extend
an existing turbo equalization approach based on symbol-wise AR modeling of channels to
channels based on CE-BEM.
Based on the subblock-wise approach, we also propose a decision-directed tracking based
on BEM, where we track the BEM coefficients using the information symbol decisions of a
decision feedback equalizer (DFE) as virtual training. The time gap between symbol deci-
sions and required channel estimates, arising from the decision-directed tracking, is bridged
by CE-BEM-based channel prediction using the estimated BEM coefficients. We also adopt
an exponentially-weighted (EW) RLS algorithm for our BEM-based decision-directed track-
ing scheme. Decision-directed tracking requires fewer training symbols compared to the
training-based tracking, for the same performance.
The contribution of the proposed BEM-based channel estimation and equalization schemes
is that we track the BEM coefficients in CE-BEM, not the channel taps directly, based on
ii
the subblock-wise approach and then generate the time-varying channels via CE-BEM. Sim-
ulation examples illustrate the superior performance of our approach over several existing
doubly-selective channel estimators.
Finally, we extend all the proposed channel estimation and equalization approaches to
multiple-input multiple-output (MIMO) systems.
iii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Previous Work and Contributions . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Wireless Channel and Background . . . . . . . . . . . . . . . . . . . . . 6
2.1 Wireless Communication Channel Characteristics . . . . . . . . . . . . . . . 6
2.1.1 Rayleigh Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Jakes? Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Representations of Wireless Channels . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Autoregressive (AR) Model . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Complex Exponential Basis Expansion Model (CE-BEM) . . . . . . . 10
2.3 Block-Adaptive Channel Estimation using CE-BEM [30,70] . . . . . . . . . . 12
2.4 Useful Equalization Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.1 Kalman Detector (KD) . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.2 Mimimum Mean Square Error Decision Feedback Equalizer (MMSE-
DFE) [40] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Turbo Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5.1 Principle of Turbo Equalization [39] . . . . . . . . . . . . . . . . . . . 20
2.5.2 Maximum A Posteriori (MAP) Decoding Algorithm for Convolutional
Codes [72] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Doubly-selective Adaptive Channel Estimation Using Exponential
Basis Models and Subblock Tracking . . . . . . . . . . . . . . . . . . . . 24
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Subblock-wise Channel Estimation using CE-BEM . . . . . . . . . . . . . . . 25
3.2.1 Subblock-wise Kalman Tracking [52] . . . . . . . . . . . . . . . . . . 26
iv
3.2.2 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Subblock-wise MIMO Channel Estimation using CE-BEM . . . . . . . . . . 38
3.3.1 Subblock-wise Kalman Tracking for MIMO Channel . . . . . . . . . . 38
3.3.2 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 AR(1) Coefficient u1D6FC for Subblock-wise Kalman Tracking . . . . . . . . . . . 49
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Recursive Least-squares Doubly-selective Channel Estimation Us-
ing Exponential Basis Models and Subblock-wise Tracking . . . . . 53
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Subblock-wise RLS Channel Estimation using CE-BEM [23] . . . . . . . . . 54
4.2.1 Exponentially-Weighted RLS Tracking . . . . . . . . . . . . . . . . . 56
4.2.2 Sliding-Window RLS Tracking . . . . . . . . . . . . . . . . . . . . . . 57
4.2.3 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Subblock-wise RLS MIMO Channel Estimation using CE-BEM . . . . . . . 63
4.3.1 RLS Tracking for MIMO Channel . . . . . . . . . . . . . . . . . . . . 63
4.3.2 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Forgetting Factor u1D706 for Subblock-wise EW-RLS Tracking . . . . . . . . . . . 72
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 TurboEqualizationfor Doubly-SelectiveChannelsUsingNonliner
Kalman Filtering and Basis Expansion Models . . . . . . . . . . . . . . 77
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Turbo Equalization using Extended Kalman Filter (EKF) and CE-BEM . . . 79
5.2.1 System Model and Turbo Equalization Receiver . . . . . . . . . . . . 79
5.2.2 Adaptive Soft-In Soft-Out Nonlinear Kalman Equalizer . . . . . . . . 84
5.2.3 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2.4 EXIT Chart Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3 MIMO Turbo Equalization using EKF and CE-BEM . . . . . . . . . . . . . 101
5.3.1 MIMO System Model and Turbo Equalization Receiver . . . . . . . . 101
5.3.2 Adaptive Soft-In Soft-Out Nonlinear Kalman Equalizer for Doubly-
Selective MIMO Channels . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3.3 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.4 EXIT Chart Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6 Decision-Directed Tracking for Doubly-Selective Channel using
Basis Expansion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
v
6.2 Decision-Directed Tracking using CE-BEM . . . . . . . . . . . . . . . . . . . 119
6.2.1 Decision-Directed Kalman Tracking . . . . . . . . . . . . . . . . . . . 122
6.2.2 Decision-Directed EW-RLS Tracking . . . . . . . . . . . . . . . . . . 123
6.2.3 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.3 Decision-Directed MIMO Tracking using CE-BEM . . . . . . . . . . . . . . . 133
6.3.1 Decision-Directed MIMO Tracking via KF or EW-RLS . . . . . . . . 136
6.3.2 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.4 Performance Analysis of Decision-Directed EW-RLS Tracking using CE-BEM 143
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.1 Summary of Original Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.2 Suggested Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
vi
List of Figures
2.1 Decision-feedback equalizer (DFE). . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 A turbo equalization receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1 Subblock-wise channel estimation: overlapping blocks and subblocks, where
one block comprises several subblocks and each subblock has an information
session followed by a training session. . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Subblock-wise Kalman channel estimation: performance comparison for SNR?s
under u1D441 = u1D43E = 1,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 20. . . . . . . . . . . . . . . . . . . . . 31
3.3 Subblock-wise Kalman channel estimation: performance comparison for Doppler
spread under u1D441 = u1D43E = 1,SNR = 20dB,u1D45Au1D44F = 20. . . . . . . . . . . . . . . . 32
3.4 Subblock-wise Kalman channel estimation: performance comparison for SNR?s
under u1D441 = u1D43E = 1,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 40. . . . . . . . . . . . . . . . . . . . . 33
3.5 Subblock-wise Kalman channel estimation: performance comparison for Doppler
spread under u1D441 = u1D43E = 1,SNR = 20dB,u1D45Au1D44F = 40. . . . . . . . . . . . . . . . 34
3.6 Subblock-wise Kalman channel estimation: performance comparison for SNR?s
under u1D43E = 1,u1D453u1D451u1D447u1D460 = 0.0025,u1D45Au1D44F = 80. . . . . . . . . . . . . . . . . . . . . . . 36
3.7 Subblock-wise Kalman channel estimation: performance comparison for SNR?s
under u1D43E = 1,u1D453u1D451u1D447u1D460 = 0.0025,u1D45Au1D44F = 100. . . . . . . . . . . . . . . . . . . . . . 37
3.8 Subblock-wise Kalman MIMO channel estimation: performance comparison
for SNR?s under u1D441 = u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 20. . . . . . . . . . . . . . . . 43
3.9 Subblock-wise Kalman MIMO channel estimation: performance comparison
for Doppler spread under u1D441 = u1D43E = 2,SNR = 20dB,u1D45Au1D44F = 20. . . . . . . . . . 44
3.10 Subblock-wise Kalman MIMO channel estimation: performance comparison
for SNR?s under u1D441 = u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 40. . . . . . . . . . . . . . . . 45
3.11 Subblock-wise Kalman MIMO channel estimation: performance comparison
for Doppler spread under u1D441 = u1D43E = 2,SNR = 20dB,u1D45Au1D44F = 40. . . . . . . . . . 46
3.12 Subblock-wise Kalman MIMO channel estimation: performance comparison
for SNR?s under u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.0025,u1D45Au1D44F = 80. . . . . . . . . . . . . . . . . 48
3.13 Subblock-wise Kalman MIMO channel estimation: performance comparison
for SNR?s under u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.0025,u1D45Au1D44F = 100. . . . . . . . . . . . . . . . . 49
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3.14 Subblock-wise Kalman channel estimation: performances for AR(1) coefficient
u1D6FC?s, under u1D441 = u1D43E = 1,u1D453u1D451u1D447u1D460 = 0.01,SNR = 20dB,u1D45Au1D44F = 20 and 40. . . . . . . 52
4.1 Subblock-wise RLS channel estimation: performance comparison for SNR?s
under u1D441 = u1D43E = 1,u1D453u1D451u1D447u1D460 = 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Subblock-wise RLS channel estimation: performance comparison for Doppler
spread under u1D441 = u1D43E = 1,SNR = 20dB. . . . . . . . . . . . . . . . . . . . . 61
4.3 Subblock-wise RLS channel estimation: performance comparison for SNR?s
under u1D43E = 1,u1D453u1D451u1D447u1D460 = 0.0025, u1D45Au1D44F = 80. . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Subblock-wise RLS MIMO channel estimation: performance comparison for
SNR?s under u1D441 = u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.01. . . . . . . . . . . . . . . . . . . . . . . 70
4.5 Subblock-wise RLS MIMO channel estimation: performance comparison for
Doppler spread under u1D441 = u1D43E = 2,SNR = 20dB. . . . . . . . . . . . . . . . . 71
4.6 Subblock-wise RLS MIMO channel estimation: performance comparison for
SNR?s under u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.0025, u1D45Au1D44F = 80. . . . . . . . . . . . . . . . . . . 72
4.7 Subblock-wise EW-RLS channel estimation: performances for forgetting fac-
tor u1D706?s, under u1D453u1D451u1D447u1D460 = 0.01,SNR = 20dB,u1D45Au1D44F = 20 and 40. . . . . . . . . . . . 75
5.1 Bit-interleaved coded modulation system model for doubly-selective fading
channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Turbo equalization receiver. Following [10,49,68] and contrary to the original
turbo principle, a posteriori LLR Lu1D44E{c(u1D45B)} = Lu1D440u1D452 {c(u1D45B)}+Lu1D437u1D452 {c(u1D45B)} instead
of the extrinsic LLR Lu1D437u1D452 {c(u1D45B)} can be input to the LLR-to-symbol block.
Inclusion of Lu1D440u1D452 {c(u1D45B)} to create a posteriori LLR is shown via dashed line.
For our proposed approach we follow [10,49,68]. . . . . . . . . . . . . . . . . 81
5.3 Structure of the adaptive SfiSfo equalizer proposed in [68] . . . . . . . . . . . 89
5.4 Turbo equalization: performance comparison for SNR?s underu1D453u1D451u1D447u1D460 = 0.01,u1D459u1D45D =
5,u1D459u1D460 = 20 (20% training overhead). . . . . . . . . . . . . . . . . . . . . . . . 91
5.5 Turbo equalization: performance comparison for normalized Doppler spread(u1D453u1D451u1D447u1D460)?s
under SNR = 10dB,u1D459u1D45D = 5,u1D459u1D460 = 20 (20% training overhead). . . . . . . . . . . 92
5.6 Turbo equalization: performance comparison for SNR?s underu1D453u1D451u1D447u1D460 = 0.004,u1D459u1D45D =
5,u1D459u1D460 = 20 (20% training overhead). . . . . . . . . . . . . . . . . . . . . . . . 93
5.7 Turbo equalization: performance comparison of TE-BEM(200) with different
interleaver lengths for SNR?s under u1D453u1D451u1D447u1D460 = 0.01,u1D459u1D45D = 5,u1D459u1D460 = 20 (20% training
overhead). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.8 Turbo equalization: performances of TE-BEM(200) for AR(1) coefficient u1D6FC?s
under u1D453u1D451u1D447u1D460 = 0.01,SNR = 10dB,u1D459u1D45D = 5,u1D459u1D460 = 20 (20% training overhead). . . . 95
viii
5.9 Turbo equalization: performances of TE-BEM(400) for AR(1) coefficient u1D6FC?s
under u1D453u1D451u1D447u1D460 = 0.01,SNR = 10dB,u1D459u1D45D = 5,u1D459u1D460 = 20 (20% training overhead). . . . 96
5.10 Simulation setup for generating extrinsic information transfer functions . . . 97
5.11 EXIT charts of the proposed turbo equalizer using CE-BEM withu1D447 = 200,u1D444 =
5 for different SNR?s underu1D453u1D451u1D447u1D460 = 0.008,u1D459u1D45D = 5,u1D459u1D460 = 20 (20% training overhead). 98
5.12 EXIT charts for TE-AR5, TE-AR9, TE-BEM(200) and TE-BEM(400) under
fixed u1D453u1D451u1D447u1D460, SNR = 10dB,u1D459u1D45D = 5,u1D459u1D460 = 20 (20% training overhead). . . . . . . . 99
5.13 EXIT charts for different u1D453u1D451u1D447u1D460?s under SNR = 10dB,u1D459u1D45D = 5,u1D459u1D460 = 20 (20%
training overhead). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.14 Bit-interleaved coded modulation system model for doubly-selective fading
MIMO channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.15 MIMO turbo equalization receiver. Following [10,49,68], a posteriori LLR
Lu1D44E{cu1D458(u1D45B)} = Lu1D440u1D452 {cu1D458(u1D45B)}+Lu1D437u1D452 {cu1D458(u1D45B)} is input to the LLR-to-symbol block.
Inclusion of Lu1D440u1D452 {c(u1D45B)} to create a posteriori LLR is shown via dashed line. . 103
5.16 MIMO turbo equalization: performance comparison for SNR?s under u1D441 =
u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.01,u1D459u1D45D = 6,u1D459u1D460 = 14 (30% training overhead). . . . . . . . . . . 109
5.17 MIMO turbo equalization: performance comparison for normalized Doppler
spread(u1D453u1D451u1D447u1D460)?s under u1D441 = u1D43E = 2,SNR = 10dB,u1D459u1D45D = 6,u1D459u1D460 = 14 (30% training
overhead). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.18 MIMO turbo equalization: performance comparison for SNR?s under u1D441 =
u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.004,u1D459u1D45D = 6,u1D459u1D460 = 14 (30% training overhead). . . . . . . . . . 111
5.19 MIMO turbo equalization: performance comparison of TE-BEM(200) with
different interleaver lengths for SNR?s under u1D441 = u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.01,u1D459u1D45D =
6,u1D459u1D460 = 14 (30% training overhead). . . . . . . . . . . . . . . . . . . . . . . . 112
5.20 Simulation setup for generating extrinsic information transfer functions for
MIMO system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.21 EXIT charts for different SNR?s under u1D441 = u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.008,u1D459u1D45D = 6,u1D459u1D460 =
14 (30% training overhead). . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.22 EXIT charts for TE-AR5, TE-AR9, TE-BEM(200) and TE-BEM(400) under
u1D441 = u1D43E = 2, fixed u1D453u1D451u1D447u1D460, SNR = 10dB,u1D459u1D45D = 6,u1D459u1D460 = 14 (30% training overhead). 115
5.23 EXIT charts for different u1D453u1D451u1D447u1D460?s under u1D441 = u1D43E = 2,SNR = 10dB,u1D459u1D45D = 6,u1D459u1D460 =
14 (30% training overhead). . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.1 Decision-directed tracking: overlapping blocks that differ by a small step size
u1D45Au1D460. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
ix
6.2 Decision-directed tracking: performance comparison for SNR?s, under u1D441 =
1,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3 Decision-directed tracking: performance comparison for SNR?s, under u1D441 =
2,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4 Decision-directed tracking: performance comparison for u1D453u1D451u1D447u1D460?s, under u1D441 =
2,SNR = 14dB,u1D45Au1D44F = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.5 Decision-directed tracking: performance comparison for SNR?s, under u1D441 =
1,u1D453u1D451u1D447u1D460 = 0.004,u1D45Au1D44F = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.6 Decision-directed tracking: performance comparison for SNR?s, under u1D441 =
2,u1D453u1D451u1D447u1D460 = 0.004,u1D45Au1D44F = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.7 Decision-directed Kalman tracking: performances with different step sizeu1D45Au1D460?s
for SNR?s, under u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 100. . . . . . . . . . . . . . . . . . . . . . 132
6.8 Decision-directed Kalman tracking: performances for AR(1) coefficient u1D6FC?s,
under u1D441 = 2,u1D453u1D451u1D447u1D460 = 0.01,SNR = 14dB,u1D45Au1D44F = 100. . . . . . . . . . . . . . . . 133
6.9 Decision-directed EW-RLS tracking: performances for forgetting factor u1D706?s,
under u1D441 = 2,u1D453u1D451u1D447u1D460 = 0.01,SNR = 14dB,u1D45Au1D44F = 100. . . . . . . . . . . . . . . . 134
6.10 Decision-directed MIMO tracking: performance comparison for SNR?s, under
u1D441 = 3,u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 100. . . . . . . . . . . . . . . . . . . . . . . 140
6.11 Decision-directed MIMO tracking: performance comparison for u1D453u1D451u1D447u1D460?s, under
u1D441 = 3,u1D43E = 2,SNR = 20dB,u1D45Au1D44F = 100. . . . . . . . . . . . . . . . . . . . . . 141
6.12 Decision-directed EW-RLS MIMO tracking: performances with different step
size u1D45Au1D460?s for SNR?s, under u1D441 = 3,u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 100. . . . . . . . 142
6.13 Decision-directed EW-RLS tracking: MSE with different step size u1D45Au1D460?s for
forgetting factor u1D706?s, under u1D441 = 2,u1D453u1D451u1D447u1D460 = 0.01,SNR = 14u1D451u1D435. . . . . . . . . . 150
x
List of Tables
3.1 Subblock-wise Kalman filtering: flop count for channel estimation over one
subblock of u1D45Au1D44F symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Block-wise least-square and subblock-wise Kalman filtering: comparative flop
count for channel estimation over one block of u1D447u1D44E = 400 symbols. . . . . . . . 42
4.1 Subblock-wise EW-RLS channel estimation: flop count over one subblock of
u1D45Au1D44F symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Block-wise least-square and subblock-wise RLS: flop count for channel esti-
mation over one block of u1D447u1D435 symbols. . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Block-wise least-square and subblock-wise RLS: comparative flop count for
channel estimation over one block of u1D447u1D435 symbols. . . . . . . . . . . . . . . . 68
4.4 Theoretical u1D706 for subblock-wise EW-RLS channel estimation. . . . . . . . . . 75
5.1 Turbo equalization: comparison between actual BER (via simulations) and
predicted BER (via EXIT charts) . . . . . . . . . . . . . . . . . . . . . . . . 100
6.1 DFE: flop count for one cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2 Subblock-wise and decision-directed tracking with DFE: comparative flop
count over one block of u1D447u1D435 symbols. . . . . . . . . . . . . . . . . . . . . . . . 126
6.3 Subblock-wise and decision-directed MIMO channel tracking with DFE: com-
parative flop count over one block of u1D447u1D435 symbols. . . . . . . . . . . . . . . . 138
xi
List of symbols
? approximately equal to
? Kronecker product
0u1D440 u1D440 ?1 all zeros vector
0u1D440?u1D441 u1D440 ?u1D441 all zeros matrix
u1D44E lower case letters for scalars
?u1D44E? integer ceiling of u1D44E
?u1D44E? integer floor of u1D44E
?u1D44E? magnitude or absolute value of u1D44E
a lower case letters in bold face for column vectors
?u1D44E? Euclidean norm of u1D44E
?u1D44E?u1D439 Frobenius norm of u1D44E
A upper case letters in bold face for matrices
A? complex conjugate of A
A? Moore-Penrose pseudo-inverse of A
A?1 inverse of A
Au1D43B complex conjugate transpose of A
Au1D447 transpose of A
arg max
u1D465
u1D453(u1D465) value of u1D465 for which u1D453(u1D465) attains its maximum
arg min
u1D465
u1D453(u1D465) value of u1D465 for which u1D453(u1D465) attains its minimum
u1D6FF(?) Kronecker delta function, defined as
u1D6FF(u1D45B) =
?
??
??
1, if u1D45B = 0
0, otherwise
u1D451u1D456u1D44Eu1D454{u1D44E1,??? ,u1D44Eu1D441} u1D441 ?u1D441 diagonal matrix with diagonal elements u1D44E1,??? ,u1D44Eu1D441
u1D438{?} expectation operator
Iu1D441 u1D441 ?u1D441 identity matrix
max(?) maximum value operator
min(?) minimum value operator
u1D4AA(?) big O notation
u1D461u1D45F{A} trace of a square matrix A
xii
List of Abbreviations
AM amplitude modulation
APP a posteriori probability
AR auto-regressive
AWGN additive white Gaussian noise
BEM basis expansion model
BER bit error rate
BICM bit-interleaved coded modulation
BPSK binary phase-shift keying
CDMA code division multiple access
CE-BEM complex exponential basis expansion model
CSI channel state information
DFE decision feedback equalizer
DFT discrete Fourier transform
DKL-BEM discrete Karhuen-Lo`eve basis expansion model
DML deterministic maximum likelihood
DPS discrete prolate spheroidal
DPS-BEM discrete prolate spheroidal basis expansion model
EKF extended Kalman filter
EXIT extrinsic information transfer
EW-RLS exponentially-weighted recursive least square
FB feed-back
FF feed-forward
FIR finite impulse response
FM frequency modulation
ID iterative decoding
i.i.d. independent and identically distributed
ISI inter-symbol interference
KD Kalman detector
KF Kalman filter
LLR log-likelihood ratio
LMS least mean squares
xiii
LS least squares
MAP maximum a posteriori
MIMO multiple-input multiple-output
ML maximum likelihood
MSE mean square error
MMSE minimum mean square error
NCMSE normalized channel mean square error
OFDM orthogonal frequency division multiplexing
OP-BEM orthogonal polynomial basis expansion model
pdf probability density function
PSAM pilot symbol aided modulation
QAM quadrature amplitude modulation
PSK phase-shift keying
QPSK quadrature phase-shift keying
RLS recursive least-square
SfiSfo soft-in soft-out
SIMO single-input multiple-output
SISO single-input single-output
SNR signal-to-noise ratio
SW-RLS sliding-window recursive least square
TE turbo equalization
TM time-multiplexed
WSS wide-sense stationary
WSSUS wide-sense stationary uncorrelated scattering
xiv
Chapter 1
Introduction
Wireless channels are challenging communication media with limited bandwidth, rel-
atively low capacity per unit bandwidth, random amplitude and phase fluctuations, and
inter-symbol interference (ISI). Due to multipath propagation and Doppler spread, wireless
channels are characterized by frequency- and time-selectivity [60]. To design a physical link
with data rates approaching the fundamental information capacity limits of the wireless
channel, accurate knowledge of the channel state information (CSI) becomes a prerequisite
for many physical layer approaches. At the receiver end, equalizers are usually used to com-
pensate for the signal distortion. One may design an equalizer based on a channel estimate
or directly using the received signal. Accurate modeling of the temporal evolution of the
channel plays a crucial role for estimation and tracking.
Among various models, the first-order autoregressive (AR) process is often regarded
as a tractable model to describe a time-varying channel, where the channel is assumed to
be Markovian; that is, for the current channel sample, the effect of channel samples other
than the immediately preceding one is negligible [16]. This Markovian assumption has been
justified for Rayleigh fading channels in [16] by considering the mutual information between
successive channel samples.
The AR models, however, have their own drawback. When time-multiplexed (TM)
training is used, channel tracking may not perform well during information-symbol trans-
missions (information sessions) since the information data are unknown. During information
sessions, channel estimates can only be obtained based on the results from training ses-
sions [76]. This channel prediction strategy, apparently, is not appropriate to a fast-varying
channel, where the AR model may lead to high estimation variance resulting in erroneous
symbol detections [6]. Potential solutions lie in exploiting the detected symbols for channel
tracking: In [64], a Kalman filter is used in a decision-feedback mode during information
sessions; per survivor processing (PSP) is used in [43], which embeds data-aided channel
estimation into the Viterbi algorithm; in [68], joint channel estimation and data detection
is implemented via extended Kalman filtering. Although channel tracking can be improved
by such means during information sessions, the phenomenon of error propagation can be
pronounced for fast-fading channels. More accurate channel modeling becomes necessary to
track fast-fading channels.
1
Recently, basis expansion models (BEM) have been widely investigated to represent
doubly-selective channels in wireless applications [8,11,35,63,70], for which, the time-varying
channel taps are expressed as superpositions of time-varying basis functions in modeling
Doppler effects, weighted by time-invariant coefficients. Candidate basis functions include
complex exponential (Fourier) functions [11,70], polynomials [8], wavelets [35], and discrete
prolate spheroidal sequences [63], etc. In contrast to AR models that describe temporal
variation on a symbol-by-symbol update basis, a BEM depicts the evolution of the channel
over a period (block) of time. Since the time-varying nature of the channel can be well
captured in a BEM by the known basis functions, the BEM coefficients should evolve much
more slowly than the channel taps, and hence are more convenient to track in a fast-fading
environment. We will discuss the adaptive channel estimation and equalization approaches
using BEM?s.
1.1 Previous Work and Contributions
In this section, we summarize the previous research work on BEM-based channel esti-
mation, equalization, and related areas.
Doubly-selective channel estimation using a complex exponential basis expansion model
(CE-BEM) and time-multiplexed training is considered in [30,70], where CE-BEM based
on Fourier basis functions is applied to represent the time-variant channel. However, since
the Fourier basis expansion has the major drawback that the rectangular window associated
with the discrete Fourier transform (DFT) introduces spectral leakage [20], the bit error
rate (BER) suffers an error floor [17,62]. In [62,63], the linear minimum mean-square-error
(MMSE) channel estimation using discrete prolate spheroidal (DPS) sequences is considered.
It is shown that DPS-BEM-based approaches outperform CE-BEM-based approaches for
doubly-selective channel estimation and data detection.
Although the CE-BEM is more convenient in theoretical analysis, its modeling error
is noticeable due to the spectral leakage. To mitigate this leakage, the over-sampled CE-
BEM has been considered in [14] where the Doppler spectrum is said to be oversampled
compared to the case in [11,70]. In this dissertation, we will use the oversampled CE-BEM
for doubly-selective channel estimation.
To acquire the channel state information at the receiver, training symbols are usually
periodically inserted during transmission, which is known as pilot symbol aided modulation
(PSAM) [22]. Optimization of the PSAM for CE-BEM based doubly-selective channel models
has been considered in [62,70] where the time-multiplexed (TM) training sequence is designed
to minimize the channel estimation mean-square-error (MSE). In the case of CE-BEM with
2
independent basis expansion coefficients, minimizing the channel estimation MSE is also
shown in [30,70] to be equivalent to maximizing a lower bound on the estimated channel-
based average capacity. No such considerations are to be found in [62,63] where the doubly-
selective channel is represented by DPS-BEM. We adopt the TM training scheme in [30,70]
for the proposed subblock-wise tracking approach.
Decision-directed channel tracking using a polynomial BEM has been investigated in [8],
where the BEM coefficients are updated every block via the RLS algorithm within a sliding
window. Channel estimation using Kalman filtering and polynomial or CE-BEM?s for OFDM
systems has been explored in [26,41,44], among which decision-directed tracking is considered
in [26,41]. All these contributions consider block-by-block updating unlike our contribution
where we exploit subblock-wise updating. The distinction is as follows: One block comprises
several subblocks. For parameter identifiability, one needs the number of subblocks at least as
large as the number of basis functions used for channel modeling. Data in just one subblock
do not satisfy the parameter identifiability requirements. However, unlike the block-wise
schemes where the receiver has to collect the whole block of data in order to generate channel
estimates and perform equalization, in the proposed subblock-wise approaches the receiver
is able to accomplish the two tasks after every subblock.
In our subblock-wise approaches, we first assume that the BEM coefficients (rather than
the time-varying channel taps) follow a first-order AR model, and then Kalman filtering is
used to track the coefficients. This first-order AR assumption, however, is not necessarily
true for a real-world channel, and possibly incurs modeling error in estimation. We then seek
adaptive channel estimation schemes with no a priori models for the BEM coefficients. Two
adaptive filtering algorithms with finite memory are considered: the exponentially-weighted
recursive least-squares (RLS) algorithm and the sliding-window RLS algorithm.
We also consider BEM-based approach to coded modulation communication systems
using turbo equalization receiver. Turbo (iterative) equalization is a powerful suboptimal
technique used in place of the computationally prohibitive but optimal maximum likelihood
(ML) or maximum a posteriori (MAP) sequence detection based on a super trellis. By
combining a MAP equalizer and a MAP decoder, and exchanging probabilistic information
about data symbols iteratively, turbo equalization usually can achieve close-to-optimal per-
formance but with much lower complexity [1,5]. In [71], a turbo-equalization-like system
using linear equalizers based on soft interference cancellation and linear minimum mean-
square error (MMSE) filtering is proposed as part of a multiuser detector for code division
multiple access (CDMA). Based on this work, a variety of soft-in soft-out equalizers employ-
ing linear MMSE and decision feedback equalization (DFE) are proposed in [38,39] and so
forth.
3
For doubly-selective channels, an adaptive equalizer has been presented in [68], using
extended Kalman filter (EKF) to incorporate channel estimation into the equalization pro-
cess. This adaptive soft nonlinear Kalman equalizer jointly optimizes the estimates of the
channel and data symbols in each iteration. Based on the turbo equalization approach pro-
posed in [68] and our subblock-wise or decision-directed BEM-based approach, we present
an adaptive turbo equalizer based on CE-BEM. This adaptive equalizer takes the decision
of data symbols provided by the decoder as its a priori information and the performance
can be improved iteratively. The extrinsic information transfer (EXIT) chart analyses of our
BEM-based turbo equalization approach are also provided.
Based on the subblock-wise approach, decision-directed tracking of doubly-selective
channels is presented using CE-BEM in Chapter 6. In [6], the decision-directed scheme
was proposed that relies on the AR model only. Acting as virtual training symbols, the
information symbol decisions, provided by a DFE (with delay u1D451? 0) utilizing the estimated
channel, are used in Chapter 6 to enhance the estimation of the BEM coefficients, so that
much of the spectrum resource allocated to training can be saved. Although a time gap still
exists between the available symbol decisions and the channel estimates required by the DFE,
it can be successfully bridged by the CE-BEM-based channel prediction, without incurring
much estimation variance. Our decision-directed scheme based on subblock tracking, updates
the BEM coefficients of much smaller size of block than BEM period. The periodic training
symbols are still necessary to recover the channel tracking from possible phase ambiguity due
to the error propagation. To circumvent an arbitrary modeling error incurred due to the AR
assumption for the BEM coefficients, an adaptive channel estimation scheme with no a pri-
ori model for the BEM coefficients is proposed, where the exponentially-weighted recursive
least-squares (EW-RLS) algorithm is considered for subblock-wise channel tracking.
Computer simulation examples demonstrate the superior performance of the proposed
channel estimation and equalization using CE-BEM, compared to several existing approaches.
1.2 Organization
The rest of this dissertation is organized as follows.
In Chapter 2, we provide system models and some background that we use for chan-
nel estimation and equalization in this dissertation. This chapter covers wireless channel
characteristics and representations, TM training scheme, symbol detection using Kalman fil-
ter and Minimum-Mean-Square-Error Decision-Feedback Equalizer (MMSE-DFE) and turbo
equalization.
4
In Chapter 3, a subblock-wise Kalman tracking approach to doubly-selective channel
estimation is presented, exploiting the oversampled CE-BEM for the overall channel varia-
tions, and an AR model to update the BEM coefficients. The channel estimates are updated
subblock-by-subblock unlike other existing block-by-block or symbol-by-symbol schemes.
In Chapter 4, we present a doubly-selective RLS channel estimator using CE-BEM,
replacing Kalman filter in Chapter 3 with RLS filter so that we can avoid the modeling error
incurred when using the arbitrary AR(1) model for BEM coefficients.
In Chapter 5, we propose an adaptive turbo equalization for doubly-selective channels
using CE-BEM and provide EXIT chart analyses. Based on the turbo equalization approach
proposed in [68], our proposed iterative turbo equalizer with nonlinear Kalman filtering
jointly optimizes the estimates of BEM channel coefficients and data symbol detection in
each iteration.
In Chapter 6, a decision-directed tracking approach for doubly-selective channels is
considered, exploiting the CE-BEM for the overall channel variations. We track the BEM
coefficients via Kalman filtering based on an AR model to update BEM coefficients, aided
by symbol decisions from a DFE. We also present EW-RLS tracking with a forgetting factor,
a finite-memory decision-directed tracking.
We extend all the proposed algorithms to multi-input multi-output (MIMO) scenario
throughout the preceding chapters. The dissertation concludes with Chapter 7, where future
research directions are also suggested.
5
Chapter 2
Wireless Channel and Background
In this chapter, we briefly review the wireless channel characteristics in Sec 2.1, that
includes Jakes? model, which will be used as the model of the ?real? channel in the simulation
examples of this dissertation. In Section 2.2, we introduce the representations of time-varying
channels, the AR models and CE-BEM, which are used for channel estimation purposes in
different schemes. We summarize the TM training scheme in Section 2.3 introduced in [30,70].
In Section 2.4, we introduce Kalman detector (KD) and MMSE-DFE that we employ for
symbol equalization at the receiver. We also present the well-known turbo equalization in
Section 2.5.
2.1 Wireless Communication Channel Characteristics
Due to multipath propagation and Doppler spread, wireless channels are characterized
by frequency- and time-selectivity [60]. A radio signal, experiencing distortions through
transmission by fading, background noises, and interferences of every sort, becomes stochastic
to an observer at the receiver. Small-scale fading, or simply fading, is the term to describe
the rapid fluctuations of the amplitudes, phases, or multipath delays of a signal over a short
period of time or travel distance, so that large-scale path loss may be ignored [60]. The goal
of channel estimation and equalization is mainly to combat small-scale fading.
Fading can be attributed to physical factors including multipath propagation, relative
motion between the transmitter and the receiver or surrounding objects, and the transmission
bandwidth of the signal, etc. [65]. The presence of reflecting objects and scatterers makes
the wireless channel constantly changing, which dissipates the signal energy and distorts the
signal in amplitude, phase, and time. Multiple versions of the transmitted signal arrive at
the receiver through different paths. The random amplitudes and phases of the different
multipath components induce fading. The relative motion between the transmitter and the
receiver, as well as the motion of the objects within the wireless channel, induces Doppler
spreads, which are typically time-varying and become a source of fading also.
6
2.1.1 Rayleigh Fading
The Rayleigh distribution is commonly used to describe the statistical time-varying
nature of the received envelope of a flat fading signal, or the envelope of an individual
multipath component [60]. If we assume that many statistically independent scattering
waves with random amplitudes and phases reach the receiver with the phases uniformly
lying in [0,2u1D70B), and there is no dominant non-fading signal component present (no line-of-
sight), by the central limit theorem, the real and imaginary parts of the sum of the scattering
waves are both Gaussian. The signal envelope A obeys a Rayleigh distribution, which has a
probability density function (pdf) given by
u1D453u1D434(u1D44E) :=
?
?
?
u1D44E
u1D70E2exp(?
u1D44E2
2u1D70E2) u1D44E? 0,
0 u1D44E< 0 (2.1)
with u1D70E2 being the time-average power of the received signal before envelope detection. The
phase u1D703 of the received signal is uniformly distributed with pdf
u1D453?(u1D703) := 12u1D70B,u1D703 ? [0,2u1D70B). (2.2)
The autocorrelation function of the received signal for two-dimensional isotropic scattering
and an omnidirectional receiving antenna is given by [15,45]
u1D445u1D434(u1D70F) = u1D70E2 cos(u1D714u1D450u1D70F)u1D43D0(u1D714u1D45Au1D70F) (2.3)
where u1D714u1D450 is the carrier radian frequency, u1D43D0(?) is the zero-order Bessel function of the first
kind and u1D714u1D45A is the maximum Doppler radian frequency spread. Any model that attempts
to model the Rayleigh flat fading narrow-band wireless channel has to exhibit the statistical
behaviors given by (2.1)-(2.3).
2.1.2 Jakes? Model
Clarke summarized the important characteristics of fading channels and provided a
useful mathematical model [45]. According to this model, Jakes proposed a sum-of-sinusoids
based simulator [65] that has been widely used and studied over the past decades. The
simulator supposes the received signal u1D446(u1D461) to be a superposition of waves
u1D446(u1D461) = u1D4380
u1D441uni2211.alt02
u1D45B=1
u1D436u1D45B cos(u1D714u1D450u1D461+u1D714u1D45Au1D461cosu1D434u1D45B + ?u1D45B) (2.4)
7
whereu1D4380 is the amplitude of the transmitted cosine wave, u1D436u1D45B is a random variable represent-
ing the attenuation of theu1D45B-th path, u1D434u1D45B is a random variable representing the angle of arrival
of theu1D45B-th ray with respect to the direction of motion of the receiver, ?u1D45B is a random variable
representing the phase shift undergone by the u1D45B-th ray. Note that the stochastic signal u1D446(u1D461)
representing the flat fading signal can be characterized byu1D441 sets of triples (u1D436u1D45B,u1D434u1D45B,?u1D45B). The
random variables u1D436u1D45B,u1D434u1D45B and ?u1D45B are assumed statistically independent.
To reduce the complexity, Jakes? model selects
u1D436u1D45B = 1?u1D441, u1D434u1D45B = 2u1D70Bu1D45Bu1D441 , ?u1D45B = 0, (2.5)
where u1D45B = 1,2,...,u1D441. Furthermore, u1D441 is of the form u1D441 = 4u1D440 + 2 where u1D440 is a positive
integer.
However, the simplification in (2.5) makes this simulation model deterministic and wide-
sense nonstationary [33,75]. In [75], a modified Jakes? simulator was proposed. It is wide-
sense stationary and its autocorrelation and cross correlation functions match the desired
reference model exactly. Following [75], the normalized low-pass fading process of the sta-
tistical sum-of-sinusoids simulation model is defined by
u1D44B(u1D461) = u1D44Bu1D450(u1D461) +u1D457u1D44Bu1D460(u1D461), (2.6a)
u1D44Bu1D450(u1D461) = 2?u1D440
u1D440uni2211.alt02
u1D45B=1
cos(u1D713u1D45B)cos(u1D714u1D45Au1D461cosu1D6FCu1D45B +u1D719), (2.6b)
u1D44Bu1D460(u1D461) = 2?u1D440
u1D440uni2211.alt02
u1D45B=1
sin(u1D713u1D45B)cos(u1D714u1D45Au1D461cosu1D6FCu1D45B +u1D719). (2.6c)
with
u1D6FCu1D45B = 2u1D70Bu1D45B?u1D70B+u1D7034u1D440 , u1D45B = 1,2,...,u1D440
where u1D703,u1D719 and u1D713u1D45B are statistically independent and uniformly distributed over [?u1D70B,u1D70B) for all
u1D45B. As u1D440 ? ?, the envelope ?u1D44B? is Rayleigh distributed and the phase ?u1D44B(u1D461) is uniformly
distributed over [?u1D70B,u1D70B), for which the pdf?s are given by
u1D453?u1D44B?(u1D465) = u1D465 exp(?u1D465
2
2 ), u1D465? 0,
u1D453?u1D44B(u1D703) = 12u1D70B, u1D703 ? [?u1D70B,u1D70B).
A minor defect, however, occurs in model (2.6) when u1D714u1D45A = 0 or the Doppler spread
is small: A Rayleigh distribution cannot be guaranteed [63]. This problem can be easily
8
resolved by replacing a common phase u1D719 by u1D719u1D45B, which is also uniformly distributed over
[?u1D70B,u1D70B) for all u1D45B. The simulation model is revised as [63]:
u1D44B(u1D461) = u1D44Bu1D450(u1D461) +u1D457u1D44Bu1D460(u1D461), (2.7a)
u1D44Bu1D450(u1D461) = 2?u1D440
u1D440uni2211.alt02
u1D45B=1
cos(u1D713u1D45B)cos(u1D714u1D45Au1D461cosu1D6FCu1D45B +u1D719u1D45B), (2.7b)
u1D44Bu1D460(u1D461) = 2?u1D440
u1D440uni2211.alt02
u1D45B=1
sin(u1D713u1D45B)cos(u1D714u1D45Au1D461cosu1D6FCu1D45B +u1D719u1D45B). (2.7c)
2.2 Representations of Wireless Channels
For channel estimation or tracking purposes, accurate modeling of the temporal evolu-
tion of the channel plays an important role. A parsimonious and accurate channel repre-
sentation is always preferred. Among various models for channel time variations, the AR
process, particularly the first-order AR model, is regarded as a tractable model to describe
a time-varying channel, where the channel is assumed to be Markovian, i.e., for the current
channel symbol, the effect of channel symbols other than the immediately preceding one is
negligible [16]. This Markovian assumption has been verified for Rayleigh fading channels
in [16], by considering the mutual information between channel symbols. The AR model has
been used for time varying channel estimation in [31,43,64,68,76].
The AR model, based on symbol-by-symbol update, is suitable for sequential time-
domain processing. When we deal with block processing schemes, it is often more convenient
to use a block-based channel models such as BEM?s. The BEM that is optimal in MSE is
the discrete Karhuen-Lo`eve BEM (DKL-BEM), which is a reduced-rank decomposition of a
certain type of Doppler spectrum [77]. The CE-BEM can be viewed as a special DKL-BEM
based on a white Doppler spectrum, and the DPS-BEM corresponds to the DKL-BEM with
a rectangular Doppler spectrum [77].
2.2.1 Autoregressive (AR) Model
It is possible to accurately represent a wide-sense stationary uncorrelated scattering
(WSSUS) channel by a large order AR model; see [6,68] and references therein. Let
?h(u1D45B) := [?(u1D45B;0) ?(u1D45B;1) ??? ?(u1D45B;u1D459)]u1D447 (2.8)
9
where ?h(u1D45B) is (u1D43F+1)?1 vector. Then a u1D443-th order AR model, AR(u1D443) for ?h(u1D45B) is given by
?h(u1D45B) = u1D443uni2211.alt02
u1D456=1
Au1D456?h(u1D45B?u1D456) +G0?w(u1D45B) (2.9)
where Au1D456s are the (u1D43F+1)?(u1D43F+1) AR coefficient matrices, G0 is also (u1D43F+1)?(u1D43F+1) and
independent and identically distributed (i.i.d.) (u1D43F+ 1)?1 driving noise ?w(u1D45B) is zero-mean
with identity covariance matrix. Suppose that we know the correlation function R?(u1D70F) =
u1D438
uni007B.alt02?
h(u1D45B+u1D70F)?hu1D43B(u1D45B)
uni007D.alt02
for lags u1D70F = 0,1,??? ,u1D443. The following Yule-Walker equation holds for
(2.9) [55]:
R?(u1D70F) =
u1D443uni2211.alt02
u1D456=0
Au1D456R?(u1D70F ?u1D456) +G0Gu1D43B0 u1D6FF(u1D70F). (2.10)
Using (2.10) for u1D70F = 1,2,??? ,u1D443, and the fact that R?(?u1D70F) = Ru1D43B? (u1D70F), one can estimate Au1D456s.
Using the estimated Au1D456s and (2.10) for u1D70F = 0, one can find G0Gu1D43B0 , from which one can find
(non-unique) G0 by computing its ?square root? [42, p.358]. In [6,68] only AR(1) or AR(2)
models have been used.
In this dissertation, we will use AR models for some simulation comparisons where
various channel taps are assumed to be mutually statistically independent. In this case we
have an independent AR process for each channel tap. Furthermore, following [6,68], we
consider the first-order AR models, given by
?(u1D45B;u1D459) = ?u1D6FCu1D459?(u1D45B?1;u1D459) + ?u1D464u1D459(u1D45B), (2.11)
where ?u1D6FCu1D459 is the AR coefficient and the driving noise ?u1D464u1D459(u1D45B) is zero-mean white with variance
u1D70E2?u1D464u1D459. If we assume that ?(u1D45B;u1D459) is also zero-mean with variance u1D70E2?u1D459, then one picks (to match
correlation functions at lags 0 and 1) [76]
?u1D6FCu1D459 = 1u1D70E2
?u1D459
u1D438{?(u1D45B;u1D459)??(u1D45B?1;u1D459)}, (2.12)
u1D70E2?u1D464u1D459 = u1D70E2?u1D459(1???u1D6FCu1D459?2). (2.13)
It must be noted that in practice, one would not know R?(u1D70F).
2.2.2 Complex Exponential Basis Expansion Model (CE-BEM)
Recently, deterministic CE-BEM have been widely investigated in wireless applications,
especially when the multipath is caused by a few strong reflectors, and path delays exhibit
variations due to the kinematics of the mobiles [11]. In these models, the time-varying taps
10
are expressed as a superposition of time-varying basis functions in modeling Doppler effects,
with time-invariant coefficients. By assigning temporal variations to basis functions, rapidly
fading channels with coherence time as small as a few tens of symbols can be captured. If
the delay spread and the Doppler spread of the channel (or at least the upper bounds of
them) are known, one can infer the basis functions of the CE-BEM [70]. Treating the basis
functions as known parameters, estimation of a time-varying process is reduced to estimate
time-invariant coefficients.
Consider a time-varying channel with impulse response ?(u1D461;u1D70F) (response at time u1D461 to a
unit impulse at time u1D461?u1D70F) which includes transmit-receive filters as well as doubly-selective
propagation effects. Let u1D460(u1D461) denote the complex baseband, continuous-time input signal
(with symbol duration u1D447u1D460), and u1D465(u1D461) denote the complex baseband, continuous-time received
signal. The noise-free received signal u1D465(u1D461) is the convolution of u1D460(u1D461) and ?(u1D461;u1D70F) [19]:
u1D465(u1D461) =
uni222B.alt02 ?
0
?(u1D461;u1D70F)u1D460(u1D461?u1D70F)u1D451u1D70F. (2.14)
Let u1D43B(u1D453;u1D70F) = uni222B.alt01??? ?(u1D461;u1D70F)u1D452?u1D4572u1D70Bu1D453u1D461u1D451u1D461 be the Fourier transform of ?(u1D461;u1D70F). If ?u1D43B(u1D453;u1D70F)? ? 0
for ?u1D70F? > u1D70Fu1D451, then u1D70Fu1D451 is defined as the delay-spread of the channel; if ?u1D43B(u1D453;u1D70F)? ? 0 for
?u1D453? > u1D453u1D451, then u1D453u1D451 is defined as the Doppler spread of the channel [70]. Sampling u1D460(u1D461),u1D465(u1D461)
and ?(u1D461;u1D70F) in (2.14) at the symbol rate, then for u1D461 = u1D45Bu1D447u1D460 ? [u1D4610,u1D4610 +u1D447u1D447u1D460), the sampled signal
u1D465(u1D45B) := u1D465(u1D461)?u1D461=u1D45Bu1D447u1D460 has the representation
u1D465(u1D45B) =
u1D43Funi2211.alt02
u1D459=0
?(u1D45B;u1D459)u1D460(u1D45B?u1D459). (2.15)
Over the block interval of [u1D4610,u1D4610 +u1D447u1D447u1D460), the channel impulse response {?(u1D45B;u1D459)}u1D447?1u1D45B=0can be
represented by u1D444 coefficients {?u1D45E(u1D459)}u1D444u1D45E=1 (which remain invariant throughout this block but
are allowed to change at the next block) and the corresponding u1D444 Fourier basis functions
that are common for each block. Then over the interval [u1D4610,u1D4610 +u1D447u1D447u1D460), the discrete-time
baseband equivalent channel model for the block can be described as [69,70]:
?(u1D45B;u1D459) =
u1D444uni2211.alt02
u1D45E=1
?u1D45E(u1D459)u1D452u1D457u1D714u1D45Eu1D45B, u1D459 = 0,1,??? ,u1D43F (2.16)
11
where u1D45B = (u1D456?1)u1D447u1D435,(u1D456?1)u1D447u1D435 + 1,??? ,u1D456u1D447u1D435 ?1 and one chooses (? is an integer)
u1D447 := ?u1D447u1D435, ? ? 1, (2.17)
u1D444? 2?u1D453u1D451u1D447u1D447u1D460?+ 1, (2.18)
u1D714u1D45E := 2u1D70Bu1D447 [u1D45E?(u1D444+ 1)/2], u1D45E = 1,2,??? ,u1D444 (2.19)
u1D43F := ?u1D70Fu1D451/u1D447u1D460?, (2.20)
u1D70Fu1D451 and u1D453u1D451 are respectively the delay spread and the Doppler spread, and u1D447u1D460 is the symbol
duration. The BEM coefficients ?u1D45E(u1D459)?s remain invariant during this block, but are allowed
to change at the next consecutive block; the Fourier basis functions {u1D452u1D457u1D714u1D45Eu1D45B} (u1D45E = 1,2,??? ,u1D444)
are common for every block. If the delay spread and the Doppler spread (or at least their
upper bounds) are known, one can infer the basis functions of the CE-BEM [70]. Treating
the basis functions as known, estimation of a time-varying process is reduced to estimating
the invariant coefficients over a block of u1D447u1D435 symbols.
If ? > 1 (e.g. ? = 2 or 3), then the Doppler spectrum is said to be oversampled [14]
compared to the case ? = 1 where the Doppler spectrum is said to be critically sampled.
In [11,70] only ? = 1 (henceforth called CE-BEM) is considered whereas [14] considers
? ? 2 (henceforth called over-sampled CE-BEM). For ? = 1, the rectangular window of
this truncated discrete Fourier transform (DFT)-based model introduces spectral leakage.
To mitigate this leakage, the over-sampled CE-BEM with ? = 2 or 3 has been considered
in [14] although the basis functions are no longer orthogonal by over-sampling.
The model (2.16) is periodic with period u1D447. Later we will use it for all time u1D45B (not just
the block size u1D447u1D435) by allowing the BEM coefficients ?u1D45E (u1D459)s to change with time. So long as
the effective ?memory? of the ?processors? used later is less than the model period (recall
that the channel is by no means periodic), there are no deleterious effects due to the use of
(2.16) for all time.
2.3 Block-Adaptive Channel Estimation using CE-BEM [30,70]
In this section, we summarize the time-multiplexed (TM) training scheme in [30,70] that
we employ in our subblock-wise tracking approach. Each transmitted block of symbols u1D460(u1D45B)
of length u1D447u1D435 symbols is segmented into ?u1D443 subblocks of length u1D45Au1D44F := u1D447u1D435/?u1D443 symbols, which
has the time-multiplexed u1D45Au1D451 information symbols and u1D45Au1D461 training symbols (u1D45Au1D44F = u1D45Au1D451 +u1D45Au1D461).
If s denotes a column-vector composed of {u1D460(u1D45B)}u1D456u1D447u1D435?1u1D45B=(u1D456?1)u1D447u1D435 for u1D456-th block, then it is arranged
as
s :=
uni005B.alt02
bu1D4470 cu1D4470 bu1D4471 cu1D4471 ??? bu1D447?u1D443?1 cu1D447?u1D443?1
uni005D.alt02u1D447
(2.21)
12
where bu1D45D is a column of u1D45Au1D451 information symbols and cu1D45D is a column of u1D45Au1D461 training symbols
for u1D45D = 0,1,??? , ?u1D443 ? 1. Based on CE-BEM (2.16), [70] has shown that for ? = 1 (the
critically-sampled CE-BEM) and ?u1D443 ? u1D444, the optimal session contains an impulse guarded
by zeros (silent periods), which has the structure
cu1D45D :=
uni005B.alt02
0u1D447u1D43F u1D6FE 0u1D447u1D43F
uni005D.alt02u1D447
, u1D6FE > 0 (2.22)
with u1D45Au1D461 = 2u1D43F + 1. Thus, given a transmission block of size u1D447u1D435, (2u1D43F + 1) ?u1D443 symbols have
to be devoted for training and the remaining are available for information symbols. Let
u1D45Bu1D45D := u1D45Du1D45Au1D44F +u1D45Au1D451 +u1D43F, (u1D45D = 0,1,??? , ?u1D443 ?1), denote the unique locations of (nonzero) u1D6FE?s in
the cu1D45D?s in the ?u1D443 subblocks. Let u1D45Bu1D45D := u1D45Du1D45Au1D44F + u1D45Au1D451 + u1D43F, (u1D45D = 0,1,??? , ?u1D443 ? 1), denote the
unique locations of (nonzero) u1D6FE?s in the cu1D45D?s in the ?u1D443 subblocks.
For multiple users, let{u1D460u1D458(u1D45B)}denoteu1D458-th user?s information sequence foru1D458 = 1,2,??? ,u1D43E.
If su1D458 denotes a column-vector composed of {u1D460u1D458 (u1D45B)}u1D456u1D447u1D435?1u1D45B=(u1D456?1)u1D447u1D435 foru1D456-th block, then it is arranged
as
su1D458 :=
uni005B.alt02
bu1D447u1D458,0 cu1D447u1D458,0 bu1D447u1D458,1 cu1D447u1D458,1 ??? bu1D447u1D458,?u1D443?1 cu1D447u1D458,?u1D443?1
uni005D.alt02u1D447
(2.23)
where bu1D458,u1D45D is a column of u1D45Au1D451 information symbols and cu1D458,u1D45D is a column of u1D45Au1D461 training
symbols for u1D45D = 0,1,??? , ?u1D443 ?1. We clearly have u1D447u1D435 = ?u1D443u1D45Au1D44F. The training session for each
user contains an impulse guarded by zeros (silent periods), which for the u1D458-th user has the
structure (u1D6FE > 0)
cu1D458,u1D45D :=
uni005B.alt02
0u1D447(u1D458?1)(u1D43F+1)+u1D43F u1D6FE 0u1D447(u1D43E?u1D458)(u1D43F+1)+u1D43F
uni005D.alt02u1D447
. (2.24)
Therefore, u1D45Au1D461 = u1D43E(u1D43F+ 1) + u1D43F symbols, which have to be devoted for training and the
remaining are available for information symbols. Let u1D45Bu1D458,u1D45D := u1D45Du1D45Au1D44F +u1D45Au1D451 +(u1D458?1)(u1D43F+1)+u1D43F,
(u1D45D = 0,1,??? , ?u1D443 ? 1) denote the unique locations of (nonzero) u1D6FE?s in the cu1D458,u1D45D?s in the ?u1D443
subblocks for u1D458-th user.
As shown in [18,70], one needs to satisfy number of subblocks ?u1D443 ?u1D444to uniquely identify
the unknown BEM parameters. By (2.18), as u1D447 increases, u1D444 = 2?u1D453u1D451u1D447u1D447u1D460? + 1 increases (in
discrete steps) for a given value of u1D453u1D451u1D447u1D460. Since the subblock size u1D45Au1D44F = u1D447u1D435/?u1D443, therefore,
?u1D443 = u1D447u1D435/u1D45Au1D44F = u1D447/(?u1D45Au1D44F), as u1D447 increases, ?u1D443 also increases. Typically, one would first pick
u1D447u1D435 (which would fix u1D447 = ?u1D447u1D435 with ? = 2 for oversampled CE-BEM), and then pick u1D45Au1D44F to
satisfy ?u1D443 ? u1D444 for a given value of u1D447 or u1D447u1D435, u1D444 does not depend upon ?u1D443. If u1D45Au1D44F turns out
to be ?too small?, training overhead is large since 2u1D43F+ 1 symbols out of u1D45Au1D44F are needed for
training. On the other hand, suppose one first picksu1D45Au1D44F to achieve a certain training overhead
(= (2u1D43F + 1)/u1D45Au1D44F). Then in order to have parameter identifiability, one needs ?u1D443 ? u1D444, i.e.,
u1D447/(?u1D45Au1D44F) ? 2?u1D453u1D451u1D447u1D447u1D460?+ 1. For example, under u1D447u1D460 = 25u1D707u1D460,u1D453u1D451 = 400u1D43Bu1D467 and u1D45Au1D44F = 40, there
13
exists no integer u1D447 with ? = 2 for which we can satisfy u1D447/(?u1D45Au1D44F) ? 2?u1D453u1D451u1D447u1D447u1D460? + 1 whereas
u1D447 = 400 satisfies this inequality when ? = 1 (u1D447u1D435 = 400, ?u1D443 = 10 and u1D444 = 9). This implies
that for block-wise updating, we cannot always use oversampled CE-BEM, only the less
accurate critically sampled CE-BEM.
2.4 Useful Equalization Techniques
In communication systems, the channel estimates given by a channel estimator are fed
into an equalizer for symbol detection. In this section, we introduce two equalizers, Kalman
detector (KD) and Minimum Mean Square Error (MMSE) Decision Feedback Equalizer
(DFE), that we employ in this paper to detect the transmitted symbols using the channel
estimates. We consider a multi-input multi-output (MIMO) system with u1D43E inputs and u1D441
outputs for both equalizers; one can adapt easily to single-input single-output (SISO) or
single-input multi-output (SIMO) systems.
Let {u1D460u1D458 (u1D45B)} denote u1D458-th user?s information sequence that is input to the time-varying
channel with discrete-time response {hu1D458 (u1D45B;u1D459)} (channel response for the u1D458-th user at time
instanceu1D45Bto a unit input at time instanceu1D45B?u1D459). We assume{u1D460u1D458 (u1D45B)}is mutually independent
and identically distributed (i.i.d.) with zero mean and variance u1D438{u1D460u1D458 (u1D45B)u1D460?u1D458 (u1D45B)} = u1D70E2u1D460u1D458 = u1D70E2u1D460
for u1D458 = 1,2,??? ,u1D43E. Then the symbol-rate noisy u1D441-column channel output vector is given
by (u1D45B = 0,1,...)
y(u1D45B) =
u1D43Euni2211.alt02
u1D458=1
u1D43Funi2211.alt02
u1D459=0
hu1D458 (u1D45B;u1D459)u1D460u1D458 (u1D45B?u1D459) +v(u1D45B) (2.25)
where the u1D441-column vector v(u1D45B) is zero-mean, white, uncorrelated with u1D460u1D458 (u1D45B), complex
Gaussian noise, with the autocorrelation u1D438{v(u1D45B+u1D70F)vu1D43B (u1D45B)} = u1D70E2u1D463Iu1D441u1D6FF(u1D70F). Define
s(u1D45B) :=
uni005B.alt02
u1D4601(u1D45B) u1D4602(u1D45B) ??? u1D460u1D43E(u1D45B)
uni005D.alt02u1D447
h(u1D45B;u1D459) :=
uni005B.alt02
h1(u1D45B;u1D459) h2(u1D45B;u1D459) ??? hu1D43E(u1D45B;u1D459)
uni005D.alt02
.
and we may rewrite (2.25) as
y(u1D45B) =
u1D43Funi2211.alt02
u1D459=0
h(u1D45B;u1D459)s(u1D45B?u1D459) +v(u1D45B). (2.26)
2.4.1 Kalman Detector (KD)
The Kalman filter, together with a quantizer, act as the symbol detector at the receiver
end. We transform the Kalman filter for joint channel estimation and equalization in [68] to
14
KD only for symbol equalizaton. The state and the measurement equations are given by
Su1D451 (u1D45B) = ?Su1D451 (u1D45B?1) +??s(u1D45B) +??s(u1D45B), (2.27)
y(u1D45B) = ?Hu1D451 (u1D45B)Su1D451 (u1D45B) +v(u1D45B), (2.28)
with the following definitions
Su1D451(u1D45B) :=
uni005B.alt02
su1D447 (u1D45B) su1D447 (u1D45B?1) ??? su1D447 (u1D45B?u1D451)
uni005D.alt02u1D447
,
?s(u1D45B) := u1D438{s(u1D45B)}, ?s(u1D45B) := s(u1D45B)??s(u1D45B),
? :=
?
?0
u1D447
u1D451 0
Iu1D451 0u1D451
?
??Iu1D43E, ? := uni005B.alt021 0u1D447u1D451uni005D.alt02u1D447 ?Iu1D43E,
?Hu1D451 (u1D45B) := uni005B.alt02?h(u1D45B;0) ?h(u1D45B;1) ??? ?h(u1D45B;u1D43F) 0u1D441?u1D43E(u1D451?u1D43F)uni005D.alt02
where s(u1D45B) is u1D43E-column vector of symbols, ?h(u1D45B;u1D459) is the estimated u1D441 ?u1D43E channel matrix
and integeru1D451?u1D43F; it will also be the equalization delay. Assume data symbols are zero-mean
and white. If u1D460(u1D45B) is a data symbol, we have ?u1D460(u1D45B) = 0,?u1D460(u1D45B) = u1D460(u1D45B) and u1D70E2u1D460(u1D45B) = u1D70E2u1D460; if u1D460(u1D45B)
is a training symbol, ?u1D460(u1D45B) = u1D460(u1D45B),?u1D460(u1D45B) = 0 and u1D70E2u1D460(u1D45B) = 0.
Kalman filtering for the system described by (2.27) and (2.28) is initialized with
?Su1D451 (?1 ? ?1) = 0u1D43E(u1D451+1) and P(?1 ? ?1) = ??u1D447,
where ?Su1D451 (u1D45D?u1D45A) denotes the estimate of Su1D451 (u1D45D) given the observations {y(u1D45B)}u1D45Au1D45B=0, and
P(u1D45D?u1D45A) denotes the error covariance matrix of ?Su1D451 (u1D45D?u1D45A), defined as
P(u1D45D?u1D45A) := u1D438{[?Su1D451 (u1D45D?u1D45A)?Su1D451 (u1D45D)][?Su1D451 (u1D45D?u1D45A)?Su1D451 (u1D45D)]u1D43B}.
Then recursive filtering (for u1D45B = 0,1,???) is applied via the following steps:
1. Time update:
?Su1D451 (u1D45B?u1D45B?1) = ??Su1D451 (u1D45B?1 ?u1D45B?1) +??s(u1D45B),
P(u1D45B?u1D45B?1) = ?P(u1D45B?1 ?u1D45B?1)?u1D447 +u1D70E2u1D460(u1D45B)??u1D447;
2. Kalman gain:
Pu1D702(u1D45B) = u1D70E2u1D463Iu1D441 + ?Hu1D451(u1D45B)P(u1D45B?u1D45B?1)?Hu1D43Bu1D451 (u1D45B),
K(u1D45B) = P(u1D45B?u1D45B?1)?Hu1D43Bu1D451 (u1D45B)P?1u1D702 (u1D45B);
15
3. Measurement update:
?Su1D451 (u1D45B?u1D45B) = ?Su1D451 (u1D45B?u1D45B?1) +K(u1D45B)uni007B.alt02y(u1D45B)? ?Hu1D451(u1D45B)?Su1D451(u1D45B?u1D45B?1)uni007D.alt02,
P(u1D45B?u1D45B) =
uni007B.alt02
Iu1D43E(u1D451+1) ?K(u1D45B) ?Hu1D451(u1D45B)
uni007D.alt02
P(u1D45B?u1D45B?1).
After Kalman filtering for every u1D45B, the estimated vector is given as
?Su1D451(u1D45B?u1D45B) = uni005B.alt02?su1D447(u1D45B?u1D45B) ?su1D447(u1D45B?1 ?u1D45B) ??? ?su1D447(u1D45B?u1D451?u1D45B)uni005D.alt02u1D447
and we extract its last (u1D43E-column vector) term ?s(u1D45B?u1D451?u1D45B) as the desired equalized output
for u1D43E-users. Finally, we hard-quantize ?s(u1D45B?u1D451?u1D45B) to acquire the detected symbols.
Using the estimated channel, one may rewrite the received signal as
y(u1D45B) =
u1D43Funi2211.alt02
u1D459=0
?h(u1D45B;u1D459)s(u1D45B?u1D459) + u1D43Funi2211.alt02
u1D459=0
uni005B.alt02
h(u1D45B;u1D459)? ?h(u1D45B;u1D459)
uni005D.alt02
s(u1D45B?u1D459) +v(u1D45B)
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=:?v(u1D45B)
(2.29)
where the ?effective? noise is ?v(u1D45B) instead of v(u1D45B). In order to compensate for this channel
estimation error, that is because the channel estimates
uni007B.alt02?
h(u1D45B;u1D459)
uni007D.alt02
obtained by a channel
estimator is not equal to the ?true? channel response {h(u1D45B;u1D459)}, we take the variance of ?v(u1D45B)
in (2.29) to be u1D70E2u1D463 + 0.01u1D70E2u1D460 instead of the variance of v(u1D45B), u1D70E2u1D463, for simulations presented in
Section 3.3.2.
2.4.2 Mimimum Mean Square Error Decision Feedback Equalizer (MMSE-
DFE) [40]
Figure 2.1: Decision-feedback equalizer (DFE).
The DFE structure is shown in Fig. 2.1 to equalize the delayed symbols s(u1D45B?u1D451), with
the feed-forward (FF) and feed-back (FB) filters. Since each measurement y(u1D45B) contains
inter-symbol-interference (ISI) caused by prior symbols, DFE is designed to reduce ISI and
16
to recover s(u1D45B) using FIR filters. The FF filter takes current and prior measurements yu1D453 as
its input to get information correlated with ISI and to remove its effect.
Stack the inputs of the FF filter with u1D459u1D453 taps at time u1D45B into a ?tall? vector
yu1D453 (u1D45B) :=
uni005B.alt02
yu1D447 (u1D45B) yu1D447 (u1D45B?1) ??? yu1D447 (u1D45B?u1D459u1D453 + 1)
uni005D.alt02u1D447
,
where y(u1D45B) is u1D441 -column vector and also define vu1D453(u1D45B) likewise. Then, the received signal is
given as
yu1D453 (u1D45B) = H(u1D45B)su1D453 (u1D45B) +vu1D453 (u1D45B), (2.30)
with
H(u1D45B) :=
?
??
??
h(u1D45B;0) ??? h(u1D45B;u1D43F)
... ... ...
h(u1D45B?u1D459u1D453 + 1;0) ??? h(u1D45B?u1D459u1D453 + 1;u1D43F)
?
??
??
su1D453 (u1D45B) :=
uni005B.alt02
su1D447 (u1D45B) su1D447 (u1D45B?1) ??? su1D447 (u1D45B?u1D459u1D453 ?u1D43F+ 1)
uni005D.alt02u1D447
.
where h(u1D45B;u1D459) is u1D441 ?u1D43E matrix channel response at time u1D45B to a unit input at time u1D45B?u1D459 and
s(u1D45B) is u1D43E-column vector.
As shown in Fig. 2.1, the input to the FB filter comes from the decision output, denoted
by ?su1D453(u1D45B?u1D451). The FB filter uses prior symbol decisions to cancel the trailing ISI by mapping
the estimate ?su1D453(u1D45B?u1D451) to the closest point in the symbol constellation. We define the input
vector of FB filter with u1D459u1D44F taps as
su1D44F (u1D45B) :=
uni005B.alt02
?su1D447 (u1D45B?u1D451) ?su1D447 (u1D45B?u1D451?1) ??? ?su1D447 (u1D45B?u1D451?u1D459u1D44F)
uni005D.alt02u1D447
.
The estimate of the information symbol, ?s(u1D45B?u1D451) is obtained by combining the outputs
of FF and FB filters and can be written at time u1D45B with delay u1D451 as
?s(u1D45B?u1D451) =
u1D459u1D453?1uni2211.alt02
u1D456=0
Fu1D447u1D456 (u1D45B)y(u1D45B?u1D456)?
u1D459u1D44Funi2211.alt02
u1D457=1
Bu1D457 (u1D45B)?s(u1D45B?u1D451?u1D457), (2.31)
where Fu1D456 (u1D45B)?s (that are u1D441 ?u1D43E matrices) and Bu1D457 (u1D45B)?s (that are u1D43E ?u1D43E matrices) are the
taps of FF and FB time-varying filters at time u1D45B, and ?s(u1D45B?u1D451?u1D457) is the hard decision of
?s(u1D45B?u1D451?u1D457). The estimate ?s(u1D45B?u1D451) is also fed into the quantizer to obtain the symbol
decision ?s(u1D45B?u1D451).
17
Let F(u1D45B) and B(u1D45B) denote the vectors of time-varying taps of FF and FB filters,
F(u1D45B) :=
uni005B.alt02
Fu1D4470 (u1D45B) Fu1D4471 (u1D45B) ??? Fu1D447u1D459u1D453?1 (u1D45B)
uni005D.alt02u1D447
,
B(u1D45B) :=
uni005B.alt02
I B1 (u1D45B) B2 (u1D45B) ??? Bu1D459u1D44F (u1D45B)
uni005D.alt02u1D447
,
then the error signal is given by
?s(u1D45B?u1D451) = s(u1D45B?u1D451)??s(u1D45B?u1D451)
= B(u1D45B)su1D44F(u1D45B)?F(u1D45B)yu1D453(u1D45B). (2.32)
Assuming the decisions {?s(u1D45B)} are correct and equal to {s(u1D45B)}, we can solve a nonlinear
optimization problem which mininmizes the variance of the error signal in (2.32),
min
{F(u1D45B),B(u1D45B)}
u1D438
uni007B.alt02
?B(u1D45B)su1D44F(u1D45B)?F(u1D45B)yu1D453(u1D45B)?2
uni007D.alt02
. (2.33)
Solve a standard linear least-mean-squares estimation problem over F(u1D45B) with B(u1D45B) fixed
and then we have a constrained optimization problem; note that the leading entry of B(u1D45B)
is the identity matrix. Therefore, the FF and the FB time-varying filters of the MMSE-DFE
are given by [40]
BMMSE (u1D45B) = R?1u1D6FF ?
uni0028.alt02
?u1D447R?1u1D6FF ?
uni0029.alt02?1
, (2.34)
FMMSE (u1D45B) = R?1u1D466u1D466 (u1D45B)Ru1D43Bu1D460u1D466 (u1D45B)BMMSE (u1D45B), (2.35)
where
? :=
uni005B.alt02
1 0 0 ??? 0
uni005D.alt02u1D447
?Iu1D43E,
Ru1D6FF := Ru1D460u1D460 (u1D45B)?Ru1D460u1D466 (u1D45B)R?1u1D466u1D466 (u1D45B)Ru1D43Bu1D460u1D466 (u1D45B)
= ?
uni005B.alt04 1
u1D70E2u1D463H(u1D45B)H
u1D43B (u1D45B) + 1
u1D70E2u1D460Iu1D441u1D459u1D453
uni005D.alt04?1
?u1D43B.
By the assumption that {s(u1D45B)} are independent and identically distributed (i.i.d.) with
variance u1D70E2u1D460, and based on (2.30), we have
Ru1D460u1D460 (u1D45B) := u1D438
uni007B.alt02
su1D44F (u1D45B)su1D43Bu1D44F (u1D45B)
uni007D.alt02
= u1D70E2u1D460Iu1D43E(u1D459u1D44F+1),
Ru1D460u1D466 (u1D45B) := u1D438
uni007B.alt02
su1D44F (u1D45B)yu1D43Bu1D453 (u1D45B)
uni007D.alt02
= u1D70E2u1D460?Hu1D43B (u1D45B),
Ru1D466u1D466 (u1D45B) := u1D438
uni007B.alt02
yu1D453 (u1D45B)yu1D43Bu1D453 (u1D45B)
uni007D.alt02
= u1D70E2u1D460H(u1D45B)Hu1D43B (u1D45B) +u1D70E2u1D463Iu1D441u1D459u1D453
18
where
su1D44F(u1D45B) = ?su1D453(u1D45B), ? :=
uni005B.alt03
0(u1D459u1D44F+1)?u1D451 Iu1D459u1D44F+1 0(u1D459u1D44F+1)?(u1D459u1D453+u1D43F?u1D451?u1D459u1D44F?1)
uni005D.alt03
?Iu1D43E.
Using (2.34), (2.35) in (2.31), we have the symbol estimate {?s(u1D45B?u1D451)}. Since the ?true?
channel response {h(u1D45B;u1D459)} is not available at the receiver, we use the channel estimates
{?h(u1D45B;u1D459)} obtained by a channel estimator to design the MMSE-DFE [See (2.29) in 2.4.1 for
a formal equation]. In order to compensate for this channel estimation error in (2.30), for
simulations where we use MMSE-DFE, we increased the variance of v(u1D45B) in (2.30) from u1D70E2u1D463
to u1D70E2u1D463 + 0.01u1D70E2u1D460 in our simulation presented later.
2.5 Turbo Equalization
In digital communications, a turbo equalizer is a type of receiver used to detect a
message corrupted by a communication channel with ISI. It approaches the performance
of a maximum a posteriori (MAP) receiver via iterative message passing between a soft-in
soft-out (SfiSfo) equalizer and a SfiSfo decoder [46]. It is closely related to turbo codes, as a
turbo equalizer may be considered a turbo decoder if the channel is viewed as a convolutional
code.
The optimal equalization methods for minimizing sequence error rate or the bit error rate
(BER) are based on maximum likelihood (ML) estimation, which turns into MAP estimation
in the presence of a priori information about the transmitted data. For instance, see Viterbi
algorithm (VA) [13,19,54] for MAP/ML sequence estimation and BCJR algorithm [29] for
MAP/ML symbol estimation.
Communicating soft information probability distribution between the equalizer and the
decoder, instead of hard information (symbol estimates only), improves the BER performance
but usually requires more complex decoding algorithms. State-of-the-art systems for a variety
of communication channels employ convolutional codes and ML equalizers together with an
interleaver after the encoder and a deinterleaver before the decoder [66,74]. Interleaving
shu?es symbols within a given time frame or block of data and thus decorrelates error
events introduced by the equalizer betweeen neigboring symbols.
An optimal joint processing of the equalization and decoding steps is usually impossible
due to complexity considerations. A number of iterative receiver algorithms repeat the
equalization and decoding tasks on the same set of received data, where feedback information
from the decoder is incorporated into the equalization process. This method, called turbo
equalization, was originally developed for concatenated convolutional codes (turbo code [4])
and is now adapted to various communication problems.
19
The MAP/ML-based solutions often suffer from high computational load for channels
with long memory or large constellation sizes (expensive equalizer) or convolutional codes
with long memory (expensive decoder). This situation is exacerbated by the need to perform
equalization and decoding several times for each block of data. A major research issue is
thus the complexity reduction of such iterative algorithms.
2.5.1 Principle of Turbo Equalization [39]
Figure 2.2: A turbo equalization receiver
We consider a simple transmitter where a sequence of datab(u1D45B?) = [u1D44F1(u1D45B?),u1D44F2(u1D45B?),??? ,u1D44Fu1D4580(u1D45B?)] ?
{1,0}u1D4580 is encoded to the code symbols c(u1D45B?) = [u1D4501(u1D45B?),u1D4502(u1D45B?),??? ,u1D450u1D45B0(u1D45B?)] ? {1,0}u1D45B0 with a
code rateu1D445u1D450 = u1D4580/u1D45B0, which is interleaved in a block and then mapped into binary phase shift
keying (BPSK) symbols. Fig. 2.2 depicts the receiver structure for turbo equalization [5],
where both the SfiSfo equalizer and SfiSfo decoder are of the MAP type. The extrinsic
log-likelihood ratios (LLR?s), Lu1D452{c(?)} are transfered iteratively between the equalizer and
decoder. [The subscript ?u1D452? for representing ?extrinsic?, the superscript ?u1D438? and ?u1D437? for out-
put of ?Equalizer? and ?Decoder? respectively.] The MAP equalizer computes the a posteriori
probabilities (APP?s) given u1D447u1D45F received symbols, u1D443 {u1D450u1D456(u1D45B) = u1D44F? y(u1D459),1 ?u1D459 ?u1D447u1D45F},u1D44F ? {1,0}
and generates the extrinsic LLR as (a posteriori LLR ? a priori LLR)
u1D43Fu1D438u1D452
uni007B.alt02
u1D450u1D456(u1D45B)
uni007D.alt02
:= ln u1D443{u1D450
u1D456(u1D45B) = 1 ? y(u1D459), 0 ?u1D459<u1D447u1D45F}
u1D443{u1D450u1D456(u1D45B) = 0 ? y(u1D459), 0 ?u1D459<u1D447u1D45F} ?ln
u1D443{u1D450u1D456(u1D45B) = 1}
u1D443{u1D450u1D456(u1D45B) = 0}bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=:u1D43F{u1D450u1D456(u1D45B)}
. (2.36)
The a priori LLR of the MAP equalizer,u1D43F{u1D450u1D456(u1D45B)}is provided by the interleaved output of the
MAP decoder at the previous iteration butu1D43F{u1D450u1D456(u1D45B)} = 0 for the first iteration. The extrinsic
LLR?s Lu1D438u1D452 {c(u1D45B)} produced by the MAP demodulator is sent to the MAP decoder as the a
priori LLR?s for channel decoding. Based on the a priori LLR?s and the channel code con-
straints, the MAP decoder computes the APPs u1D443
uni007B.alt02
u1D450u1D456(u1D45B?) = u1D44F? Lu1D438u1D452 {c(u1D459)},1 ?u1D459 ?u1D447u1D45F
uni007D.alt02
,u1D44F ?
20
{1,0} and generates the extrinsic LLR as
u1D43Fu1D437u1D452
uni007B.alt02
u1D450u1D456(u1D45B?)
uni007D.alt02
:= ln u1D443{u1D450
u1D456(u1D45B?) = 1 ? L{c(u1D459)}, 0 ?u1D459<u1D447u1D45F}
u1D443{u1D450u1D456(u1D45B?) = 0 ? L{c(u1D459)}, 0 ?u1D459<u1D447u1D45F} ?ln
u1D443{u1D450u1D456(u1D45B?) = 1}
u1D443{u1D450u1D456(u1D45B?) = 0}bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=:u1D43F{u1D450u1D456(u1D45B?)}
. (2.37)
The extrinsic output of the MAP decoder is fedback to the MAP equalizer iteratively. Note
that (2.36) and (2.37) are valid only if the a posteriori outputs are independent of the a
priori inputs for the equalizer and the decoder. Assuming ideal interleaver between the
equalizer and the decoder, we can apply the turbo principle and the correct ordering of
the LLRs L{c(u1D45B?)} = u1D70B?1
uni005B.alt02
Lu1D438u1D452 {c(u1D45B)}
uni005D.alt02
and L{c(u1D45B)} = u1D70B
uni005B.alt02
Lu1D437u1D452 {c(u1D45B?)}
uni005D.alt02
, which are input to
the MAP decoder and the MAP equalizer repectively. The MAP decoder also compute
the a posteriori probabilities of the input data bit and then the data estimate ?b(u1D45B?) =
[?u1D44F1(u1D45B?),?u1D44F2(u1D45B?),??? ,?u1D44Fu1D4580(u1D45B?)] as
?u1D44Fu1D456(u1D45B?) = arg max
u1D44F?{1,0}
u1D443
uni007B.alt02
u1D44Fu1D456(u1D45B?) = u1D44F? L{c(u1D459)},1 ?u1D459 ?u1D447u1D45F
uni007D.alt02
. (2.38)
2.5.2 Maximum A Posteriori (MAP) Decoding Algorithm for Convolutional
Codes [72]
In this section we introduce a recursive forward and backward algorithm for the MAP
decoding that operates on a trellis description for the code, based on BCJR algorithm [29].
The MAP decoder updates the extrinsic LLR?s Lu1D437u1D452 {c(u1D45B?)} of the coded bits at each iteration
and generates the a posteriori LLR?s L{b(u1D45B?)} of the information bits at last iteration, based
on the a priori LLR?s L{c(u1D45B?)} and code trellis structure.
A convolutional code with memory length u1D459u1D45A has finite 2u1D459u1D45A states and its trellis struc-
ture describes the transitions between states at time u1D45B and u1D45B + 1. In Chapter 5, we con-
sider a recursive systematic convolutional encoder with u1D459u1D45A = 2 (constraint length of 3), a
4-state machine. Denote the input information bits at time u1D45B that cause the state tran-
sition from ?(u1D45B? 1) = u1D713? to ?(u1D45B) = u1D713 by b(u1D45B;u1D713? ? u1D713) and the corresponding output
coded bits by c(u1D45B;u1D713? ? u1D713). The MAP decoder computes the state transition probability
u1D443 {?(u1D45B?1) = u1D713?,?(u1D45B) = u1D713} using the a priori probabilities of the coded bits.
Assume that the encoder starts in state ?(0) = 0 and also ends in state ?(u1D70F) = 0,u1D70F =
u1D447u1D45F + u1D459u1D45A, where we have at least u1D459u1D45A additional zero inputs following the information bit
21
sequence {b(u1D45B)}u1D447u1D45Fu1D45B=1. Then the state transition probability can be given as
u1D443 {?(u1D45B?1) = u1D713?,?(u1D45B) = u1D713} = u1D719(u1D45B?1;u1D713?)u1D703(u1D45B;u1D713)u1D443 {c(u1D45B;u1D713? ?u1D713)}, (2.39)
= u1D719(u1D45B?1;u1D713?)u1D703(u1D45B;u1D713)
u1D45B0uni220F.alt02
u1D457=1
u1D443
uni007B.alt02
u1D450u1D457(u1D45B;u1D713? ?u1D713)
uni007D.alt02
,
where u1D719(u1D45B;u1D713) denotes the total probability of all path segments starting from the origin of
the trellis that terminate in state u1D713 at time u1D45B; and u1D703(u1D45B;u1D713) denotes the total probability of
all path segments terminating at the end of the trellis that originate from state u1D713 at time
u1D45B. The probability u1D719(u1D45B;u1D713) and u1D703(u1D45B;u1D713) can be computed through the following forward and
backward recursions:
u1D719(u1D45B;u1D713) = uni2211.alt02
??
u1D719(u1D45B?1;u1D713?)u1D443 {c(u1D45B;u1D713? ?u1D713)}, u1D45B = 1,2,??? ,u1D70F, (2.40)
u1D703(u1D45B;u1D713) = uni2211.alt02
??
u1D703(u1D45B+ 1;u1D713?)u1D443 {c(u1D45B+ 1;u1D713 ?u1D713?)}, u1D45B = u1D70F ?1,u1D70F ?2,??? ,0, (2.41)
where initial conditions are given as
u1D719(0,0) = 1, u1D719(0,u1D713) = 0 for u1D713 ?= 0,
u1D703(u1D70F,0) = 1, u1D703(u1D70F,u1D713) = 0 for u1D713 ?= 0,
and ?? is a set of the state u1D713? for all possible transition from u1D713? to u1D713. At each recursion,
since u1D719(u1D45B,u1D713) and u1D703(u1D45B;u1D713) are the probabilities for u1D713 and time u1D45B, we can normalize as
u1D719(u1D45B;u1D713) = u1D719(u1D45B;u1D713)uni2211.alt01
?u1D719(u1D45B;u1D713)
, u1D703(u1D45B;u1D713) = u1D703(u1D45B;u1D713)uni2211.alt01
?u1D703(u1D45B;u1D713)
,
where ? is a set of all the possible state u1D713.
22
Given the a priori LLR?s L{c(u1D45B?)} and the trellis structure of convolution code, we
have the extrinsic a posteriori LLR for the coded bit u1D450u1D456(u1D45B?) as
u1D43Fu1D437u1D452
uni007B.alt02
u1D450u1D456(u1D45B?)
uni007D.alt02
= log u1D443 {u1D450
u1D456(u1D45B?) = 1 ? L{c(u1D45B?)}}
u1D443 {u1D450u1D456(u1D45B?) = 0 ? L{c(u1D45B?)}} ?log
u1D443 {u1D450u1D456(u1D45B?) = 1}
u1D443 {u1D450u1D456(u1D45B?) = 0}
= log
uni2211.alt01
?(u1D456,1)u1D443 {?(u1D45B? ?1) = u1D713?,?(u1D45B?) = u1D713}/u1D443 {u1D450u1D456(u1D45B?) = 1}uni2211.alt01
?(u1D456,0)u1D443 {?(u1D45B? ?1) = u1D713?,?(u1D45B?) = u1D713}/u1D443 {u1D450u1D456(u1D45B?) = 0}
= log
uni2211.alt01
?(u1D456,1)u1D719(u1D45B? ?1;u1D713?)u1D703(u1D45B?;u1D713)u1D443 {c(u1D45B?;u1D713? ?u1D713)}/u1D443 {u1D450u1D456(u1D45B?) = 1}uni2211.alt01
?(u1D456,0)u1D719(u1D45B? ?1;u1D713?)u1D703(u1D45B?;u1D713)u1D443 {c(u1D45B?;u1D713? ?u1D713)}/u1D443 {u1D450u1D456(u1D45B?) = 0}
= log
uni2211.alt01
?(u1D456,1)u1D719(u1D45B? ?1;u1D713?)u1D703(u1D45B?;u1D713)
uni220F.alt01u1D45B0
u1D457=1u1D443 {u1D450u1D457(u1D45B?;u1D713? ?u1D713)}/u1D443 {u1D450u1D456(u1D45B?) = 1}uni2211.alt01
?(u1D456,0)u1D719(u1D45B? ?1;u1D713?)u1D703(u1D45B?;u1D713)
uni220F.alt01u1D45B0
u1D457=1u1D443 {u1D450u1D457(u1D45B?;u1D713? ?u1D713)}/u1D443 {u1D450u1D456(u1D45B?) = 0}
= log
uni2211.alt01
?(u1D456,1)u1D719(u1D45B? ?1;u1D713?)u1D703(u1D45B?;u1D713)
uni220F.alt01u1D45B0
u1D457=1,u1D457?=u1D456u1D443 {u1D450
u1D457(u1D45B?) = u1D450u1D457(u1D713? ?u1D713)}
uni2211.alt01
?(u1D456,0)u1D719(u1D45B? ?1;u1D713?)u1D703(u1D45B?;u1D713)
uni220F.alt01u1D45B0
u1D457=1,u1D457?=u1D456u1D443 {u1D450u1D457(u1D45B?) = u1D450u1D457(u1D713? ?u1D713)}
where ?(u1D456,u1D44F) is a set of all possible state pairs (u1D713?,u1D713) for u1D450u1D456(u1D713? ?u1D713) = u1D44F? {1,0}.
The MAP decoder also generates the decoded bits ?b(u1D45B?) at the last iteration. We can
compute the a posteriori LLR of the information bit by
u1D43F
uni007B.alt02
u1D44Fu1D456(u1D45B?)
uni007D.alt02
= log
uni2211.alt01
u1D4B0(u1D456,1)u1D719(u1D45B? ?1;u1D713?)u1D703(u1D45B?;u1D713)
uni220F.alt01u1D45B0
u1D457=1u1D443 {u1D450u1D457(u1D45B?) = u1D450u1D457(u1D713? ?u1D713)}uni2211.alt01
u1D4B0(u1D456,0)u1D719(u1D45B? ?1;u1D713?)u1D703(u1D45B?;u1D713)
uni220F.alt01u1D45B0
u1D457=1u1D443 {u1D450u1D457(u1D45B?) = u1D450u1D457(u1D713? ?u1D713)}
, (2.42)
where u1D4B0(u1D456,u1D44F) is a set of all possible state pairs (u1D713?,u1D713) for u1D44Fu1D456(u1D713? ?u1D713) = u1D44F? {1,0}. Counting
on the sign of LLR, u1D43F{u1D44Fu1D456(u1D45B?)} in (2.42), we can decide the information bit ?u1D44Fu1D456(u1D45B?) ? {1,0}.
[See more details in [72].]
2.6 Conclusions
In this chapter, we reviewed characteristics and representations of wireless channels.
We will use the Jakes? model that follows [75] (with a correction in the appendix of [63]) as
the model of the ?real? channel in all the simulations in this dissertation. To describe the
temporal variation of the channel for the purpose of channel estimation, we will use CE-BEM,
a block-based channel model that is often more convenient than the symbol-based AR models.
We also provided some background that we use for channel estimation and equalization in
the following chapters. We summarized the TM training scheme in [30,70] which is useful to
estiamte channel in subblock-wise tracking approaches in Chapter 3 and 4. We will employ
the Kalman detector (KD) and MMSE-DFE to detect the transmitted symbols using channel
estimates provided by a channel estimator. For the coded communication systems, the turbo
equalization is a powerful technique that we will investigate in Chapter 5.
23
Chapter 3
Doubly-selective Adaptive Channel Estimation Using Exponential Basis
Models and Subblock Tracking
We present a subblock-wise tracking approach to doubly-selective channel estimation,
exploiting the oversampled CE-BEM for the overall channel variations, and an AR model to
update the BEM coefficients. The time-varying nature of the channel is well captured by the
CE-BEM while the time-variations of the (unknown) BEM coefficients are likely much slower
than that of the channel. We track the BEM coefficients via Kalman filtering, based on time-
multiplexed periodically transmitted training symbols. Simulation examples demonstrate its
superior performance over some existing doubly-selective channel tracking schemes.
3.1 Introduction
The first-order AR model, which has been very popular to describe a time-varying
channel, is not appropriate for a fast-fading channel, since the channel estimates can only
be obtained from training symbols [76]. Potential solutions lie in exploiting the detected
symbols for channel tracking [6,43,64], where error propagation due to incorrect detections
can be pronounced for fast-fading channels. Recently, BEM has been widely investigated to
represent doubly-selective channels in wireless applications [8,11,35,63,70], for which, the
time-varying channel taps are expressed as superpositions of time-varying basis functions in
modeling Doppler effects, weighted by time-invariant coefficients. In contrast to AR models
that describe temporal variation on a symbol-by-symbol update basis, a BEM depicts the
evolution of the channel over a period (block) of time. Intuitively, the BEM coefficients are
more convenient to track in a fast-fading environment, since they evolve much more slowly
in time (based on a block-by-block update) than the real channel.
We propose a subblock-wise tracking approach for doubly-selective channels using time-
multiplexed (TM) training. It exploits the CE-BEM for the overall channel variations of
each (overlapping) block and a first-order AR model to describe the evolutions of the BEM
coefficients. The time-varying nature of the channel can be well captured in the CE-BEM
by (known) Fourier basis functions, and the slow-varying BEM coefficients are updated via
Kalman filtering at each training session; during information sessions, channel estimates are
generated by CE-BEM using the estimated BEM coefficients. This approach achieves better
24
performance in fast-fading environments, than using conventional symbol-wise AR models
or block-wise BEM representations.
Channel tracking based on BEMs has been considered in [8,26,41] using block-by-block
updating, unlike our contribution where we exploit subblock-wise updating. The distinction
is as follows. One block comprises several subblocks. For parameter identifiability, one needs
the number of subblocks at least as large as the number of basis functions used for channel
modeling. Unlike the block-wise schemes where the receiver has to collect the whole block
of data in order to generate channel estimates and perform equalization, in the proposed
approaches the receiver is able to accomplish the two tasks after every subblock.
3.2 Subblock-wise Channel Estimation using CE-BEM
Figure 3.1: Subblock-wise channel estimation: overlapping blocks and subblocks, where one
block comprises several subblocks and each subblock has an information session followed by
a training session.
Suppose that we collect the received signal over a time period ?u1D447 symbols. We wish
to estimate the time-variant channel using a useful channel model and time-multiplexed
training, and then detect the information symbols using this estimated channel. For the
CE-BEM, if we choose ?u1D447 as the block size, then in general by (3.4) we have a large number
of basis functions to be estimated, thereby degrading the channel estimation performance.
If we divide ?u1D447 into blocks of (much smaller) size u1D447u1D435 each, and then fit the CE-BEM block by
block, we have smaller u1D444; however, estimation of ?u1D45E(u1D459)?s is now based on shorter observation
size of u1D447u1D435 symbols which might also degrade channel estimation performance. Thus one has
to strike a balance between the estimation variance and the block size. Such considerations
do not apply to the symbol-wise AR channel model fitting. In the sequel, we propose a novel
subblock-wise tracking approach to CE-BEM channel estimation where we update estimates
of ?u1D45E(u1D459)?s every subblock based on all past training sessions.
25
As we shall see later, use of subblock updating avoids some of the limitations of blockwise
estimation. In subblock tracking there is no longer a strict definition of the block size u1D447u1D435
unlike blockwise estimation, allowing for a more flexible choice of parameters such as subblock
size and training overhead. Intuitively, updating more often as in subblock tracking should
be superior to (or at least as good as) less frequent updating as in block-wise estimation. As
noted earlier in Section 2.3, the parameter identifiability requirement of ?u1D443 ? u1D444 for block-
wise approach impose certain restrictions on the selection of the block and subblock sizes,
which, in turn, may preclude the use oversampled CE-BEM, only the less accurate critically
sampled CE-BEM may be used. Our simulation results will illustrate this fact together with
the fact that subblock-wise updating encounters no such problem.
We also note that the same considerations as above prevent the use of u1D447u1D435 = u1D45Au1D44F in the
block-wise scheme because this would lead to ?u1D443 = 1. Alternatively, to make the block-wise
scheme update the BEM parameters as frequently as the subblock-wise schemes, one may
take u1D447u1D435 = u1D45Au1D44F but to achieve parameter identifiability, one has to insert at least u1D444 ?clusters?
of zero-padded training impulses (as Section 2.3). To make this more concrete, let us consider
an example where we take u1D45Au1D44F = 80,u1D453u1D451u1D447u1D460 = 0.0025 and u1D43F = 8. With u1D447u1D435 = u1D45Au1D44F = 80, one
obtains u1D444 = 3; note that for doubly selective channels, when using CE-BEM, minimum
u1D444 = 3. Thus, we need 3 clusters of zero-padded training impulses with each cluster of
length 2u1D43F+ 1 = 17. Therefore, one assigns 51 symbols out 80 for training leaving just 29
information symbols - a poor choice resulting in poor spectral efficiency. Such a design has
63.75% training overhead whereas our proposed subblock schemes have a training overhead
of 21.25% (17 out of 80 symbols for training).
By exploiting the invariance of the coefficients of CE-BEM over each block (and hence
each of the ?u1D443 subblocks per block of length u1D447u1D435 symbols), we seek subblock-wise tracking of
the BEM coefficients of the doubly-selective channel. Consider two overlapping blocks (each
of u1D447u1D435 symbols) that differ by only one subblock of size u1D45Au1D44F: the ?past? block and ?current?
block begin at timeu1D45B0 and u1D45B0+u1D45Au1D44F respectively. Since the two blocks overlap so significantly,
one would expect the BEM coefficients to vary only a little from the past block to the current
one. Therefore, we can track ?u1D45E (u1D459) in (3.2) subblock-by-subblock via a first-order AR model
for their variations, rather than the anew with every non-overlapping block as in [70]. See
Fig. 3.1 that illustrates the overlapping blocks and subblocks in our approach.
3.2.1 Subblock-wise Kalman Tracking [52]
Consider a doubly-selective (time- and frequency-selective), single-input multi-output
(SIMO), finite impulse response (FIR) linear channel with u1D441 outputs. Let {u1D460(u1D45B)} denote
a scalar information sequence that is input to the time-varying channel with discrete-time
26
response {?(u1D45B;u1D459)} (u1D441-column vector channel response at time instance u1D45B to a unit input at
time instance u1D45B?u1D459). We assume {u1D460(u1D45B)} is independent and identically distributed (i.i.d.)
with zero mean and variance u1D70E2u1D460. Then the symbol-rate noisy u1D441-column channel output
vector is given by (u1D45B = 0,1,...)
y(u1D45B) =
u1D43Funi2211.alt02
u1D459=0
h(u1D45B;u1D459)u1D460(u1D45B?u1D459) +v(u1D45B) (3.1)
where the u1D441-column vector v(u1D45B) is zero-mean, white complex Gaussian noise, with variance
u1D70E2u1D463Iu1D441. We assume that {h(u1D45B;u1D459)} represents a wide-sense stationary uncorrelated scattering
(WSSUS) channel [60].
The time-varying channel with discrete-time response {h(u1D45B;u1D459)} is represented by CE-
BEM as (u1D45B = (u1D456?1)u1D447u1D435,(u1D456?1)u1D447u1D435 + 1,??? ,u1D456u1D447u1D435 ?1 for u1D456-th block and u1D459 = 0,1,??? ,u1D43F )
h(u1D45B;u1D459) =
u1D444uni2211.alt02
u1D45E=1
hu1D45E(u1D459)u1D452u1D457u1D714u1D45Eu1D45B, (3.2)
where hu1D45E (u1D459) is the u1D441-column time-invariant BEM coefficient vector and one chooses (? is
an integer)
u1D447 := ?u1D447u1D435, ? ? 1, (3.3)
u1D444? 2?u1D453u1D451u1D447u1D447u1D460?+ 1, (3.4)
u1D714u1D45E := 2u1D70Bu1D447 [u1D45E?(u1D444+ 1)/2], u1D45E = 1,2,??? ,u1D444 (3.5)
u1D43F := ?u1D70Fu1D451/u1D447u1D460?, (3.6)
u1D70Fu1D451 and u1D453u1D451 are respectively the delay spread and the Doppler spread, and u1D447u1D460 is the symbol
duration.
Let ?u1D45F,u1D45E(u1D459) denote the u1D45F-th component of the u1D441 column vector hu1D45E(u1D459). Stack the BEM
coefficients in (3.2) into vectors as
h(u1D459)u1D45F :=
uni005B.alt02
?u1D45F,1 (u1D459) ?u1D45F,2 (u1D459) ??? ?u1D45F,u1D444 (u1D459)
uni005D.alt02u1D447
(3.7)
hu1D45F :=
uni005B.alt02
h(0)u1D447u1D45F h(1)u1D447u1D45F ??? h(u1D43F)u1D447u1D45F
uni005D.alt02u1D447
(3.8)
h :=
uni005B.alt02
hu1D4471 hu1D4472 ??? hu1D447u1D441
uni005D.alt02u1D447
(3.9)
of size u1D444,u1D440 := u1D444(u1D43F+1) and u1D441u1D444(u1D43F+1) respectively. The coefficient vectors in (3.7), (3.8)
and (3.9) of the u1D45D-th overlapping block (u1D45D = 0,1,???) will be denoted by h(u1D459)u1D45F (u1D45D), hu1D45F (u1D45D), and
27
h(u1D45D) respectively. Note that the u1D45D-th block and the following (u1D45D+ 1)-st block differ by just
one subblock. Since a fading channel can follow well a Markov model [16], we further assume
that the BEM coefficients over each overlapping block are also Markovian. Although one
could fit a general AR model of (high) order u1D443 (as in Section 2.2.1), here we seek a ?simple?
formulation given by the first-order AR model, i.e.,
h(u1D45D) = A1h(u1D45D?1) +w(u1D45D), (3.10)
where we assume A1 = u1D6FCIu1D441u1D444(u1D43F+1) is the AR coefficient matrix (implying that all taps have
the same Doppler spectrum), and the driving noise vector w(u1D45D) is zero-mean white complex
Gaussian with variance u1D70E2u1D464Iu1D441u1D444(u1D43F+1) and statistically independent of h(u1D45D?1). We provide a
details about this simple AR(1) model and choice of u1D6FC in Section 3.4. Assuming the channel
is wide-sense stationary (WSS) and coefficients ?u1D45F,u1D45E (u1D459)?s are independent, we have
u1D70E2u1D464 = u1D70E2?(1??u1D6FC?2)/u1D444 (3.11)
where u1D70E2? := u1D438{?u1D45F (u1D45B;u1D459)??u1D45F (u1D45B;u1D459)}.
Under this formulation, we do not need a ?strict? definition of the block size u1D447u1D435 ?
although the channel is still represented by (3.2) for arbitrary time u1D45B, the BEM coefficients
?u1D45F,u1D45E(u1D459)?s are updated every subblock based on the training symbols. A key parameter now is
the CE-BEM period u1D447, not the block size u1D447u1D435. Later we use (3.2) for all times u1D45B, not just
the particular block of size u1D447u1D435 symbols, by allowing the coefficients ?u1D45F,u1D45E(u1D459)?s to change with
time (subblock-wise).
Based on CE-BEM given by (3.2), define
uD4D4 (u1D45B) :=
uni005B.alt02
u1D452?u1D457u1D7141u1D45B u1D452?u1D457u1D7142u1D45B ??? u1D452?u1D457u1D714u1D444u1D45B
uni005D.alt02u1D447
,
s(u1D45B) :=
uni005B.alt02
u1D460(u1D45B) u1D460(u1D45B?1) ??? u1D460(u1D45B?u1D43F)
uni005D.alt02u1D447
,
and then the received signal can be written as
y(u1D45B) = [s(u1D45B)?Iu1D441]u1D447
uni005B.alt02
Iu1D441(u1D43F+1) ?uD4D4 (u1D45B)
uni005D.alt02u1D43B
h(u1D45D) +v(u1D45B) (3.12)
when the u1D45D-th subblock is being received at time u1D45B, by (3.1), (3.2)-(3.6) and (3.7)-(3.9).
Treating (3.10) and (3.12) as the state and the measurement equations respectively, Kalman
filtering can be applied to track the BEM coefficient vector h(u1D45D) for each subblock.
We will employ the time-multiplexed (TM) training scheme in Section 2.3 [30] where
each subblock (of equal length u1D45Au1D44F symbols) consist of a data session (of length u1D45Au1D451 symbols)
28
and a succeeding training session (of lengthu1D45Au1D461 = 2u1D43F+1 symbols). Then the received signal at
u1D45F-th output at time u1D45Bu1D45D+u1D459 with u1D459 = 0,1,??? ,u1D43F is given by (assuming timing synchronization)
u1D466u1D45F (u1D45Bu1D45D +u1D459) = u1D6FE?u1D45F (u1D45Bu1D45D +u1D459;u1D459) +u1D463u1D45F (u1D45Bu1D45D +u1D459). (3.13)
for u1D45F = 1,2,??? ,u1D441. Using CE-BEM (3.2) in these u1D466u1D45F (u1D45Bu1D45D +u1D459)?s, one can uniquely solve for
?u1D45F,u1D45E(u1D459)?s via a least-square approach. Then the channel estimates are given by the CE-BEM
(3.2) using the estimated BEM coefficients. Using CE-BEM in (3.2) and (3.7)-(3.9), we can
simplify (3.12) at time u1D45Bu1D45D +u1D459 as (u1D45D = 0,1,??? and u1D459 = 0,1,??? ,u1D43F)
u1D466u1D45F (u1D45Bu1D45D +u1D459) = u1D6FEuD4D4u1D43B (u1D45Bu1D45D +u1D459)h(u1D459)u1D45F (u1D45D) +u1D463u1D45F (u1D45Bu1D45D +u1D459) (3.14)
Note that each (sub-)channel atu1D45F-th output can be tracked usingu1D45F-th received signal in (3.14)
respectively without interference. We intend to use only training sessions for subblock-wise
channel tracking regardless of data sessions. Further define
yu1D45F (u1D45D) :=
uni005B.alt02
u1D466u1D45F (u1D45Bu1D45D) u1D466u1D45F (u1D45Bu1D45D + 1) ??? u1D466u1D45F (u1D45Bu1D45D +u1D43F)
uni005D.alt02u1D447
,
vu1D45F (u1D45D) :=
uni005B.alt02
u1D463u1D45F (u1D45Bu1D45D) u1D463u1D45F (u1D45Bu1D45D + 1) ??? u1D463u1D45F (u1D45Bu1D45D +u1D43F)
uni005D.alt02u1D447
,
and
?(u1D45D) :=
?
??
??
??
?
uD4D4 (u1D45Bu1D45D)
uD4D4 (u1D45Bu1D45D + 1)
...
uD4D4 (u1D45Bu1D45D +u1D43F)
?
??
??
??
?
u1D43B
.
Then by (3.14),
yu1D45F (u1D45D) = u1D6FE?(u1D45D)hu1D45F (u1D45D) +vu1D45F (u1D45D), (3.15)
which gives us a formulation to estimate the BEM coefficients using training symbols (ses-
sions).
Thus we have obtained a linear discrete-time system represented by (3.10) and (3.15),
with which Kalman filtering can be applied to track the BEM coefficient vector hu1D45F (u1D45D) for
each subblock. Kalman tracking in the training mode is initialized with
?hu1D45F (?1 ? ?1) = 0u1D440 and R?
u1D45F (?1 ? ?1) = u1D70E
2
u1D464Iu1D440,
29
where ?hu1D45F (u1D45D?u1D45A) denotes the estimate of hu1D45F (u1D45D) given the observations {yu1D45F (u1D45B)}u1D45Au1D45B=0, and
R?u1D45F (u1D45D?u1D45A) denotes the error covariance matrix of ?hu1D45F (u1D45D?u1D45A), defined as being equal to
u1D438{[?hu1D45F (u1D45D?u1D45A)?hu1D45F (u1D45D)][?hu1D45F (u1D45D?u1D45A)?hu1D45F (u1D45D)]u1D43B}.
Kalman recursive filtering (for u1D45D = 0,1,???) one by one (for each u1D45F-th output) via the
following steps [32] (we have omitted the subscript u1D45F in the following steps):
1. Time update:
?h(u1D45D?u1D45D?1) = u1D6FC?h(u1D45D?1 ?u1D45D?1),
R? (u1D45D?u1D45D?1) = ?u1D6FC?2R? (u1D45D?1 ?u1D45D?1) +u1D70E2u1D464Iu1D440;
2. Kalman gain:
Ru1D702 (u1D45D) = ?u1D6FE?2?(u1D45D)R? (u1D45D?u1D45D?1)?u1D43B (u1D45D) +u1D70E2u1D463Iu1D43F+1,
K(u1D45D) = u1D6FER? (u1D45D?u1D45D?1)?u1D43B (u1D45D)R?1u1D702 (u1D45D);
3. Measurement update:
?h(u1D45D?u1D45D) = ?h(u1D45D?u1D45D?1) +K(u1D45D)uni005B.alt02y(u1D45D)?u1D6FE?(u1D45D)?h(u1D45D?u1D45D?1)uni005D.alt02,
R? (u1D45D?u1D45D) = [Iu1D440 ?u1D6FEK(u1D45D)?(u1D45D)]R? (u1D45D?u1D45D?1).
After Kalman filtering for every u1D45D, we generate the channel for the u1D45D-th subblock by the
estimate ?h(u1D45D?u1D45D) via the CE-BEM (3.2) as
??u1D45F (u1D45B;u1D459) = uD4D4u1D43B (u1D45B)?h(u1D459)u1D45F (u1D45D?u1D45D) (3.16)
for u1D45B = u1D45Du1D45Au1D44F,u1D45Du1D45Au1D44F + 1,??? ,(u1D45D+ 1)u1D45Au1D44F ?1. The definition of ?h(u1D459)u1D45F (u1D45D?u1D45D) is similar to (3.7).
3.2.2 Simulation Examples
Example 1
A random time- and frequency-selective Rayleigh fading channel is considered. We
assume h(u1D45B;u1D459) are zero-mean, complex Gaussian, and spatially white. We take u1D43F = 2 (3
taps) in (3.1), and u1D70E2? = u1D438{?u1D45F(u1D45B;u1D459)??u1D45F(u1D45B;u1D459)} = 1/(u1D43F+ 1). For different u1D459?s, h(u1D45B;u1D459)?s are
30
0 5 10 15 20 25 30?30
?25
?20
?15
?10
?5
0
SNR (dB)
NCMSE(dB)
N1K1,L=2,mb=20,fd=400Hz,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
(a) NCMSE vs SNR
0 5 10 15 20 25 3010
?4
10?3
10?2
10?1
100
SNR (dB)
Bit Error Rate
KD,N1K1,L=2,mb=20,fd=400Hz,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
TrueCH
(b) BER vs SNR, with Kalman detector of equalization
delay u1D451 = 5
Figure 3.2: Subblock-wise Kalman channel estimation: performance comparison for SNR?s
under u1D441 = u1D43E = 1,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 20.
mutually independent and satisfy Jakes? model. To this end, we simulate each single tap
following [75] (with a correction in the appendix of [63]).
We consider a communication system with carrier frequency of 2GHz, data rate of
40kBd (kilo-Bauds), therefore u1D447u1D460 = 25u1D707s, and a varying Doppler spread u1D453u1D451 in the range
of 0 to 400Hz, or the normalized Doppler spread u1D453u1D451u1D447u1D460 from 0 to 0.01 (corresponding to a
maximum mobile velocity from 0 to 216km/h). The additive noise is zero-mean complex
white Gaussian. The (receiver) SNR refers to the average energy per symbol over one-sided
noise spectral density.
The TM training scheme of [70], which is optimal for channels satisfying the critically
sampled CE-BEM representation, is adopted, where each subblock of equal length u1D45Au1D44F sym-
bols consists of an information session of u1D45Au1D451 symbols and a succeeding training session of u1D45Au1D461
symbols (u1D45Au1D44F = u1D45Au1D451 +u1D45Au1D461). We assume that each information symbol has unit power, while
at every training session given by (2.22), we set u1D6FE = ?2u1D43F+ 1 so that the average power per
symbol at training sessions is equal to that of information sessions. In the simulations, we
set u1D6FE = ?5 for u1D45Au1D44F = 20 (u1D45Au1D451 = 15 and u1D45Au1D461 = 5) or u1D45Au1D44F = 40 (u1D45Au1D451 = 35 and u1D45Au1D461 = 5).
We compared the following five schemes:
1. The block-adaptive channel estimation in [70], where the transmitted symbols are
segmented into consecutive blocks ofu1D447u1D435 symbols each. Every block consists of ?u1D443 (?u1D444)
subblocks as introduced in Section 2.3. For each non-overlapping block, we estimate the
BEM coefficients anew via a least-squares approach, and obtain the channel estimates
31
0 50 100 150 200 250 300 350 400?30
?25
?20
?15
?10
?5
fd (Hz)
NCMSE(dB)
N1K1,L=2,mb=20,SNR=20dB,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
(a) NCMSE vs u1D453u1D451
0 50 100 150 200 250 300 350 40010
?4
10?3
10?2
10?1
100
fd (Hz)
Bit Error Rate
KD,N1K1,L=2,mb=20,SNR=20dB,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
TrueCH
(b) BER vs u1D453u1D451, with Kalman detector of equalization
delay u1D451 = 5
Figure 3.3: Subblock-wise Kalman channel estimation: performance comparison for Doppler
spread under u1D441 = u1D43E = 1,SNR = 20dB,u1D45Au1D44F = 20.
over this block by the CE-BEM. We consider two models corresponding to u1D447 = 200
and 400 respectively, so that u1D444 = 5 and 9 by (3.4). We use an oversampled CE-BEM
with u1D447u1D435 = u1D447/2 for u1D45Au1D44F = 20 in order to suppress spectral leakage; whereas for u1D45Au1D44F = 40,
since an oversampled CE-BEM is not possible to satisfy ?u1D443 ? u1D444, we take u1D447u1D435 = u1D447.
[For parameter identifiability, one needs ?u1D443 ? u1D444, i.e., u1D447/(?u1D45Au1D44F) ? 2?u1D453u1D451u1D447u1D447u1D460?+ 1. With
u1D447u1D460 = 25u1D707u1D460,u1D453u1D451 = 400u1D43Bu1D467 and u1D45Au1D44F = 40, there exists no integer with ? = 2 for which we
can satisfy u1D447/(?u1D45Au1D44F) ? 2?u1D453u1D451u1D447u1D447u1D460? + 1. This implies that for blockwise updating with
u1D45Au1D44F = 40, we cannot use oversampled CE-BEM, only the less accurate critically sampled
CE-BEM.] Considering the oversampled CE-BEM basis functions are not orthogonal,
we apply a regularized least-square approach with a regularization parameter u1D6FD = 1
to estimate of BEM coefficients, and then obtain the channel estimates over this block
via the CE-BEM. Furthermore, the results were worse in the oversampled CE-BEM
case when no regularization was used, caused by possible ill-conditioning of certain
matrices when the basis functions are not orthogonal. For u1D456-th (non-overlapping) block
which has ?u1D443 subblocks, the estimate of u1D440 ?1 BEM coefficient vector ?h(u1D456) is given by
(u1D440 = u1D444(u1D43F+ 1) is the number of unknown BEM coefficients)
?h(u1D456) = uni005B.alt02u1D6FE2?u1D43Bu1D435(u1D456)?u1D435(u1D456) +u1D6FDIu1D440uni005D.alt02?1u1D6FE?u1D43Bu1D435(u1D456)yu1D435(u1D456), (3.17)
32
0 5 10 15 20 25 30?20
?18
?16
?14
?12
?10
?8
?6
?4
?2
0
SNR (dB)
NCMSE(dB)
N1K1,L=2,mb=40,fd=400Hz,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
(a) NCMSE vs SNR
0 5 10 15 20 25 3010
?3
10?2
10?1
100
SNR (dB)
Bit Error Rate
KD,N1K1,L=2,mb=40,fd=400Hz,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
TrueCH
(b) BER vs SNR, with Kalman detector of equalization
delay u1D451 = 5
Figure 3.4: Subblock-wise Kalman channel estimation: performance comparison for SNR?s
under u1D441 = u1D43E = 1,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 40.
where
yu1D435(u1D456) = u1D6FE?u1D435(u1D456)h(u1D456) +vu1D435(u1D456), (3.18)
defining
yu1D435(u1D456) =
uni005B.alt02
yu1D460
uni0028.alt02?
u1D443
uni0029.alt02
yu1D460
uni0028.alt02?
u1D443 ?1
uni0029.alt02
??? yu1D460 (1)
uni005D.alt02u1D447
,
?u1D435(u1D456) =
uni005B.alt02
?
uni0028.alt02?
u1D443
uni0029.alt02
?
uni0028.alt02?
u1D443 ?1
uni0029.alt02
??? ?(1)
uni005D.alt02u1D447
,
and then obtain the channel estimates over this block by the CE-BEM. In the figures,
this scheme is denoted by ?BA-LS?.
2. The channel tracking scheme in [76] using the symbol-wise first-order AR model and
Kalman filtering. Channel tracking is performed at training sessions only. For the
information sessions, the receiver updates the channel via ??(u1D45B;u1D459) = u1D6FCu1D450??(u1D45B?1;u1D459). We
assume that only the upper-bound of the Doppler spread is known. Then for Jakes?
model, u1D6FCu1D450 = u1D43D0 (2u1D70Bu1D453u1D451u1D447u1D460) = 0.999 for u1D453u1D451u1D447u1D460 = 0.01, where u1D43D0 (?) denotes the zero-th
Bessel function of the first kind. This scheme is denoted by ?AR(1)-KF?.
3. We also compared the approach of joint channel estimation and data detection via
extended Kalman filtering (EKF) in [68], where the channel taps are also described
by AR(1) model. For fairness, the turbo equalization (and channel coding) procedure
in [68] is omitted. This scheme is denoted by ?Joint-KF?.
33
0 50 100 150 200 250 300 350 400?30
?25
?20
?15
?10
?5
0
fd (Hz)
NCMSE(dB)
N1K1,L=2,mb=40,SNR=20dB,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
(a) NCMSE vs u1D453u1D451
0 50 100 150 200 250 300 350 40010
?4
10?3
10?2
10?1
100
fd (Hz)
Bit Error Rate
KD,N1K1,L=2,mb=40,SNR=20dB,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
TrueCH
(b) BER vs u1D453u1D451, with Kalman detector of equalization
delay u1D451 = 5
Figure 3.5: Subblock-wise Kalman channel estimation: performance comparison for Doppler
spread under u1D441 = u1D43E = 1,SNR = 20dB,u1D45Au1D44F = 40.
4. Our subblock-wise tracking using CE-BEM, which is denoted by ?SB-KF?, takes u1D447 =
200 and 400 for two different settings of CE-BEM, and u1D444 = 5 and 9 correspondingly.
We also take AR coefficient u1D6FC = 0.99 for u1D447 = 200 and u1D6FC = 0.9944 for u1D447 = 400 when
u1D45Au1D44F = 20, while u1D6FC = 0.94 for u1D447 = 200 and u1D6FC = 0.9667 for u1D447 = 400 when u1D45Au1D44F = 40(these
values were selected empirically via simulations).
5. Perfect channel estimates are available at the receiver, which is denoted by ?TrueCH?.
We evaluate the performances of various schemes by considering their normalized chan-
nel mean square error (NCMSE) and their bit error rates (BER). The NCMSE is defined
as
NCMSE :=
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0
vextenddoublevextenddouble
vextenddouble?h(u1D456) (u1D45B;u1D459)?h(u1D456) (u1D45B;u1D459)
vextenddoublevextenddouble
vextenddouble2
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0?h(u1D456) (u1D45B;u1D459)?
2
where h(u1D456) (u1D45B;u1D459) is the true channel and ?h(u1D456) (u1D45B;u1D459) is the estimated channel at the u1D456-th Monte
Carlo run, among total u1D440u1D45F runs, and u1D447u1D441 is the total observation length in symbols. The
Kalman detector in Section 2.4.1 with delay u1D451 = 5 is used using the channel estimates
obtained by each scheme. In each run, a symbol sequence of length u1D447u1D441 = 4000 is modulated
by QPSK and the first 200 symbols are discarded in evaluations. All the simulation results
are based on 500 runs.
In Fig. 3.2, the performances of the five schemes with u1D45Au1D44F = 20 (u1D45Au1D451 = 15 and u1D45Au1D461 =
5) are compared for different SNR?s under normalized Doppler spread u1D453u1D451u1D447u1D460 = 0.01. The
34
block-adaptive scheme with the regularized LS approach (BA-LS) and subblock-wise Kalman
tracking (SB-KF), based on CE-BEM, has better performance than the symbol-wise schemes,
AR(1)-based Kalman filtering (AR(1)-KF) and joint channel estimation and data detection
(Joint-KF). For CE-BEM-based schemes, more basis functions (u1D447 = 400,u1D444 = 9) in (3.2)
translate into better performance although the larger block size (u1D447u1D435 = 200) may increase
estimation variance.
In Fig. 3.3, the performances are compared for different Doppler spread u1D453u1D451?s under fixed
SNR = 20dB. It is clear that the symbol-wise AR(1)-KF and Joint-KF deteriorate sharply
due to the modeling inadequacies of the symbol-wise AR representation. In contrast, the
NCMSE?s and BER?s of the CE-BEM-based BA-LS and SB-KF vary more gradually with
increasing Doppler spreads, because the modeling error of the CE-BEM is much smaller.
Noting that the CE-BEM used in the schemes is based solely on the maximum Doppler
shift u1D453u1D451u1D447u1D460 = 0.01, it follows that the performances of the CE-BEM-based schemes are not
sensitive to the ?actual? Doppler spread if it is no larger than the assumed maximum value.
That is, we do not have to know the exact Doppler spread of the channel ? an upper
bound of it suffices. Since more unknown parameters are involved in the CE-BEM-based
schemes and hence result in higher estimation variance, these schemes are slightly inferior to
the symbol-wise AR-based schemes for small u1D453u1D451?s; but as u1D453u1D451 increases, the CE-BEM-based
schemes outperform the AR-based schemes.
In Fig. 3.4, we have longer information sessions to achieve a more spectrally-efficient
transmission, where the subblock size u1D45Au1D44F = 40 (u1D45Au1D451 = 35 and u1D45Au1D461 = 5) so that only 12.5%
of the transmitted symbols are dedicated to training; for the small subblock with u1D45Au1D44F = 20,
25% of the transmitted symbols are dedicated to training. Similarly, symbol-wise AR(1)-KF
and Joint-KF cannot capture the fast-varying channel even with high SNR?s. For the BA-
LS, we cannot use the oversampled CE-BEM with this longer subblock and hence results in
worse performance in NCMSE?s and BER?s due to the spectral leakage. [It is when we use
the oversampled CE-BEM that block-adaptive LS has similar performance as our proposed
subblock-wise Kalman tracking.] However, our proposed subblock-wise Kalman tracking
is insensitive to this problem and maintain satisfactory performances. In Fig. 3.5, four
schemes with u1D45Au1D44F = 40 are compared over different Doppler spread u1D453u1D451?s. Definitely, block-
adaptive scheme based on critically-sampled CE-BEM deteriorates except for small (nearby
zero) Doppler spread; the spectral leakage is also small with small u1D453u1D451. Our subblock-wise
Kalman tracking, where it updates the channel estimates subblock-by-subblock using CE-
BEM, outperforms other existing schemes.
35
0 5 10 15 20 25 30?25
?20
?15
?10
?5
0
SNR(dB)
NCMSE (dB)
K=1,L=8,mb=80,T=800,Q=5,fd=250Hz,Ts=10?s,500Runs
 
 
N=1
N=2
BA?LS
AR(1)?KF
Joint?KF
SB?KF
(a) NCMSE vs SNR
0 5 10 15 20 25 30
10?4
10?3
10?2
10?1
100
SNR(dB)
Bit Error Rate
K=1,L=8,mb=80,T=800,Q=5,fd=250Hz,Ts=10?s,500Runs
 
 
N=1
N=2
BA?LS
AR(1)?KF
Joint?KF
SB?KF
TrueCH
(b) BER vs SNR, with Kalman detector of equalization
delay u1D451 = 10
Figure 3.6: Subblock-wise Kalman channel estimation: performance comparison for SNR?s
under u1D43E = 1,u1D453u1D451u1D447u1D460 = 0.0025,u1D45Au1D44F = 80.
Example 2
The random Rayleigh fading channel in this example is as in Example 1, except that
we change the channel length u1D43F from 2 to 8 leading to 9 taps, and the power delay profile
is exponential satisfying u1D438{??u1D45F(u1D45B : u1D459)?2} = u1D70E2?(u1D459) = u1D44Eu1D452?u1D459/u1D43F where the constant u1D44E is picked
to satisfy uni2211.alt01u1D43Fu1D459=0u1D70E2?(u1D459) = 1. We consider a communication system with carrier frequency of
2GHz, data rate of 0.1MBd (mega-Bauds), therefore u1D447u1D460 = 10u1D707s, and a varying Doppler
spread u1D453u1D451 = 250Hz, or the normalized Doppler spread u1D453u1D451u1D447u1D460 = 0.0025 (corresponding to
a maximum mobile velocity of 135km/h). The training session is described by (2.22) of
length u1D45Au1D461 = 2u1D43F+ 1 = 17 symbols with u1D6FE = ?2u1D43F+ 1 so that the average symbol power of
training sessions is equal to that of information sessions. We select the period of the CE-
BEM u1D447 = 800 symbols, and hence u1D444 = 5 by (3.4) and a block size u1D447u1D435 = 400 for oversampled
CE-BEM.
Four estimation and tracking schemes are compared: block-adaptive channel estimation
in [70] using the regularized LS approach (?BA-LS?), channel tracking scheme in [76] using
a first-order AR model (?AR(1)-KF?), joint channel estimation and data detection via EKF
in [68] (?Joint-KF?) and our subblock-wise Kalman tracking (?SB-KF?). The BER?s are eval-
uated by employing the Kalman detector with delay u1D451 = 10, using the channel estimates
obtained by each scheme.
In Figs. 3.6, the performances of the four schemes with the subblock size u1D45Au1D44F = 80 are
compared for different SNR?s. For the subblock-wise Kalman tracking, we picked the AR
36
0 5 10 15 20 25 30?25
?20
?15
?10
?5
0
SNR(dB)
NCMSE (dB)
K=1,L=8,mb=100,T=800,Q=5,fd=250Hz,Ts=10?s,500Runs
 
 
N=1
N=2
BA?LS
AR(1)?KF
Joint?KF
SB?KF
(a) NCMSE vs SNR
0 5 10 15 20 25 30
10?4
10?3
10?2
10?1
100
SNR(dB)
Bit Error Rate
K=1,L=8,mb=100,T=800,Q=5,fd=250Hz,Ts=10?s,500Runs
 
 
N=1
N=2
BA?LS
AR(1)?KF
Joint?KF
SB?KF
TrueCH
(b) BER vs SNR, with Kalman detector of equalization
delay u1D451 = 10
Figure 3.7: Subblock-wise Kalman channel estimation: performance comparison for SNR?s
under u1D43E = 1,u1D453u1D451u1D447u1D460 = 0.0025,u1D45Au1D44F = 100.
coefficient u1D6FC = 0.993. Based on CE-BEM, block-adaptive scheme [70] with regularized LS
approach and subblock-wise Kalman filtering has superior performance to the symbol-wise
AR-based AR(1)-KF [76] and Joint-KF [68]. It is sure that CE-BEM is more effective to
estimate channel than AR model, especially in fast-varying channel. We also consider single-
input multiple-output system with 2 receivers to distinguish the performances of BA-LS and
SB-KF. Due to the assumption that the channel is spatially uncorrelated for different outputs,
the NCMSE curves foru1D441 = 1 and 2 coincide with each other but Joint-KF estimating channel
and data symbols jointly. It is worth pointing out that superior average channel NCMSE does
not necessarily translate into superior BER since a monotone relationship between average
NCMSE and BER does not necessarily exist. The BER?s with multi-receivers show the
proposed subblock-wise Kalman filtering outperforms block-adaptive LS approach. [One can
replace Kalman detector with MMSE-DFE in Section 2.4.2 that makes more distinguishable
BER performances. We will use DFE later.]
In Figs. 3.7, we have a little longer information sessions, where we set the subblock
size u1D45Au1D44F = 100 and the AR coefficient u1D6FC = 0.992 for the subblock-wise Kalman tracking.
Admitting the worse performance with the less training overhead (21.3% for u1D45Au1D44F = 80 and
17% for u1D45Au1D44F = 100), the deteriorations of BA-LS in NCMSE and BER are severe due to
the identifiability problem. For the block-adaptive schemes, we consider the oversampling
(? = 2), although parameter identifiability does not hold for oversampling since the number
of subblocks ( ?u1D443 = u1D447u1D435/u1D45Au1D44F = (u1D447/?)/u1D45Au1D44F = 4) is less than one of unknown parameters (u1D444 = 5).
37
[One can use the critically-sampling (? = 1) to satisfy the parameter identifiability. However,
our experiments for this example show the BA-LS solution with critically-sampled CE-BEM
is even worse than with oversampled one.] Our proposed subblock-wise Kalman tracking
updates the channel estimates subblock-by-subblock and hence the block size u1D447u1D435 is not a
key parameter as in BA-LS. As a result shown in Figs, the subblock-wise approach has more
advantages than block-wise or symbol-wise ones.
3.3 Subblock-wise MIMO Channel Estimation using CE-BEM
3.3.1 Subblock-wise Kalman Tracking for MIMO Channel
Consider a doubly-selective multi-input multi-output (MIMO), finite impulse response
(FIR) linear channel with u1D43E inputs and u1D441 outputs. Let {u1D460u1D458 (u1D45B)} denote u1D458-th user?s in-
formation sequence that is input to the time-varying channel with discrete-time response
{hu1D458 (u1D45B;u1D459)} (channel response for the u1D458-th user at time instance u1D45B to a unit input at time
instance u1D45B?u1D459). We assume {u1D460u1D458 (u1D45B)} is mutually independent and identically distributed
(i.i.d.) with zero mean and variance u1D438{u1D460u1D458 (u1D45B)u1D460?u1D458 (u1D45B)} = u1D70E2u1D460u1D458 = u1D70E2u1D460 for u1D458 = 1,2,??? ,u1D43E. Then
the symbol-rate noisy u1D441-column channel output vector is given by (u1D45B = 0,1,...)
y(u1D45B) =
u1D43Euni2211.alt02
u1D458=1
u1D43Funi2211.alt02
u1D459=0
hu1D458 (u1D45B;u1D459)u1D460u1D458 (u1D45B?u1D459) +v(u1D45B) (3.19)
where the u1D441-column vector v(u1D45B) is zero-mean, white, uncorrelated with u1D460u1D458 (u1D45B), complex
Gaussian noise, with the autocorrelation u1D438{v(u1D45B+u1D70F)vu1D43B (u1D45B)} = u1D70E2u1D463Iu1D441u1D6FF(u1D70F). We assume that
{hu1D458 (u1D45B;u1D459)} represents a wide-sense stationary uncorrelated scattering (WSSUS) channel [60],
independent for different u1D458?s. Define
s(u1D45B) :=
uni005B.alt02
u1D4601(u1D45B) u1D4602(u1D45B) ??? u1D460u1D43E(u1D45B)
uni005D.alt02u1D447
h(u1D45B;u1D459) :=
uni005B.alt02
h1(u1D45B;u1D459) h2(u1D45B;u1D459) ??? hu1D43E(u1D45B;u1D459)
uni005D.alt02
.
and we may rewrite (3.19) as
y(u1D45B) =
u1D43Funi2211.alt02
u1D459=0
h(u1D45B;u1D459)s(u1D45B?u1D459) +v(u1D45B). (3.20)
In CE-BEM [11,14,70], over the u1D456-th block consisting of an observation window of
u1D447u1D435 symbols, the channel is represented as (u1D45B = (u1D456? 1)u1D447u1D435,(u1D456? 1)u1D447u1D435 + 1,??? ,u1D456u1D447u1D435 ? 1 and
38
u1D459 = 0,1,??? ,u1D43F )
hu1D458(u1D45B;u1D459) =
u1D444uni2211.alt02
u1D45E=1
hu1D458,u1D45E(u1D459)u1D452u1D457u1D714u1D45Eu1D45B, (3.21)
where hu1D458,u1D45E (u1D459) is the u1D441-column time-invariant BEM coefficient vector for u1D458-th user and one
chooses (? is an integer)
u1D447 := ?u1D447u1D435, ? ? 1, (3.22)
u1D444? 2?u1D453u1D451u1D447u1D447u1D460?+ 1, (3.23)
u1D714u1D45E := 2u1D70Bu1D447 [u1D45E?(u1D444+ 1)/2], u1D45E = 1,2,??? ,u1D444 (3.24)
u1D43F := ?u1D70Fu1D451/u1D447u1D460?, (3.25)
u1D70Fu1D451 and u1D453u1D451 are respectively the delay spread and the Doppler spread, and u1D447u1D460 is the symbol
duration.
Now extend the proposed subblock-wise approach in Section 3.2 to MIMO systems. Let
?u1D45Fu1D458,u1D45E(u1D459) denote the u1D45F?th component of the column hu1D458,u1D45E(u1D459). Stack the BEM coefficients in
(3.21) into vectors as
h(u1D459)u1D45Fu1D458 :=
uni005B.alt02
?u1D45Fu1D458,1(u1D459) ?u1D45Fu1D458,2(u1D459) ??? ?u1D45Fu1D458,u1D444(u1D459)
uni005D.alt02u1D447
(3.26)
hu1D45Fu1D458 :=
uni005B.alt02
h(0)u1D447u1D45Fu1D458 h(1)u1D447u1D45Fu1D458 ??? h(u1D43F)u1D447u1D45Fu1D458
uni005D.alt02u1D447
(3.27)
h :=
uni005B.alt02
hu1D44711 ??? hu1D447u1D4411 ??? hu1D4471u1D43E ??? hu1D447u1D441u1D43E
uni005D.alt02u1D447
(3.28)
of size u1D444, u1D440 := u1D444(u1D43F+ 1) and u1D441u1D43Eu1D444(u1D43F+ 1), respectively. The coefficient vectors in (3.26),
(3.27) and (3.28) of the u1D45D-th subblock will be denoted by h(u1D459)u1D45Fu1D458 (u1D45D), hu1D45Fu1D458 (u1D45D), and h(u1D45D). Since
a fading channel can follow well a Markov model [16], we further assume that the BEM
coefficients over each overlapping block are also Markovian. A simplified formulation is
given by the first-order AR model, i.e.,
h(u1D45D) = A1h(u1D45D?1) +w(u1D45D), (3.29)
where we assume A1 = u1D6FCIu1D441u1D43Eu1D444(u1D43F+1) is the AR coefficient matrix, and the driving noise vector
w(u1D45D) is zero-mean white complex Gaussian with varianceu1D70E2u1D464Iu1D441u1D43Eu1D444(u1D43F+1) and statistically inde-
pendent of h(u1D45D?1). Assuming the channel is wide-sense stationary (WSS) and coefficients
?u1D45Fu1D458,u1D45E(u1D459)?s are independent, we have
u1D70E2u1D464 = u1D70E2?(1??u1D6FC?2)/u1D444 (3.30)
39
where u1D70E2? := u1D438{?u1D45Fu1D458 (u1D45B;u1D459)??u1D45Fu1D458 (u1D45B;u1D459)}.
Based on CE-BEM given by (3.21), define (where s(u1D45B) is u1D43E-column vector)
uD4D4 (u1D45B) :=
uni005B.alt02
u1D452?u1D457u1D7141u1D45B u1D452?u1D457u1D7142u1D45B ??? u1D452?u1D457u1D714u1D444u1D45B
uni005D.alt02u1D447
, (3.31)
S(u1D45B) :=
uni005B.alt02
su1D447 (u1D45B) su1D447 (u1D45B?1) ??? su1D447 (u1D45B?u1D43F)
uni005D.alt02u1D447
, (3.32)
and then the received signal can be written as
y(u1D45B) = [S(u1D45B)?Iu1D441]u1D447
uni005B.alt02
Iu1D441u1D43E(u1D43F+1) ?uD4D4 (u1D45B)
uni005D.alt02u1D43B
h(u1D45D) +v(u1D45B) (3.33)
when theu1D45D-th subblock is being received at timeu1D45B, by (3.20), (3.21)-(3.25) and (3.26)-(3.28).
Treating (3.29) and (3.33) as the state and the measurement equations respectively, Kalman
filtering can be applied to track the BEM coefficient vector h(u1D45D) for each subblock.
We will employ the TM training scheme proposed in Section 2.3 [30]. Then by design,
a given user?s training does not affect that of any other user during training session. Hence
the received signal between u1D45F-th output and u1D458-th input can be written as (assuming timing
synchronization)
u1D466u1D45F (u1D45Bu1D458,u1D45D +u1D459) = u1D6FE?u1D45Fu1D458 (u1D45Bu1D458,u1D45D +u1D459;u1D459) +u1D463u1D45F (u1D45Bu1D458,u1D45D +u1D459) (3.34)
for u1D459 = 0,1,??? ,u1D43F, u1D458 = 1,2,??? ,u1D43E and u1D45F = 1,2,??? ,u1D441. Using (3.21) in these u1D466u1D45F (u1D45Bu1D458,u1D45D +u1D459)?s,
one can uniquely solve for ?u1D45Fu1D458,u1D45E(u1D459)?s via a least-square (LS) approach. Then the channel
estimates are given by the CE-BEM (3.21) using the estimated BEM coefficients. Therefore,
every element of u1D441 ?u1D43E channel matrix,
h(u1D45B;u1D459) =
?
??
??
?11 (u1D45B;u1D459) ?12 (u1D45B;u1D459) ??? ?1u1D43E (u1D45B;u1D459)
... ... ?
u1D45Fu1D458 (u1D45B;u1D459)
...
?u1D4411 (u1D45B;u1D459) ?u1D4412 (u1D45B;u1D459) ??? ?u1D441u1D43E (u1D45B;u1D459)
?
??
??,
can be tracked independently one by one. Using (3.34), in order to track every (sub-)channel
between u1D441 outputs and u1D43E inputs respectively, we can simplify (3.33) at time u1D45Bu1D458,u1D45D + u1D459 as
(u1D45D = 0,1,??? and u1D459 = 0,1,??? ,u1D43F)
u1D466u1D45F (u1D45Bu1D458,u1D45D +u1D459) = u1D6FEuD4D4u1D43B (u1D45Bu1D458,u1D45D +u1D459)h(u1D459)u1D45Fu1D458 (u1D45D) +u1D463u1D45F (u1D45Bu1D458,u1D45D +u1D459) (3.35)
40
We intend to use only training sessions for subblock-wise channel tracking regardless of data
sessions. Further define
yu1D45F (u1D45D) :=
uni005B.alt02
u1D466u1D45F (u1D45Bu1D458,u1D45D) u1D466u1D45F (u1D45Bu1D458,u1D45D + 1) ??? u1D466u1D45F (u1D45Bu1D458,u1D45D +u1D43F)
uni005D.alt02u1D447
,
vu1D45F (u1D45D) :=
uni005B.alt02
u1D463u1D45F (u1D45Bu1D458,u1D45D) u1D463u1D45F (u1D45Bu1D458,u1D45D + 1) ??? u1D463u1D45F (u1D45Bu1D458,u1D45D +u1D43F)
uni005D.alt02u1D447
,
?(u1D45D) :=
?
??
??
??
?
uD4D4 (u1D45Bu1D458,u1D45D)
uD4D4 (u1D45Bu1D458,u1D45D + 1)
...
uD4D4 (u1D45Bu1D458,u1D45D +u1D43F)
?
??
??
??
?
u1D43B
.
Then by (3.35),
yu1D45F (u1D45D) = u1D6FE?(u1D45D)hu1D45Fu1D458 (u1D45D) +vu1D45F (u1D45D). (3.36)
Thus we have obtained a linear discrete-time system represented by (3.29) and (3.36).
Kalman filtering can be applied to track hu1D45Fu1D458 (u1D45D), each (sub-)channel?s BEM coefficients be-
tween u1D45F-th receiver and u1D458-th user. Since Kalman recursive filtering (for u1D45D = 0,1,???) is
applied one by one (for each pair u1D45Fu1D458) independently, we can just follow the same algorithms
in Section 3.2. After Kalman filtering for every u1D45D, we generate the channel for the u1D45D-th
subblock by the estimate ?h(u1D45D?u1D45D) via the CE-BEM (3.21) as
??u1D45Fu1D458 (u1D45B;u1D459) = uD4D4u1D43B (u1D45B)?h(u1D459)u1D45Fu1D458 (u1D45D?u1D45D) (3.37)
for u1D45B = u1D45Du1D45Au1D44F,u1D45Du1D45Au1D44F + 1,??? ,(u1D45D+ 1)u1D45Au1D44F ? 1. The definition of ?h(u1D459)u1D45Fu1D458 (u1D45D?u1D45D) is similar to (3.26).
Thus we obtain the entire u1D441 ?u1D43E channel matrix for the current subblock.
Computational Complexity
Here we consider computational complexity of channel estimation algorithm using the
floating point operation (flop) count for the channel estimation algorithm of each simulation
program. One floating point multiply/divide and associated adds/subtracts is called a flop
since the number of multiplies and divides often determines the CPU time; general multi-
plication of (u1D45A?u1D45D) matrix A and (u1D45D?u1D45B) matrix B has u1D45Au1D45Bu1D45D flops. For the explicit inverse
of (u1D45B?u1D45B) matrix A with (u1D45B?u1D45D) or (u1D45D?u1D45B) matrix B, i.e., A?1B or BA?1 is solved by
LU factorization using Gaussian elimination with partial pivoting and hence has u1D45B3/3+u1D45Du1D45B2
flops.
The detailed flops counts for one iteration of subblock-wise tracking algorithm are shown
in Table 3.1. Note that we have ?u1D443 = u1D447u1D435/u1D45Au1D44F iterations for one block length of u1D447u1D435. Defining
41
the number of flops for one cycle of subblock-wise tracking and block-wise LS one as (u1D440 =
u1D444(u1D43F+ 1),u1D43Fu1D45D = u1D43F+ 1)
u1D453u1D450(SB-KF) = u1D4403 +u1D4402(3u1D43Fu1D45D + 3) +u1D440(2u1D43F2u1D45D + 5u1D43Fu1D45D + 1) +u1D43F3u1D45D/3 +u1D43F2u1D45D,
u1D453u1D450(BA-LS) = u1D4403/3 +u1D4402(2 ?u1D443u1D43Fu1D45D + 1) +u1D440( ?u1D443u1D43Fu1D45D + 1),
we compare the overall flops counting for u1D447 = 200,u1D444 = 5 and u1D447 = 400,u1D444 = 9 with a block
of length u1D447u1D44E = 400 in Table 3.2. We set the overall flops for BA-LS(u1D447 = 200,u1D444 = 5) to be
one and take u1D447u1D435 = u1D447/?,? = 2(oversampling),u1D45Au1D44F = 20, ?u1D443 = u1D447u1D435/u1D45Au1D44F,u1D43Fu1D45D = u1D43F+ 1 = 3.
Note that the computational complexity of MIMO channels with u1D43E inputs and u1D441 out-
puts is obtained by simply u1D441u1D43E times of its single-input single-output (SISO) case since
the algorithm for MIMO channels are implemented one-by-one independently for every pair
between u1D45F-th output and u1D458-th input by the design of TM training scheme.
Table 3.1: Subblock-wise Kalman filtering: flop count for channel estimation over one sub-
block of u1D45Au1D44F symbols.
Operation Number of flops
?h(u1D45D?u1D45D?1) = u1D6FC?h(u1D45D?1 ?u1D45D?1) u1D440
R? (u1D45D?u1D45D?1) = ?u1D6FC?2R? (u1D45D?1 ?u1D45D?1) +u1D70E2u1D464Iu1D440 u1D4402(2)
Ru1D702 (u1D45D) = ?u1D6FE?2?(u1D45D)R? (u1D45D?u1D45D?1)?u1D43B (u1D45D) +u1D70E2u1D463Iu1D43F+1 u1D4402u1D43Fu1D45D +u1D440(u1D43F2u1D45D +u1D43Fu1D45D) +u1D43F2u1D45D
K(u1D45D) = u1D6FER? (u1D45D?u1D45D?1)?u1D43B (u1D45D)R?1u1D702 (u1D45D) u1D4402(u1D43Fu1D45D + 1) +u1D440u1D43F2u1D45D +u1D43F3u1D45D/3
?h(u1D45D?u1D45D) = ?h(u1D45D?u1D45D?1) +K(u1D45D)uni005B.alt02y(u1D45D)?u1D6FE?(u1D45D)?h(u1D45D?u1D45D?1)uni005D.alt02 u1D440(3u1D43Fu1D45D)
R? (u1D45D?u1D45D) = [Iu1D440 ?u1D6FEK(u1D45D)?(u1D45D)]R? (u1D45D?u1D45D?1) u1D4403 +u1D4402u1D43Fu1D45D +u1D440u1D43Fu1D45D
Total : u1D4403 +u1D4402(3u1D43Fu1D45D + 3) +u1D440(2u1D43F2u1D45D + 5u1D43Fu1D45D + 1) +u1D43F3u1D45D/3 +u1D43F2u1D45D
Table 3.2: Block-wise least-square and subblock-wise Kalman filtering: comparative flop
count for channel estimation over one block of u1D447u1D44E = 400 symbols.
Scheme Number of flops relative flops
BA-LS(T=200,Q=5) u1D453u1D450(BA-LS)(u1D447u1D44E/u1D447u1D435) 1.00
BA-LS(T=400,Q=9) u1D453u1D450(BA-LS)(u1D447u1D44E/u1D447u1D435) 3.11
SB-KF(T=200,Q=5) ?u1D443u1D453u1D450(SB-KF)(u1D447u1D44E/u1D447u1D435), ?u1D443 = 5 3.96
SB-KF(T=400,Q=9) ?u1D443u1D453u1D450(SB-KF)(u1D447u1D44E/u1D447u1D435), ?u1D443 = 10 17.61
42
3.3.2 Simulation Examples
Example 1
0 5 10 15 20 25 30?30
?25
?20
?15
?10
?5
0
SNR (dB)
NCMSE(dB)
N2K2,L=2,mb=20,fd=400Hz,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
(a) NCMSE vs SNR
0 5 10 15 20 25 3010
?4
10?3
10?2
10?1
100
SNR (dB)
Bit Error Rate
KD,N2K2,L=2,mb=20,fd=400Hz,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
TrueCH
(b) BER vs SNR, with Kalman detector of equalization
delay u1D451 = 5
Figure 3.8: Subblock-wise Kalman MIMO channel estimation: performance comparison for
SNR?s under u1D441 = u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 20.
A random time- and frequency-selective Rayleigh fading MIMO channel is considered.
We assume h(u1D45B;u1D459) are zero-mean, complex Gaussian, and spatially white. We take u1D43F = 2
(3 taps) in (3.20), and u1D70E2? = u1D438{?u1D45Fu1D458(u1D45B;u1D459)??u1D45Fu1D458(u1D45B;u1D459)} = 1/(u1D43F+ 1) between u1D458-th user and u1D45F-th
receiver. For different u1D459?s, h(u1D45B;u1D459)?s are mutually independent and satisfy Jakes? model. To
this end, we simulate each single tap following [75] (with a correction in the appendix of [63]).
We consider a communication system with carrier frequency of 2GHz, data rate of
40kBd (kilo-Bauds), therefore u1D447u1D460 = 25u1D707s, and a varying Doppler spread u1D453u1D451 in the range
of 0 to 400Hz, or the normalized Doppler spread u1D453u1D451u1D447u1D460 from 0 to 0.01 (corresponding to a
maximum mobile velocity from 0 to 216km/h). The additive noise is zero-mean complex
white Gaussian. The (receiver) SNR refers to the average energy per symbol over one-sided
noise spectral density.
The TM training scheme of [30], which is optimal for channels satisfying critically sam-
pled CE-BEM representation, is adopted, where each subblock of equal length u1D45Au1D44F symbols
consists of an information session of u1D45Au1D451 symbols and a succeeding training session of u1D45Au1D461
symbols (u1D45Au1D44F = u1D45Au1D451+u1D45Au1D461). We assume that each information symbol has unit power, while at
every training session given by (2.24), we set u1D6FE =
uni221A.alt02
u1D43E(u1D43F+ 1) +u1D43F so that the average power
per symbol at training sessions is equal to that of information sessions. In the simulations,
43
0 50 100 150 200 250 300 350 400?30
?25
?20
?15
?10
?5
fd (Hz)
NCMSE(dB)
N2K2,L=2,mb=20,SNR=20dB,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
(a) NCMSE vs u1D453u1D451
0 50 100 150 200 250 300 350 40010
?4
10?3
10?2
10?1
100
fd (Hz)
Bit Error Rate
KD,N2K2,L=2,mb=20,SNR=20dB,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
TrueCH
(b) BER vs u1D453u1D451, with Kalman detector of equalization
delay u1D451 = 5
Figure 3.9: Subblock-wise Kalman MIMO channel estimation: performance comparison for
Doppler spread under u1D441 = u1D43E = 2,SNR = 20dB,u1D45Au1D44F = 20.
we consider a simple two-receiver and two-user scenario, i.e., u1D441 = 2,u1D43E = 2 with the same
transmitted power. We set u1D45Au1D44F = 20, u1D45Au1D451 = 12 and u1D45Au1D461 = 8 with u1D6FE = ?8 for every user
following the TM training scheme.
We compared the following five schemes:
1. The block-adaptive channel estimation in [30], where the transmitted symbols are
segmented into consecutive blocks ofu1D447u1D435 symbols each. Every block consists of ?u1D443 (?u1D444)
subblocks as introduced in Section 2.3. We consider two models corresponding to
u1D447 = 200 and 400 respectively, so that u1D444 = 5 and 9 by (3.23). We use an oversampled
CE-BEM with u1D447u1D435 = u1D447/2 for u1D45Au1D44F = 20 since the over-sampled CE-BEM approximates
WSSUS channels much better than critically sampled CE-BEM [14]; whereas for u1D45Au1D44F =
40, since an oversampled CE-BEM is not possible to satisfy ?u1D443 ? u1D444, we take u1D447u1D435 = u1D447.
Considering the oversampled CE-BEM basis functions are not orthogonal, we apply a
regularized least-square approach with a regularization parameter u1D6FD = 1; furthermore,
the results were worse in the oversampled CE-BEM case when no regularization was
used, caused by possible ill-conditioning of certain matrices when the basis functions
are not orthogonal. In the figures, this scheme is denoted by ?BA-LS?.
2. The channel tracking scheme in [76] using a first-order AR model. Channel tracking is
performed at training sessions only. For data sessions, the receiver updates the channel
via ?h(u1D45B;u1D459) = u1D6FCu1D450?h(u1D45B?1;u1D459). We assume that only the upper-bound of the Doppler
44
0 5 10 15 20 25 30?20
?18
?16
?14
?12
?10
?8
?6
?4
?2
0
SNR (dB)
NCMSE(dB)
N2K2,L=2,mb=40,fd=400Hz,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
(a) NCMSE vs SNR
0 5 10 15 20 25 3010
?3
10?2
10?1
100
SNR (dB)
Bit Error Rate
KD,N2K2,L=2,mb=40,fd=400Hz,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
TrueCH
(b) BER vs SNR, with Kalman detector of equalization
delay u1D451 = 5
Figure 3.10: Subblock-wise Kalman MIMO channel estimation: performance comparison for
SNR?s under u1D441 = u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 40.
spread is known. Then for Jakes? model, u1D6FCu1D450 = u1D43D0 (2u1D70Bu1D453u1D451u1D447u1D460) = 0.999 for u1D453u1D451u1D447u1D460 = 0.01,
whereu1D43D0 (?) denotes the zero-th Bessel function of the first kind. This scheme is denoted
by ?AR(1)-KF?.
3. We also compared the approach of joint channel estimation and data detection via
extended Kalman filtering (EKF) in [68] without the turbo equalization (and channel
coding) procedure, using an AR(1) model for channel taps as above. This scheme is
denoted by ?Joint-KF?.
4. Our subblock-wise tracking using CE-BEM, which is denoted by ?SB-KF?, takes u1D447 =
200 and 400 for two different settings of CE-BEM, and u1D444 = 5 and 9 correspondingly.
We also take AR coefficient, u1D6FC = 0.99 for u1D447 = 200 and u1D6FC = 0.9944 for u1D447 = 400.
5. Perfect channel estimates are available at the receiver, which is denoted by ?TrueCH?.
We evaluate the performances of various schemes by considering their normalized chan-
nel mean square error (NCMSE) and their bit error rates (BER). The NCMSE is defined
as
NCMSE :=
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0
vextenddoublevextenddouble
vextenddouble?h(u1D456) (u1D45B;u1D459)?h(u1D456) (u1D45B;u1D459)
vextenddoublevextenddouble
vextenddouble2
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0?h(u1D456) (u1D45B;u1D459)?
2
where h(u1D456) (u1D45B;u1D459) is the true channel and ?h(u1D456) (u1D45B;u1D459) is the estimated channel at the u1D456-th Monte
Carlo run, among total u1D440u1D45F runs. The Kalman detector in Section 2.4.1 with delay u1D451 = 5 is
45
0 50 100 150 200 250 300 350 400?30
?25
?20
?15
?10
?5
0
fd (Hz)
NCMSE(dB)
N2K2,L=2,mb=40,SNR=20dB,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
(a) NCMSE vs u1D453u1D451
0 50 100 150 200 250 300 350 40010
?4
10?3
10?2
10?1
100
fd (Hz)
Bit Error Rate
KD,N2K2,L=2,mb=40,SNR=20dB,Ts=25?s,500Runs
 
 
BA?LS(T=200)
BA?LS(T=400)
AR(1)?KF
Joint?KF
SB?KF(T=200)
SB?KF(T=400)
TrueCH
(b) BER vs u1D453u1D451, with Kalman detector of equalization
delay u1D451 = 5
Figure 3.11: Subblock-wise Kalman MIMO channel estimation: performance comparison for
Doppler spread under u1D441 = u1D43E = 2,SNR = 20dB,u1D45Au1D44F = 40.
used using the channel estimates obtained by each scheme. In each run, a symbol sequence
of length, u1D447u1D441 = 4000 for each user is modulated by QPSK and the first 200 symbols are
discarded in evaluations. All the simulation results are based on 500 runs.
In Fig. 3.8, the performances of the five schemes are compared for different SNR?s under
fixed normalized Doppler spread u1D453u1D451u1D447u1D460 = 0.01. It is seen that the CE-BEM-based tracking
schemes, BA-LS [30] and SB-KF, have superior performance compared to the symbol-wise
AR-based AR(1)-KF and Joint-KF. The symbol-wise AR-model cannot capture the fast-
varying channel even with high SNR?s. Using the oversampled CE-BEM, more basis functions
(u1D447 = 400,u1D444 = 9) in (3.21) result in better performance. For the block-adaptive scheme,
larger block size (u1D447u1D435 = 200 = u1D447/?,? = 2 for oversampled CE-BEM) may induce larger
estimation variance. However, this strategy can well improve the performance of the proposed
SB-KF scheme since all past data are implicitly utilized in Kalman filtering-based subblock
tracking and we also update every subblock.
In Fig. 3.9, we compare the same schemes over different Doppler spreadu1D453u1D451?s with SNR =
20dB. For slowly fading channels, the CE-BEM-based trackings are slightly inferior to the
symbol-wise AR(1)-KF or Joint-KF, since more unknown parameters (BEM coefficients) are
involved in and therefore result in higher estimation variance. As u1D453u1D451 increases, the BA-LS
and SB-KF gradually outperforms the AR(1)-model-based schemes, since the time variations
of the channel have been well captured in CE-BEM while AR(1)-KF and Joint-KF schemes
cannot track the fast channel variations. Since the CE-BEM used in the schemes is based
46
solely on the maximum Doppler shift u1D453u1D451u1D447u1D460 = 0.01, the performances of the CE-BEM-based
schemes are not sensitive to the ?actual? Doppler spread if it is no larger than the assumed
maximum value. For the purpose of channel estimation, we need not the exact Doppler
spread but an upper bound of it when using CE-BEM.
In Fig. 3.10, we increase the subblock size u1D45Au1D44F = 20 to 40 and hence reduce the training
ratio 40% to 20%, so that we can save the transmitted symbols used for the additive training.
For the block-adaptive LS approach, the critically-sampled CE-BEM is considered and we
have u1D447u1D435 = u1D447(? = 1), ?u1D443 = u1D447u1D435/u1D45Au1D44F ? u1D444. Note that, for this longer subblock size u1D45Au1D44F =
40, BA-LS scheme with the oversampled CE-BEM cannot satisfy the unknown parameter
identifiability condition. The performances in NCMSE and BER of BA-LS with critically-
sampled CE-BEM deteriorate significantly due to the spectral leakage. However, unlike
block-adaptive scheme, the proposed subblock-wise approach has no identifiability condition
to estimate unknown BEM coefficients since we update subblock-by-subblock and the block
size u1D447u1D435 is not a strict factor.
Fig. 3.5 show the performances of this longer subblock size u1D45Au1D44F = 40 over different
Doppler spread u1D453u1D451?s. It is seen that CE-BEM is more adequate to keep track of the wire-
less channel than symbol-wise AR model; AR(1)-KF and Joint-KF deteriorate as u1D453u1D451 in-
creases. The block-adaptive LS based on the critically-sampled CE-BEM also deteriorates
with Doppler spread; it has a good performance only when u1D453u1D451 ? 0. [One can try BA-LS with
the oversampled CE-BEM, neglecting the identifiability. However, the latter performances
is similar as the former ones.] On the other hand, while SB-KF has some variations and
slightly goes worse as u1D453u1D451 increases, the overall performance is satisfactory.
Example 2
The random Rayleigh fading MIMO channel in this example is as in Example 1, except
that we change the channel length u1D43F from 2 to 8 leading to 9 taps, and the power delay
profile is exponential satisfying u1D438{??u1D45Fu1D458(u1D45B : u1D459)?2} = u1D70E2?(u1D459) = u1D44Eu1D452?u1D459/u1D43F between u1D458-th user and u1D45F-th
receiver where the constant u1D44E is pick to satisfyuni2211.alt01u1D43Fu1D459=0u1D70E2?(u1D459) = 1. We consider a communication
system with carrier frequency of 2GHz, data rate of 0.1MBd (mega-Bauds), therefore u1D447u1D460 =
10u1D707s, and a varying Doppler spread u1D453u1D451 = 250Hz, or the normalized Doppler spread u1D453u1D451u1D447u1D460 =
0.0025 (corresponding to a maximum mobile velocity of 135km/h). The training session is
described by (2.24) of length u1D45Au1D461 = u1D43E(u1D43F+ 1)+u1D43F = 26 symbols with u1D6FE =
uni221A.alt02
u1D43E(u1D43F+ 1) +u1D43F so
that the average symbol power of training sessions is equal to that of information sessions.
We select the period of the CE-BEM u1D447 = 800 symbols, and hence u1D444 = 5 by (3.23) and a
block size u1D447u1D435 = 400 for oversampled CE-BEM.
47
0 5 10 15 20 25 30?25
?20
?15
?10
?5
0
SNR(dB)
NCMSE (dB)
K=2,L=8,mb=80,T=800,Q=5,fd=250Hz,Ts=10?s,500Runs
 
 
N=2
N=3
BA?LS
AR(1)?KF
Joint?KF
SB?KF
(a) NCMSE vs SNR
0 5 10 15 20 25 30
10?4
10?3
10?2
10?1
100
SNR(dB)
Bit Error Rate
K=2,L=8,mb=80,T=800,Q=5,fd=250Hz,Ts=10?s,500Runs
 
 
N=2
N=3
BA?LS
AR(1)?KF
Joint?KF
SB?KF
TrueCH
(b) BER vs SNR, with Kalman detector of equalization
delay u1D451 = 10
Figure 3.12: Subblock-wise Kalman MIMO channel estimation: performance comparison for
SNR?s under u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.0025,u1D45Au1D44F = 80.
Four estimation and tracking schemes are compared: block-adaptive channel estimation
in [30] using the regularized LS approach (?BA-LS?), channel tracking scheme in [76] using
a first-order AR model (?AR(1)-KF?), joint channel estimation and data detection via EKF
in [68] (?Joint-KF?) and our subblock-wise Kalman tracking (?SB-KF?). The BER?s are eval-
uated by employing the Kalman detector with delay u1D451 = 10, using the channel estimates
obtained by each scheme.
In Fig. 3.12, the performances of the above schemes with the subblock size u1D45Au1D44F = 80 are
compared for different SNR?s. For the subblock-wise Kalman tracking, we picked the AR
coefficient u1D6FC = 0.993. Similarly as Example 1, the symbol-wise AR(1)-KF [76] and Joint-
KF [68] approaches is not appropriate for the fast-varying channels. Using the oversampled
CE-BEM, the block-adaptive scheme [30] with regularized LS approach (BA-LS) has similar
performances as the subblock-wise Kalman filtering (SB-KF). Clearly, the CE-BEM-based
schemes, where BEM coefficients slowly change block by block, outperform AR-based ones.
To compare BA-LS and SB-KF schemes more clearly, we also have the examples with u1D441 = 3
receivers. Since we assume the channel is spatially uncorrelated for different outputs, the
NCMSE curves show the same performances for u1D441 = 2 and u1D441 = 3. [Joint-KF has different
NCMSE curves for u1D441 = 2 and u1D441 = 3 since it estimates channel and data symbols jointly.]
The superiority in NCMSE does not guarantee the superiority in BER since NCMSE and
BER have no monotone relationship each other. When increasing the number of receivers,
48
0 5 10 15 20 25 30?25
?20
?15
?10
?5
0
SNR(dB)
NCMSE (dB)
K=2,L=8,mb=100,T=800,Q=5,fd=250Hz,Ts=10?s,500Runs
 
 
N=2
N=3
BA?LS
AR(1)?KF
Joint?KF
SB?KF
(a) NCMSE vs SNR
0 5 10 15 20 25 30
10?4
10?3
10?2
10?1
100
SNR(dB)
Bit Error Rate
K=2,L=8,mb=100,T=800,Q=5,fd=250Hz,Ts=10?s,500Runs
 
 
N=2
N=3
BA?LS
AR(1)?KF
Joint?KF
SB?KF
TrueCH
(b) BER vs SNR, with Kalman detector of equalization
delay u1D451 = 10
Figure 3.13: Subblock-wise Kalman MIMO channel estimation: performance comparison for
SNR?s under u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.0025,u1D45Au1D44F = 100.
we have better performance in BER?s. In particular, the proposed subblock-wise Kalman
filtering outperforms block-adaptive LS approach with more receivers.
In Fig. 3.13, we increase the subblock size u1D45Au1D44F = 80 to u1D45Au1D44F = 100 and hence have more
spectrally-efficient transmission; the training overhead decreases from 15% to 12%. We set
the AR coefficient u1D6FC = 0.992 for the subblock-wise Kalman tracking. The performances of
all schemes become worse due to the lower frequency of training sessions. But the relative
superiority of the subblock-wise schemes over block-wise schemes is unchanged.
3.4 AR(1) Coefficient u1D6FC for Subblock-wise Kalman Tracking
Here we discuss choice of the AR coefficient u1D6FC for subblock-wise channel estimation
using Kalman filter investigated in this chapter for a class of (true) channel models. We
assume that the components of the MIMO channel h(u1D45B;u1D459) are mutually independent and
identically distributed (i.i.d.) with zero-mean, complex Gaussian, and spatially white with
autocorrelation u1D438
uni007B.alt02
h(u1D45B;u1D459)hu1D43B (u1D45B;u1D459)
uni007D.alt02
= u1D70E2?Iu1D441. Then a simplified AR(1) model for the BEM
coefficient vector is given by
h(u1D45D) = A1h(u1D45D?1) +w(u1D45D), (3.38)
49
where A1 is the AR coefficient matrix, and the driving noise vector w(u1D45D) is zero-mean white
complex Gaussian with variance u1D70E2u1D464I and statistically independent of h(u1D45D?1). Defining
hu1D435(u1D45B;u1D459) := [h(u1D45B;u1D459) h(u1D45B+ 1;u1D459) ??? h(u1D45B+u1D447u1D435 ?1;u1D459)]u1D447 , (3.39)
consider two overlapping blocks that differ by just one subblock: hu1D435(u1D45Du1D45Au1D44F;u1D459) and hu1D435((u1D45D +
1)u1D45Au1D44F;u1D459) with h(u1D459)(u1D45D) and h(u1D459)(u1D45D+1) as the corresponding BEM coefficient vector respectively.
Further define
E(u1D45B) :=
uni005B.alt02
uD4D4(u1D45B) uD4D4(u1D45B+ 1) ??? uD4D4(u1D45B+u1D447u1D435 + 1)
uni005D.alt02u1D43B
?Iu1D441u1D43E, (3.40)
? := u1D451u1D456u1D44Eu1D454
uni007B.alt02
u1D452u1D457u1D7141u1D45Au1D44F,u1D452u1D457u1D7142u1D45Au1D44F,??? ,u1D452u1D457u1D714u1D444u1D45Au1D44F
uni007D.alt02
?Iu1D441u1D43E, (3.41)
where E(u1D45B) is u1D441u1D43Eu1D447u1D435 ?u1D441u1D43Eu1D444 and ? is u1D441u1D43Eu1D444?u1D441u1D43Eu1D444. It follows that
hu1D435(u1D45Du1D45Au1D44F;u1D459) = E(u1D45Du1D45Au1D44F)h(u1D459)(u1D45D), (3.42)
hu1D435((u1D45D+ 1)u1D45Au1D44F;u1D459) = E(u1D45Du1D45Au1D44F)?h(u1D459)(u1D45D+ 1). (3.43)
Defining
?hu1D435(u1D45B) := uni005B.alt02hu1D447u1D435(u1D45B;0) hu1D447u1D435(u1D45B;1) ??? hu1D447u1D435(u1D45B;u1D43F)uni005D.alt02u1D447 , (3.44)
we have
?hu1D435(u1D45Du1D45Au1D44F) = ?E(u1D45Du1D45Au1D44F)h(u1D45D), (3.45)
?hu1D435((u1D45D+ 1)u1D45Au1D44F) = ?E(u1D45Du1D45Au1D44F)??h(u1D45D+ 1), (3.46)
where ?E(u1D45Du1D45Au1D44F) := u1D451u1D456u1D44Eu1D454{E(u1D45Du1D45Au1D44F), E(u1D45Du1D45Au1D44F), ??? E(u1D45Du1D45Au1D44F)}, and ?? := u1D451u1D456u1D44Eu1D454{?, ?, ??? , ?}.
If (3.38) holds, then using the Yule-Walker equation
A1 = u1D438
uni007B.alt02
h(u1D45D+ 1)hu1D43B(u1D45D)
uni007D.alt02uni005B.alt02
u1D438
uni007B.alt02
h(u1D45D)hu1D43B(u1D45D)
uni007D.alt02uni005D.alt02?1
, (3.47)
where using (3.45) and (3.46) we have
u1D438
uni007B.alt02
h(u1D45D+ 1)hu1D43B(u1D45D)
uni007D.alt02
= ???1?E?(u1D45Du1D45Au1D44F)u1D438
uni007B.alt02?
hu1D435((u1D45D+ 1)u1D45Au1D44F)?hHB(pmb)
uni007D.alt02?
E?u1D43B(u1D45Du1D45Au1D44F), (3.48)
u1D438
uni007B.alt02
h(u1D45D)hu1D43B(u1D45D)
uni007D.alt02
= ?E?(u1D45Du1D45Au1D44F)u1D438
uni007B.alt02?
hu1D435(u1D45Du1D45Au1D44F)?hHB(pmb)
uni007D.alt02?
E?u1D43B(u1D45Du1D45Au1D44F), (3.49)
and u1D438
uni007B.alt02?
hu1D435((u1D45D+ 1)u1D45Au1D44F)?hu1D43Bu1D435(u1D45Du1D45Au1D44F)
uni007D.alt02
and u1D438
uni007B.alt02?
hu1D435(u1D45Du1D45Au1D44F)?hu1D43Bu1D435(u1D45Du1D45Au1D44F)
uni007D.alt02
can be calculated using (3.39)
and (3.44) if we know the channel correlation function R?(u1D70F).
50
Since typically R?(u1D70F) will not be available, in order to simplify we assumed A1 = u1D6FCI
and u1D438
uni007B.alt02
w(u1D45D)wu1D43B(u1D45D)
uni007D.alt02
= u1D70E2u1D464I, leading to
h(u1D45D) = u1D6FCh(u1D45D?1) +w(u1D45D). (3.50)
If the channel is wide-sense stationary uncorrelated scattering (WSSUS) and coefficients
?u1D45E(u1D459)?s are independent (as assumed in [70]), then by (3.50) and Yule-Walker equations, we
have
u1D6FC =
?
?u1D438
uni007B.alt02
hu1D43B(u1D45D+ 1)h(u1D45D)
uni007D.alt02
u1D438{hu1D43B(u1D45D)h(u1D45D)}
?
?
?
= u1D461u1D45F
uni007B.alt02?
??1?E?(u1D45Du1D45Au1D44F)u1D438
uni007B.alt02?
hu1D435((u1D45D+ 1)u1D45Au1D44F)?hu1D43Bu1D435(u1D45Du1D45Au1D44F)
uni007D.alt02?
E?u1D43B(u1D45Du1D45Au1D44F)
uni007D.alt02
u1D461u1D45F
uni007B.alt02?
E?(u1D45Du1D45Au1D44F)u1D438
uni007B.alt02?
hu1D435(u1D45Du1D45Au1D44F)?hu1D43Bu1D435(u1D45Du1D45Au1D44F)
uni007D.alt02?
E?u1D43B(u1D45Du1D45Au1D44F)
uni007D.alt02 , (3.51)
and u1D70E2u1D464 = u1D70E2?(1 ? ?u1D6FC?2)/u1D444. Assuming for different u1D459?s, h(u1D45B;u1D459)?s are mutually independent
and satisfy Jakes? model, we know the correlation of ?(u1D45B;u1D459) and then we can calculate u1D6FC via
(3.51). For the over-sampled CE-BEM, ?E(u1D45B) may be ill-conditioned since the basis functions
are not orthogonal. Note that (3.51) also requires knowledge of R?(u1D70F). In order to avoid
this, one can somewhat arbitrarily pick a value of u1D6FC; this has been done in, e.g. [28] (in a
different but similar context). Since the coefficients evolve slowly, we will have u1D6FC ? 1 (but
u1D6FC < 1 for tracking uses). The results of (3.51) are similar to (2.12) in Section 2.2.1 except
that in the former changes occur every subblock whereas in the latter the changes occur
every symbol.
To gain more insight, let us consider a specific channel tap ?(u1D45B;u1D459) following the Jakes?
spectrum (also used in all simulation examples in this paper). Whenu1D447 = 400,u1D447u1D435 = 200,u1D444 =
9, and u1D453u1D451u1D447u1D460 = 0.01, one gets u1D6FC = 0.9814 and 0.9425 for u1D45Au1D44F = 20 and 40, respectively; i.e.,
dependence between subblocks decreases with increasing u1D45Au1D44F. With u1D453u1D451u1D447u1D460 held at 0.01, when
u1D447 and u1D447u1D435 are halved to 200 and 100, respectively, (leading to u1D444 = 5), one gets u1D6FC = 0.9605
and 0.8791 for u1D45Au1D44F = 20 and 40, respectively; i.e., as u1D447u1D435 increases, u1D6FC increases, and vice versa.
When u1D447 = 400,u1D447u1D435 = 200, and u1D453u1D451u1D447u1D460 = 0.005 (leading to u1D444 = 5), one gets u1D6FC = 0.9893 and
0.9605 for u1D45Au1D44F = 20 and 40, respectively; i.e., as u1D453u1D451 decreases, u1D6FC increases and vice versa.
These numerical results are consistent with one?s intuition: faster channel variations call for
smaller u1D6FC values (less dependence), and vice versa.
In Figs. 3.14, the performances of subblock-wise Kalman tracking are shown for different
values of u1D6FC based on 500 Monte Carlo runs with the same environment as the simulation
example in Section 3.2.2 and 3.3.2. In our simulations, we picked the AR(1) coefficient
u1D6FC = 0.9944 and 0.9667 for u1D45Au1D44F = 20 and 40 respectively, where u1D447 = 400,u1D444 = 9 in CE-BEM.
51
It is seen that while the performance is not sensitive to the value of u1D6FC over a relatively wide
range of values, it does deteriorate as u1D6FC approaches one. Note that u1D6FC is close to one (u1D6FC? 1)
but it is not one since u1D6FC = 1 in AR(1) model implies time-invariance and u1D6FC < 1 permits
tracking by discounting older values of the channel BEM coefficients.
0.9 0.92 0.94 0.96 0.98 1?30
?25
?20
?15
?10
?5
0
?
NCMSE(dB)
N1K1,L=2,T=400,Q=9,fdTs=0.01,SNR=20dB,500runs
 
 
mb=40
mb=20
(a) NCMSE vs u1D6FC
0.9 0.92 0.94 0.96 0.98 110
?4
10?3
10?2
10?1
100
KD,N1K1,L=2,T=400,Q=9,fdTs=0.01,SNR=20dB,500runs
?
Bit Error Rate
 
 
mb=40
mb=20
(b) BER vs u1D6FC, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and delay
u1D451 = 5
Figure 3.14: Subblock-wise Kalman channel estimation: performances for AR(1) coefficient
u1D6FC?s, under u1D441 = u1D43E = 1,u1D453u1D451u1D447u1D460 = 0.01,SNR = 20dB,u1D45Au1D44F = 20 and 40.
3.5 Conclusions
We presented a subblock-wise tracking approach to doubly-selective channel estimation,
exploiting the oversampled complex exponential basis expansion model (CE-BEM) for the
overall channel variations, and an autoregressive (AR) model to update the BEM coefficients.
We tracked the BEM coefficients via Kalman filtering, based on time-multiplexed periodically
transmitted training symbols. Rather than track each channel tap gain, we estimated the
BEM coefficients subblock by subblock, and then (re-)generated the channel via the CE-BEM
with the estimated BEM coefficients. In this way, the modeling mismatch introduced by the
conventionally used symbol-wise AR channel model can be greatly reduced, and hence better
performance can be achieved in fast-fading environments. We also extended the subblock-
wise tracking to MIMO systems. Simulation examples demonstrated its superior performance
over some existing doubly-selective channel tracking schemes, over a wide range of Doppler
spreads.
52
Chapter 4
Recursive Least-squares Doubly-selective Channel Estimation Using
Exponential Basis Models and Subblock-wise Tracking
We present the subblock-wise channel estimation scheme, exploiting the oversampled
complex exponential basis expansion model (CE-BEM) for the overall channel variations.
The time-varying nature of the channel is well captured by the CE-BEM while the time-
variations of the (unknown) BEM coefficients are likely much slower than that of the channel.
We apply the exponentially-weighted (EW) and sliding-window (SW) recursive least-squares
(RLS) algorithms to track the BEM coefficients, using time-multiplexed periodically trans-
mitted training symbols. Simulation examples demonstrate its superior performance over
the conventional block-wise channel estimator.
4.1 Introduction
In Chapter 3, a subblock-wise tracking approach was proposed for doubly-selective chan-
nels using time-multiplexed (TM) training. It exploits the CE-BEM for the overall channel
variations, and a first-order AR model to describe the evolutions of the BEM coefficients.
The slow-varying BEM coefficients, rather than the fast-varying channel, are tracked and up-
dated at each training session; during information sessions, channel estimates are generated
by the CE-BEM using the estimated BEM coefficients. The BEM coefficients are updated
via Kalman filtering at each training session; during information sessions, channel estimates
are generated by the CE-BEM using the estimated BEM coefficients. The subblock-wise
approaches achieve better performance in fast-fading environments, than using conventional
symbol-wise AR models [6] or block-wise BEM representations [30,70].
However, the approach in Chapter 3 used the first-order AR model for BEM coefficients,
which is not necessarily suited to a real channel environment, and possibly causes modeling
error in estimation. To solve this problem, we propose an adaptive channel estimation
scheme without an arbitrary a priori model for the BEM coefficients. Two finite-memory
adaptive filtering algorithms, the EW- and SW-RLS algorithm, are considered for subblock-
wise channel tracking.
A decision-directed channel tracking via the RLS algorithm within a sliding window
using polynomial BEM has been investigated in [8] and channel estimation via Kalman
filtering using polynomial or complex exponential BEM for OFDM systems has been explored
53
in [26,41,44]; all these approaches use block-by-block updating (for single input systems).
Our contribution exploits subblock-wise updating, where several subblocks comprise one
block. For parameter identifiability one needs the number of subblocks at least as large as
the number of basis functions used for channel modeling. Data in just one subblock do not
satisfy the parameter identifiability requirements. We also extend the proposed schemes to
multi-input multi-output (MIMO) systems.
4.2 Subblock-wise RLS Channel Estimation using CE-BEM [23]
Suppose that we collect the received signal over a time period u1D447u1D435, a block size. We esti-
mate the time-variant channel using the useful channel models and time-multiplexed training,
and then detect the information symbols using this estimated channel. By exploiting the
invariance of the coefficients of CE-BEM over each block, hence, each of the ?u1D443 subblocks per
block of length u1D447u1D435 symbols, we seek to adaptively estimate the doubly-selective channel via
subblock-wise tracking.
Consider two overlapping blocks (each consisting of sequences of u1D447u1D435 symbols) that differ
by only one subblock of length u1D45Au1D44F: the ?past? block beginning at time u1D45B0 and the ?current?
block beginning at time u1D45B0 +u1D45Au1D44F. Since the two blocks overlaps so significantly, one would
expect the BEM coefficients to vary only a little from the past block to the current over-
lapping block. Therefore, we propose a subblock-wise channel tracking as in [52] where we
estimate hu1D45E (u1D459) in (4.2) subblock-by-subblock for their variations, rather than the anew with
every non-overlapping block as in [70].
Consider a doubly-selective (time- and frequency-selective), single-input multi-output
(SIMO), finite impulse response (FIR) linear channel with u1D441 outputs. Let {u1D460(u1D45B)} denote
a scalar information sequence that is input to the time-varying channel with discrete-time
response {h(u1D45B;u1D459)} (u1D441-column vector channel response at time instance u1D45B to a unit input at
time instance u1D45B?u1D459), assuming {u1D460(u1D45B)} is independent and identically distributed (i.i.d.) with
zero mean and variance u1D70E2u1D460. Then the symbol-rate noisy u1D441-column channel output is given
by (u1D45B = 0,1,...)
y(u1D45B) =
u1D43Funi2211.alt02
u1D459=0
h(u1D45B;u1D459)u1D460(u1D45B?u1D459) +v(u1D45B) (4.1)
where the u1D441-column vector v(u1D45B) is zero-mean, white complex Gaussian noise, with variance
u1D70E2u1D463Iu1D441. We assume that {h(u1D45B;u1D459)} represents a wide-sense stationary uncorrelated scattering
(WSSUS) channel [60].
54
The time-varying channel with discrete-time response {h(u1D45B;u1D459)} is represented by CE-
BEM as (u1D45B = (u1D456?1)u1D447u1D435,(u1D456?1)u1D447u1D435 + 1,??? ,u1D456u1D447u1D435 ?1 for u1D456-th block and u1D459 = 0,1,??? ,u1D43F )
h(u1D45B;u1D459) =
u1D444uni2211.alt02
u1D45E=1
hu1D45E(u1D459)u1D452u1D457u1D714u1D45Eu1D45B, (4.2)
where hu1D45E(u1D459) is theu1D441-column time-invariant BEM coefficient and one chooses (? is an integer)
u1D447 := ?u1D447u1D435, ? ? 1, (4.3)
u1D444? 2?u1D453u1D451u1D447u1D447u1D460?+ 1, (4.4)
u1D714u1D45E := 2u1D70Bu1D447 [u1D45E?(u1D444+ 1)/2], u1D45E = 1,2,??? ,u1D444 (4.5)
u1D43F := ?u1D70Fu1D451/u1D447u1D460?, (4.6)
u1D70Fu1D451 and u1D453u1D451 are respectively the delay spread and the Doppler spread, and u1D447u1D460 is the symbol
duration.
Let ?u1D45F,u1D45E(u1D459) denote the u1D45F-th component of the u1D441 column vector hu1D45E(u1D459). Stack the BEM
coefficients in (4.2) into vectors as
h(u1D459)u1D45F :=
uni005B.alt02
?u1D45F,1 (u1D459) ?u1D45F,2 (u1D459) ??? ?u1D45F,u1D444 (u1D459)
uni005D.alt02u1D447
(4.7)
hu1D45F :=
uni005B.alt02
h(0)u1D447u1D45F h(1)u1D447u1D45F ??? h(u1D43F)u1D447u1D45F
uni005D.alt02u1D447
(4.8)
h :=
uni005B.alt02
hu1D4471 hu1D4472 ??? hu1D447u1D441
uni005D.alt02u1D447
(4.9)
of size u1D444,u1D440 := u1D444(u1D43F+1) and u1D441u1D444(u1D43F+1) respectively. The coefficient vectors in (4.7), (4.8)
and (4.9) of the u1D45D-th subblock will be denoted by h(u1D459)u1D45F (u1D45D),hu1D45F (u1D45D) and h(u1D45D).
Based on CE-BEM given by (4.2), define
uD4D4 (u1D45B) :=
uni005B.alt02
u1D452?u1D457u1D7141u1D45B u1D452?u1D457u1D7142u1D45B ??? u1D452?u1D457u1D714u1D444u1D45B
uni005D.alt02u1D447
,
s(u1D45B) :=
uni005B.alt02
u1D460(u1D45B) u1D460(u1D45B?1) ??? u1D460(u1D45B?u1D43F)
uni005D.alt02u1D447
,
and then the received signal can be written as
y(u1D45B) = [s(u1D45B)?Iu1D441]u1D447
uni005B.alt02
Iu1D441(u1D43F+1) ?uD4D4 (u1D45B)
uni005D.alt02u1D43B
h(u1D45D) +v(u1D45B) (4.10)
when the u1D45D-th subblock is being received at time u1D45B, by (4.1), (4.2)-(4.6) and (4.7)-(4.9).
We will employ the time-multiplexed (TM) training scheme as Section 2.3, proposed
in [70]. Then the received signal at u1D45F-th output at time u1D45Bu1D45D +u1D459 with u1D459 = 0,1,??? ,u1D43F is given
55
by (assuming timing synchronization)
u1D466u1D45F (u1D45Bu1D45D +u1D459) = u1D6FE?u1D45F (u1D45Bu1D45D +u1D459;u1D459) +u1D463u1D45F (u1D45Bu1D45D +u1D459). (4.11)
for u1D45F = 1,2,??? ,u1D441. We also can simplify (4.10) at time u1D45Bu1D45D +u1D459 as (u1D459 = 0,1,??? ,u1D43F)
u1D466u1D45F (u1D45Bu1D45D +u1D459) = u1D6FEuD4D4u1D43B (u1D45Bu1D45D +u1D459)h(u1D459)u1D45F (u1D45D) +u1D463u1D45F (u1D45Bu1D45D +u1D459) (4.12)
Note that each (sub-)channel atu1D45F-th output can be tracked usingu1D45F-th received signal in (4.12)
respectively without interference. We intend to use only training sessions for subblock-wise
channel tracking regardless of data sessions. Further define
yu1D45F (u1D45D) :=
uni005B.alt02
u1D466(u1D45Bu1D45D) u1D466(u1D45Bu1D45D + 1) ??? u1D466(u1D45Bu1D45D +u1D43F)
uni005D.alt02u1D447
,
vu1D45F (u1D45D) :=
uni005B.alt02
u1D463(u1D45Bu1D45D) u1D463(u1D45Bu1D45D + 1) ??? u1D463(u1D45Bu1D45D +u1D43F)
uni005D.alt02u1D447
,
and
?(u1D45D) :=
?
??
??
??
?
uD4D4 (u1D45Bu1D45D)
uD4D4 (u1D45Bu1D45D + 1)
...
uD4D4 (u1D45Bu1D45D +u1D43F)
?
??
??
??
?
u1D43B
.
Then by (4.12),
yu1D45F (u1D45D) = u1D6FE?(u1D45D)hu1D45F (u1D45D) +vu1D45F (u1D45D). (4.13)
Since CE-BEM is periodic with period u1D447 in (4.2), the algorithm memory should be less
than u1D447 in order to avoid periodicity of the BEM coefficients. Note that the real channel is
not periodic. Therefore, an adaptive algorithm with ?finite-memory? is preferred which we
implement via either exponentially-weighted RLS or sliding-window RLS approaches.
4.2.1 Exponentially-Weighted RLS Tracking
Given an (u1D43F+1)?1 measurement vector yu1D45F (u1D45D), a training impulseu1D6FE and an (u1D43F+1)?u1D440
basis function matrix ?(u1D45D) in (4.13), we can apply exponentially-weighted regularized RLS
(EW-RLS) algorithm [2, Chapter 12] to track an unknown u1D440 ? 1 BEM coefficient vector
hu1D45F (u1D45D). Choose hu1D45F (u1D45D) to minimize the cost function
u1D706u1D45D+1u1D6FD?hu1D45F?2 +
u1D45Duni2211.alt02
u1D456=0
u1D706u1D45D?u1D456?yu1D45F(u1D456)?u1D6FE?(u1D456)hu1D45F?2 (4.14)
56
where u1D6FD > 0 is a regularization parameter and 0 < u1D706 < 1 is the forgetting factor. Note
that the forgetting factor is used to give more weight to recent data and less weight to past
data. For the subblock-wise updating, we take u1D706 to be (much) smaller than one (e.g., 0.65)
although it is very close to one (e.g., 0.99) for the symbol-wise updating.
EW-RLS tracking is initialized with ?hu1D45F (?1) = 0u1D440?1 andP?u1D45F (?1) = u1D6FD?1Iu1D440. Mimicking
[2, Algorithm 12.3.1], EW-RLS recursion (for u1D45D = 0,1,???) is applied one by one for each
u1D45F-th output via the following steps (we have omitted the subscript u1D45F for convenience):
?(u1D45D) = u1D706Iu1D43F+1 +u1D6FE2?(u1D45D)P(u1D45D?1)?u1D43B(u1D45D),
G(u1D45D) = u1D6FEP(u1D45D?1)?u1D43B(u1D45D)??1(u1D45D),
?h(u1D45D) = ?h(u1D45D?1) +G(u1D45D)uni005B.alt02yu1D460(u1D45D)?u1D6FE?(u1D45D)?h(u1D45D?1)uni005D.alt02,
P(u1D45D) = u1D706?1 [Iu1D440 ?u1D6FEG(u1D45D)?(u1D45D)]P(u1D45D?1)
where ?h(u1D45D) denotes the estimate of h(u1D45D) given the observations {y(0),yu1D460 (1),??? ,yu1D460 (u1D45D)}.
After RLS recursion for every u1D45D, we generate the channel for the u1D45D-th subblock by the
estimate ?hu1D45F (u1D45D) via the CE-BEM (4.2) as
??u1D45F (u1D45B;u1D459) = uD4D4u1D43B (u1D45B)?h(u1D459)u1D45F (u1D45D) (4.15)
for u1D45B = u1D45Du1D45Au1D44F,u1D45Du1D45Au1D44F + 1,??? ,(u1D45D+ 1)u1D45Au1D44F ?1. The definition of ?h(u1D459)u1D45F (u1D45D) is similar to (4.7).
4.2.2 Sliding-Window RLS Tracking
Compared with EW-RLS algorithm that weakens the effects of all past data using the
exponential weights, the sliding-window RLS (SW-RLS) algorithm removes the effect of
earlier data using the ?downdating? least-squares recursions. The downdating removes the
oldest received subblock sample from the window and the updating inserts the next received
subblock sample into the window. Combining the update and downdate procedures, SW-
RLS algorithm only utilizes the data in a sliding window of length u1D44A, a fixed length of
data [2, Chapter 12].
Since the BEM coefficients are invariant within a block of u1D447u1D435 symbols in (4.2), we set
u1D44A = ?u1D447u1D435/u1D45Au1D44F? subblocks so that only the current subblock and the past (u1D44A ?1) subblocks
within one block are used for adaptation. The cost function for SW-RLS is given by
u1D6FD?hu1D45F?2 +
u1D45Duni2211.alt02
u1D456=u1D45D?u1D44A+1
?yu1D45F(u1D45D)?u1D6FE?(u1D456)hu1D45F?2 (4.16)
where if u1D45D?u1D44A + 1 < 0, we set u1D456 = 0.
57
SW-RLS tracking is initialized using EW-RLS algorithm with u1D706 = 1 to get the first
sliding window foru1D45D = 0,1,??? ,u1D44A?1. Then set ?hu1D45F,u1D462(u1D44A?1) = ?hu1D45F(u1D44A?1) andP?u1D45F,u1D462(u1D44A?1) =
P?u1D45F(u1D44A ? 1). Mimicking [2, Problem 12.7], SW-RLS recursion (for u1D45D = u1D44A,u1D44A + 1,???) is
applied independently for each u1D45F-th output via the following steps (we have omitted the
subscript u1D45F since the following steps are common for every output):
1. Downdating:
?u1D451(u1D45D?1) = Iu1D43F+1 ?u1D6FE2?(u1D45D?u1D44A)Pu1D462(u1D45D?1)?u1D43B(u1D45D?u1D44A),
Gu1D451(u1D45D?1) = u1D6FEPu1D462(u1D45D?1)?u1D43B(u1D45D?u1D44A)??1u1D451 (u1D45D?1),
?hu1D451(u1D45D?1) = ?hu1D462(u1D45D?1)?Gu1D451(u1D45D?1)uni005B.alt02y(u1D45D?u1D44A)?u1D6FE?(u1D45D?u1D44A)?hu1D462(u1D45D?1)uni005D.alt02,
Pu1D451(u1D45D?1) = [Iu1D440 +u1D6FEGu1D451(u1D45D?1)?(u1D45D?u1D44A)]Pu1D462(u1D45D?1)
2. Updating:
?u1D462(u1D45D) = Iu1D43F+1 +u1D6FE2?(u1D45D)Pu1D451(u1D45D?1)?u1D43B(u1D45D),
Gu1D462(u1D45D) = u1D6FEPu1D451(u1D45D?1)?u1D43B(u1D45D)??1u1D462 (u1D45D),
?hu1D462(u1D45D) = ?hu1D451(u1D45D?1) +Gu1D462(u1D45D)uni005B.alt02y(u1D45D)?u1D6FE?(u1D45D)?hu1D451(u1D45D?1)uni005D.alt02,
Pu1D462(u1D45D) = [Iu1D440 ?u1D6FEGu1D462(u1D45D)?(u1D45D)]Pu1D451(u1D45D?1).
Now ?hu1D45F,u1D462 (u1D45D) is the estimate of hu1D45F (u1D45D) based on the observations {yu1D45F (u1D456)}u1D45Du1D456=u1D45D?u1D44A+1. The channel
estimates for every u1D45D-th subblock are also generated using (4.15) by setting ?hu1D45F(u1D45D) = ?hu1D45F,u1D462(u1D45D).
4.2.3 Simulation Examples
Example 1
A random time- and frequency-selective Rayleigh fading channel is considered. We
assume h(u1D45B;u1D459) are zero-mean, complex Gaussian, and spatially white. We take u1D43F = 2 (3
taps) in (4.1), and u1D70E2? = u1D438{?u1D45F(u1D45B;u1D459)??u1D45F(u1D45B;u1D459)} = 1/(u1D43F+ 1). For different u1D459?s, h(u1D45B;u1D459)?s are
mutually independent and satisfy Jakes? model. To this end, we simulate each single tap
following [75] (with a correction in the appendix of [63]).
We consider a communication system with carrier frequency of 2GHz, data rate of
40kBd (kilo-Bauds), therefore u1D447u1D460 = 25u1D707s, and a varying Doppler spread u1D453u1D451 in the range
of 0 to 400Hz, or the normalized Doppler spread u1D453u1D451u1D447u1D460 from 0 to 0.01 (corresponding to a
maximum mobile velocity from 0 to 216km/h). The additive noise is zero-mean complex
58
white Gaussian. The (receiver) SNR refers to the average energy per symbol over one-sided
noise spectral density.
The TM training scheme of [70], which is optimal for channels satisfying critically sam-
pled CE-BEM representation, is adopted, where each subblock of equal length u1D45Au1D44F symbols
consists of an information session of u1D45Au1D451 symbols and a succeeding training session of u1D45Au1D461
symbols (u1D45Au1D44F = u1D45Au1D451 +u1D45Au1D461). We assume that each information symbol has unit power, while
at every training session, we set u1D6FE = ?2u1D43F+ 1 so that the average power per symbol at
training sessions is equal to that of information sessions. In the simulations, we set u1D6FE = ?5
for u1D45Au1D44F = 20 (u1D45Au1D451 = 15 and u1D45Au1D461 = 5) or u1D45Au1D44F = 40 (u1D45Au1D451 = 35 and u1D45Au1D461 = 5).
For CE-BEM, we select the period u1D447 = 400 and hence u1D444 = 9 by (4.4). In Chap-
ter 3, we see that, for the fast-fading channels, the symbol-wise approaches based on AR
model (AR(1)-KF and Joint-KF in Section 3.2.2) have worse performances than block-wise
or subblock-wise ones based on CE-BEM (BA-LS and SB-KF in Section 3.2.2). Here we
compared the following four schemes:
1. The block-adaptive channel estimation in [70], where the transmitted symbols are
segmented into consecutive blocks of u1D447u1D435 symbols each. Every block consists of ?u1D443 (?
u1D444) subblocks as introduced in Section 2.3. We use an oversampled CE-BEM with
u1D447u1D435 = u1D447/2 for u1D45Au1D44F = 20 in order to suppress spectral leakage; whereas for u1D45Au1D44F = 40,
since an oversampled CE-BEM is not possible to satisfy ?u1D443 ? u1D444, we take u1D447u1D435 = u1D447.
Considering the oversampled CE-BEM basis functions are not orthogonal, we apply
a regularized least-square approach with a regularization parameter u1D6FD = 1; so it is
fair to compare with our proposed schemes. In the figures, this scheme is denoted by
?BA-LS?.
2. Our subblock-wise Kalman filtering in Chapter 3, which uses the first order AR model
as a priori model for BEM coefficients. We take the AR coefficient u1D6FC = 0.9944 and
0.9667 for u1D45Au1D44F = 20 and 40 respectively. In the figures, this scheme is denoted by
?SB-KF?.
3. Proposed subblock-wise EW-RLS algorithm with u1D6FD = 1. For u1D45Au1D44F = 20 and 40, we
take the forgetting factor u1D706 = 0.65 and 0.5 respectively (those values were determined
empirically : see the Section 4.4 for the choices). In the figures, this scheme is denoted
by ?SB-EWRLS?.
4. Proposed subblock-wise SW-RLS algorithm with u1D6FD = 1. We take u1D447u1D435 = u1D447/2 = 200, so
that for u1D45Au1D44F = 20 and 40, the window size u1D44A = 10 and 5 respectively. In the figures,
this scheme is denoted by ?SB-SWRLS?.
59
We evaluate the performances of those schemes by considering their normalized channel
mean square error (NCMSE) and their bit error rates (BER). The NCMSE is defined as
NCMSE :=
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0
vextenddoublevextenddouble
vextenddouble?h(u1D456) (u1D45B;u1D459)?h(u1D456) (u1D45B;u1D459)
vextenddoublevextenddouble
vextenddouble2
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0?h(u1D456) (u1D45B;u1D459)?
2
where h(u1D456) (u1D45B;u1D459) is the true channel and ?h(u1D456) (u1D45B;u1D459) is the estimated channel at the u1D456-th Monte
Carlo run, among total u1D440u1D45F runs, and u1D447u1D441 is the total observation length in symbols. For each
of the above four schemes, the MMSE-DFE described in Section 2.4.2 [40] is employed at
the receiver, using the obtained channel estimates, with u1D459u1D453 = 8, u1D459u1D44F = 2 and the delay u1D451 = 5
symbols. In each run, a symbol sequence of length, u1D447u1D441 = 4000 for each user is modulated
by QPSK and the first 200 symbols are discarded in evaluations. All the simulation results
are based on 500 runs.
0 5 10 15 20 25 30?35
?30
?25
?20
?15
?10
?5
0
SNR(dB)
NCMSE (dB)
N1K1,L=2,T=400,Q=9,fd=400Hz,Ts=25?s,500Runs
 
 
mb=20
mb=40
BA?LS
SB?KF
SB?EWRLS
SB?SWRLS
(a) NCMSE vs SNR
0 5 10 15 20 25 30
10?4
10?3
10?2
10?1
100
SNR(dB)
BER
DFE,N1K1,L=2,T=400,Q=9,fd=400Hz,Ts=25?s,500Runs
 
 
mb=20
mb=40
BA?LS
SB?KF
SB?EWRLS
SB?SWRLS
(b) BER vs SNR, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and delay
u1D451 = 5
Figure 4.1: Subblock-wise RLS channel estimation: performance comparison for SNR?s under
u1D441 = u1D43E = 1,u1D453u1D451u1D447u1D460 = 0.01.
In Fig. 4.1, the performances of the four schemes are compared for different SNR?s
under the normalized Doppler spread u1D453u1D451u1D447u1D460 = 0.01. Since we use only training session for
channel estimation, the smaller subblock with more training (u1D45Au1D44F = 20) results in better
performance than the larger subblock (u1D45Au1D44F = 40). Since the subblock-wise approaches update
every subblock, they have superior performance to the block-wise estimation scheme in
[70]. Assuming no a priori models of the BEM coefficients, the two proposed subblock-wise
RLS tracking schemes (SB-EWRLS and SB-SWRLS) outperform the subblock-wise Kalman
60
0 50 100 150 200 250 300 350 400?30
?28
?26
?24
?22
?20
?18
?16
?14
?12
?10
N1K1,L=2,mb=20,T=400,Q=9,Ts=25?s,SNR=20dB,500Runs
fd(Hz)
NCMSE(dB)
 
 
BA?LS
SB?KF
SB?EWRLS
SB?SWRLS
(a) NCMSE vs u1D453u1D451u1D447u1D460
0 50 100 150 200 250 300 350 400
10?4
10?3
10?2
10?1
DFE,N1K1,L=2,mb=20,T=400,Q=9,Ts=25?s,SNR=20dB,500Runs
fd(Hz)
Bit Error Rate
 
 
BA?LS
SB?KF
SB?EWRLS
SB?SWRLS
(b) BER vs u1D453u1D451u1D447u1D460, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and delay
u1D451 = 5
Figure 4.2: Subblock-wise RLS channel estimation: performance comparison for Doppler
spread under u1D441 = u1D43E = 1,SNR = 20dB.
tracking scheme (SB-KF) we proposed earlier in Chapter 3 and they have similar NCMSE
and BER performances. Noting that when u1D45Au1D44F = 20, the subblock-wise Kalman tracking
scheme is better than the block-wise LS scheme when BER is the performance criterion and
is worse when average channel MSE is the performance criterion, superior average channel
MSE does not necessarily translate into superior average BER since a monotone relationship
between average MSE (averaged over all taps and runs) and average BER does not necessarily
exist. [To further investigate this ?counter-intuitive? behavior, we plotted histograms (not
shown here) of the channel MSE and the BER for the two schemes (BA-LS and SB-KF)
with u1D45Au1D44F = 20 for a specific case corresponding to SNR = 28dB in Fig. 4.1a and 4.1b. The
histograms show that the channel MSE of the BA-LS scheme has a spread around the mean
value that is more than twice that for the SB-KF scheme. This, in turn, translates into a
larger BER for the BA-LS scheme in quite a few runs, compared to the SB-KF scheme.]
In Fig. 4.2, we compare the same schemes over different Doppler spread u1D453u1D451?s under
SNR = 20dB, the subblock size u1D45Au1D44F = 20. Since all these schemes are based on CE-BEM,
their performances are not sensitive to the actual Doppler spread. However, the proposed
SB-EWRLS and SB-SWRLS have more stable and reliable performance than BA-LS and
SB-KF over all the range of Doppler spread. Note that an arbitrary model for channel is
not necessarily suited to a real channel. The SB-KF assume the BEM coefficients follow the
first-order AR process, which may introduce additional errors in channel estimation.
61
Example 2
The random Rayleigh fading channel in this example is as in Example 1, except that
we change the channel length u1D43F from 2 to 8 leading to 9 taps, and the power delay profile
is exponential satisfying u1D438{??u1D45F(u1D45B : u1D459)?2} = u1D70E2?(u1D459) = u1D44Eu1D452?u1D459/u1D43F where the constant u1D44E is pick to
satisfy uni2211.alt01u1D43Fu1D459=0u1D70E2?(u1D459) = 1. We consider a communication system with carrier frequency of
2GHz, data rate of 0.1MBd (mega-Bauds), therefore u1D447u1D460 = 10u1D707s, and a varying Doppler
spread u1D453u1D451 = 250Hz, or the normalized Doppler spread u1D453u1D451u1D447u1D460 = 0.0025 (corresponding to
a maximum mobile velocity of 135km/h). The training session is described by (2.22) of
length u1D45Au1D461 = 2u1D43F+ 1 = 17 symbols with u1D6FE = ?2u1D43F+ 1 so that the average symbol power of
training sessions is equal to that of information sessions. We select the period of the CE-
BEM u1D447 = 800 symbols, and hence u1D444 = 5 by (4.4) and a block size u1D447u1D435 = 400 for oversampled
CE-BEM.
Five estimation and tracking schemes are compared: block-adaptive channel estimation
in [70] using the regularized LS approach (?BA-LS?), the subblock-wise Kalman tracking
(?SB-KF?) introduced in Chapter 3, the proposed subblock-wise EW-RLS tracking (?SB-
EWRLS?) and SW-RLS tracking (?SB-SWRLS?). With the subblock size u1D45Au1D44F = 80, we take
the AR coefficient u1D6FC = 0.993 for SB-KF, the forgetting factor u1D706 = 0.5 for SB-EWRLS and
the window size u1D44A = u1D447u1D435/u1D45Au1D44F = 5 for SB-SWRLS. [See Section 3.4 and 4.4 for the choices of
the AR coefficient u1D6FC and the forgetting factor u1D706 respectively.] We also consider the perfect
channel estimates(?TrueCH?). The BER?s are evaluated by employing the MMSE-DFE with
u1D459u1D453 = 14, u1D459u1D44F = 8 and the delay u1D451 = 10, using the channel estimates obtained by each scheme.
In Figs. 4.3, the performances of the five schemes are compared for different SNR?s. Due
to the assumption that the channel is spatially uncorrelated for different outputs, the NCMSE
curves for u1D441 = 1 and 2 coincide with each other. The three subblock-wise schemes (SB-
KF, SB-EWRLS and SB-SWRLS) outperform the block-wise LS estimation (BA-LS) in [70].
Definitely, the proposed two finite-memory RLS schemes, SB-EWRLS and SB-SWRLS, have
similar NCMSE and BER performance, and they are slightly better than the subblock-wise
Kalman tracking scheme since the latter scheme assumes the BEM coefficients follow a first-
order AR model and this may introduce additional modeling errors. Note that SB-EWRLS
and SB-SWRLS schemes track the channel subblock-by-subblock using the regularized least-
square solution with no a priori models for the BEM coefficients.
62
0 5 10 15 20 25 30?30
?25
?20
?15
?10
?5
0
SNR(dB)
NCMSE (dB)
K=1,L=8,mb=80,T=800,Q=5,fd=250Hz,Ts=10?s,500Runs
 
 
N=1
N=2
BA?LS
SB?KF
SB?EWRLS
SB?SWRLS
(a) NCMSE vs SNR
0 5 10 15 20 25 3010
?6
10?5
10?4
10?3
10?2
10?1
100
DFE,K=1,L=8,mb=80,T=800,Q=5,fd=250Hz,Ts=10?s,500Runs
SNR(dB)
Bit Error Rate
 
 
N=1
N=2
BA?LS
SB?KF
SB?EWRLS
SB?SWRLS
TrueCH
(b) BER vs SNR, with DFE of u1D459u1D453 = 14,u1D459u1D44F = 8 and delay
u1D451 = 10
Figure 4.3: Subblock-wise RLS channel estimation: performance comparison for SNR?s under
u1D43E = 1,u1D453u1D451u1D447u1D460 = 0.0025, u1D45Au1D44F = 80.
4.3 Subblock-wise RLS MIMO Channel Estimation using CE-BEM
4.3.1 RLS Tracking for MIMO Channel
Consider a doubly-selective multi-input multi-output (MIMO), finite impulse response
(FIR) linear channel with u1D43E inputs and u1D441 outputs. Let {u1D460u1D458 (u1D45B)} denote u1D458-th user?s in-
formation sequence that is input to the time-varying channel with discrete-time response
{hu1D458 (u1D45B;u1D459)} (channel response for the u1D458-th user at time instance u1D45B to a unit input at time
instance u1D45B?u1D459), assuming {u1D460u1D458 (u1D45B)} is mutually independent and identically distributed (i.i.d.)
with zero mean and variance u1D438{u1D460u1D458 (u1D45B)u1D460?u1D458 (u1D45B)} = u1D70E2u1D460u1D458 = u1D70E2u1D460 for u1D458 = 1,2,??? ,u1D43E. Then the
symbol-rate noisy u1D441-column channel output vector is given by (u1D45B = 0,1,...)
y(u1D45B) =
u1D43Euni2211.alt02
u1D458=1
u1D43Funi2211.alt02
u1D459=0
hu1D458 (u1D45B;u1D459)u1D460u1D458 (u1D45B?u1D459) +v(u1D45B) (4.17)
where the u1D441-column vector v(u1D45B) is zero-mean, white, uncorrelated with u1D460u1D458 (u1D45B), complex
Gaussian noise, with the autocorrelation u1D438{v(u1D45B+u1D70F)vu1D43B (u1D45B)} = u1D70E2u1D463Iu1D441u1D6FF(u1D70F). We assume that
{hu1D458 (u1D45B;u1D459)} represents a wide-sense stationary uncorrelated scattering (WSSUS) channel [60],
63
independent for different u1D458?s. Define
s(u1D45B) :=
uni005B.alt02
u1D4601(u1D45B) u1D4602(u1D45B) ??? u1D460u1D43E(u1D45B)
uni005D.alt02u1D447
h(u1D45B;u1D459) :=
uni005B.alt02
h1(u1D45B;u1D459) h2(u1D45B;u1D459) ??? hu1D43E(u1D45B;u1D459)
uni005D.alt02
.
and we may rewrite (4.17) as
y(u1D45B) =
u1D43Funi2211.alt02
u1D459=0
h(u1D45B;u1D459)s(u1D45B?u1D459) +v(u1D45B). (4.18)
In CE-BEM [11,14,70], over the u1D456-th block consisting of an observation window of
u1D447u1D435 symbols, the channel is represented as (u1D45B = (u1D456? 1)u1D447u1D435,(u1D456? 1)u1D447u1D435 + 1,??? ,u1D456u1D447u1D435 ? 1 and
u1D459 = 0,1,??? ,u1D43F )
hu1D458(u1D45B;u1D459) =
u1D444uni2211.alt02
u1D45E=1
hu1D458,u1D45E(u1D459)u1D452u1D457u1D714u1D45Eu1D45B, (4.19)
where hu1D458,u1D45E (u1D459) is the u1D441-column time-invariant BEM coefficient vector for u1D458-th user and one
chooses (? is an integer)
u1D447 := ?u1D447u1D435, ? ? 1, (4.20)
u1D444? 2?u1D453u1D451u1D447u1D447u1D460?+ 1, (4.21)
u1D714u1D45E := 2u1D70Bu1D447 [u1D45E?(u1D444+ 1)/2], u1D45E = 1,2,??? ,u1D444 (4.22)
u1D43F := ?u1D70Fu1D451/u1D447u1D460?, (4.23)
u1D70Fu1D451 and u1D453u1D451 are respectively the delay spread and the Doppler spread, and u1D447u1D460 is the symbol
duration.
Now extend the proposed RLS subblock-wise approach in Section 4.2 to MIMO systems.
Let ?u1D45Fu1D458,u1D45E(u1D459) denote the u1D45F?th component of the column hu1D458,u1D45E(u1D459). Stack the BEM coefficients in
(4.19) into vectors as
h(u1D459)u1D45Fu1D458 :=
uni005B.alt02
?u1D45Fu1D458,1(u1D459) ?u1D45Fu1D458,2(u1D459) ??? ?u1D45Fu1D458,u1D444(u1D459)
uni005D.alt02u1D447
(4.24)
hu1D45Fu1D458 :=
uni005B.alt02
h(0)u1D447u1D45Fu1D458 h(1)u1D447u1D45Fu1D458 ??? h(u1D43F)u1D447u1D45Fu1D458
uni005D.alt02u1D447
(4.25)
h :=
uni005B.alt02
hu1D44711 ??? hu1D447u1D4411 ??? hu1D4471u1D43E ??? hu1D447u1D441u1D43E
uni005D.alt02u1D447
(4.26)
of size u1D444, u1D440 := u1D444(u1D43F+ 1) and u1D441u1D43Eu1D444(u1D43F+ 1), respectively. The coefficient vectors in (4.24),
(4.25) and (4.26) of the u1D45D-th subblock will be denoted by h(u1D459)u1D45Fu1D458 (u1D45D), hu1D45Fu1D458 (u1D45D), and h(u1D45D).
64
Based on CE-BEM given by (4.19), define (where s(u1D45B) is u1D43E-column vector)
uD4D4 (u1D45B) :=
uni005B.alt02
u1D452?u1D457u1D7141u1D45B u1D452?u1D457u1D7142u1D45B ??? u1D452?u1D457u1D714u1D444u1D45B
uni005D.alt02u1D447
,
S(u1D45B) :=
uni005B.alt02
su1D447 (u1D45B) su1D447 (u1D45B?1) ??? su1D447 (u1D45B?u1D43F)
uni005D.alt02u1D447
,
and then the received signal can be written as
y(u1D45B) = [S(u1D45B)?Iu1D441]u1D447
uni005B.alt02
Iu1D441u1D43E(u1D43F+1) ?uD4D4 (u1D45B)
uni005D.alt02u1D43B
h(u1D45D) +v(u1D45B) (4.27)
when theu1D45D-th subblock is being received at timeu1D45B, by (4.18), (4.19)-(4.23) and (4.24)-(4.26).
We will employ the time-multiplexed training scheme in Section 2.3 [30]. Then by
design, a given user?s training does not affect that of any other user during training session.
Hence the received signal between u1D458-th input and u1D45F-th output can be written as (assuming
timing synchronization)
u1D466u1D45F (u1D45Bu1D458,u1D45D +u1D459) = u1D6FE?u1D45Fu1D458 (u1D45Bu1D458,u1D45D +u1D459;u1D459) +u1D463u1D45F (u1D45Bu1D458,u1D45D +u1D459) (4.28)
for u1D459 = 0,1,??? ,u1D43F, u1D458 = 1,2,??? ,u1D43E and u1D45F = 1,2,??? ,u1D441. The (unknown) MIMO channel
between u1D43E inputs and u1D441 outputs can be tracked independently one by one. Using (4.28),
in order to track every (sub-)channel between u1D43E inputs and u1D441 outputs respectively, we can
simplify (4.27) at time u1D45Bu1D458,u1D45D +u1D459 as (u1D45D = 0,1,??? and u1D459 = 0,1,??? ,u1D43F)
u1D466u1D45F (u1D45Bu1D458,u1D45D +u1D459) = u1D6FEuD4D4u1D43B (u1D45Bu1D458,u1D45D +u1D459)h(u1D459)u1D45Fu1D458 (u1D45D) +u1D463u1D45F (u1D45Bu1D458,u1D45D +u1D459) (4.29)
We intend to use only training sessions for subblock-wise channel tracking regardless of data
sessions. Further define
?yu1D45F (u1D45D) :=
uni005B.alt02
u1D466u1D45F (u1D45Bu1D458,u1D45D) u1D466u1D45F (u1D45Bu1D458,u1D45D + 1) ??? u1D466u1D45F (u1D45Bu1D458,u1D45D +u1D43F)
uni005D.alt02u1D447
,
vu1D45F (u1D45D) :=
uni005B.alt02
u1D463u1D45F (u1D45Bu1D458,u1D45D) u1D463u1D45F (u1D45Bu1D458,u1D45D + 1) ??? u1D463u1D45F (u1D45Bu1D458,u1D45D +u1D43F)
uni005D.alt02u1D447
,
and
?(u1D45D) :=
?
??
??
??
?
uD4D4 (u1D45Bu1D458,u1D45D)
uD4D4 (u1D45Bu1D458,u1D45D + 1)
...
uD4D4 (u1D45Bu1D458,u1D45D +u1D43F)
?
??
??
??
?
u1D43B
.
Then by (4.29),
?yu1D45F (u1D45D) = u1D6FE?(u1D45D)hu1D45Fu1D458 (u1D45D) +vu1D45F (u1D45D). (4.30)
65
Given an (u1D43F+1)?1 measurement vector ?yu1D45F (u1D45D), a training impulseu1D6FE and an (u1D43F+1)?u1D440
basis function matrix ?(u1D45D) in (4.30), we can apply exponentially-weighted regularized RLS
(EW-RLS) algorithm [2, Chapter 12] to track an unknown u1D440 ? 1 BEM coefficient vector
hu1D45Fu1D458 (u1D45D) between u1D45F-th receiver and u1D458-th user. Choose hu1D45Fu1D458 (u1D45D) to minimize the cost function
u1D706u1D45D+1u1D6FD?hu1D45Fu1D458?2 +
u1D45Duni2211.alt02
u1D456=0
u1D706u1D45D?u1D456??yu1D45F(u1D456)?u1D6FE?(u1D456)hu1D45Fu1D458?2 (4.31)
where u1D6FD > 0 is a regularization parameter and 0 <u1D706< 1 is the forgetting factor.
EW-RLS tracking is initialized with ?hu1D45Fu1D458 (?1) = 0u1D440?1 and P?u1D45Fu1D458 (?1) = u1D6FD?1Iu1D440. Mim-
icking [2, Algorithm 12.3.1], EW-RLS recursion (for u1D45D = 0,1,???) is applied one-by-one for
each pair between u1D45F-th receiver and u1D458-th user independently via the same steps in Section
4.2.1. After RLS recursion for every u1D45D, we generate the channel for the u1D45D-th subblock by the
estimate ?h(u1D45D) via the CE-BEM (4.19) as
??u1D45Fu1D458 (u1D45B;u1D459) = uD4D4u1D43B (u1D45B)?h(u1D459)u1D45Fu1D458 (u1D45D) (4.32)
for u1D45B = u1D45Du1D45Au1D44F,u1D45Du1D45Au1D44F +1,??? ,(u1D45D+ 1)u1D45Au1D44F ?1. The definition of ?h(u1D459)u1D45Fu1D458 (u1D45D) is similar to (4.24). Thus
we obtain the entire u1D441 ?u1D43E channel matrix for the current subblock.
Similarly, the cost function for SW-RLS is given by
u1D6FD?hu1D45Fu1D458?2 +
u1D45Duni2211.alt02
u1D456=u1D45D?u1D44A+1
??yu1D45F(u1D45D)?u1D6FE?(u1D456)hu1D45Fu1D458?2 (4.33)
where if u1D45D?u1D44A + 1 < 0, we set u1D456 = 0.
SW-RLS tracking is initialized using EW-RLS algorithm with u1D706 = 1 to get the first
sliding window for u1D45D = 0,1,??? ,u1D44A?1. Then set ?hu1D45Fu1D458,u1D462(u1D44A?1) = ?hu1D45Fu1D458(u1D44A?1) and P?u1D45Fu1D458,u1D462(u1D44A?
1) = P?u1D45Fu1D458(u1D44A ?1). Mimicking [2, Problem 12.7], SW-RLS recursion (for u1D45D = u1D44A,u1D44A +1,???)
has the same algorithm in Section 4.2.2 since it is applied for each pair u1D45Fu1D458 independently.
Now ?hu1D462 (u1D45D) is the estimate of h(u1D45D) based on the observations {y(u1D456)}u1D45Du1D456=u1D45D?u1D44A+1. The channel
estimates for every u1D45D-th subblock are also generated using (4.32) by setting ?h(u1D45D) = ?hu1D462(u1D45D).
Computational Complexity
We compare computational complexity of each scheme using flops counting for the
channel estimation algorithm of each simulation program. We use the same flops counting
for multiplication and inverse as in Section 3.3.1. The detailed flops for one iteration of EW-
RLS tracking algorithm is shown in Table 4.1 and the overall flop counts for each scheme
are compared in Table 4.2 (u1D440 = u1D444(u1D43F + 1), ?u1D443 = u1D447u1D435/u1D45Au1D44F,u1D43Fu1D45D = u1D43F + 1). We also calculate
66
the relative complexity for specific examples in Table 4.3. Note that subblock-wise EW-RLS
scheme appears to provide a good complexity-vs-performance trade-off at higher SNR?s at
the cost of an increase of flop counts whereas at lower SNR values such an increase in flop
count may not warrant the use of subblock-wise tracking schemes.
Table 4.1: Subblock-wise EW-RLS channel estimation: flop count over one subblock of u1D45Au1D44F
symbols.
Operation Number of flops
?(u1D45D) = u1D706Iu1D43F+1 +u1D6FE2?(u1D45D)P(u1D45D?1)?u1D43B(u1D45D) u1D4402u1D43Fu1D45D +u1D440(u1D43F2u1D45D +u1D43Fu1D45D) +u1D43F2u1D45D
G(u1D45D) = u1D6FEP(u1D45D?1)?u1D43B(u1D45D)??1(u1D45D) u1D4402(u1D43Fu1D45D + 1) +u1D440u1D43F2u1D45D +u1D43F3u1D45D/3
?h(u1D45D) = ?h(u1D45D?1) +G(u1D45D)uni005B.alt02y(u1D45D)?u1D6FE?(u1D45D)?h(u1D45D?1)uni005D.alt02 u1D440(3u1D43Fu1D45D)
P(u1D45D) = u1D706?1 [Iu1D440 ?u1D6FEG(u1D45D)?(u1D45D)]P(u1D45D?1) u1D4403 +u1D4402(u1D43Fu1D45D + 1) +u1D440u1D43Fu1D45D
Total : u1D4403 +u1D4402(3u1D43Fu1D45D + 2) +u1D440(2u1D43F2u1D45D + 5u1D43Fu1D45D) +u1D43F3u1D45D/3 +u1D43F2u1D45D
Table 4.2: Block-wise least-square and subblock-wise RLS: flop count for channel estimation
over one block of u1D447u1D435 symbols.
Scheme Number of flops
BA-LS u1D4403/3 +u1D4402(2 ?u1D443u1D43Fu1D45D + 1) +u1D440( ?u1D443u1D43Fu1D45D + 1)
EW-RLS ?u1D443
uni007B.alt02
u1D4403 +u1D4402(3u1D43Fu1D45D + 2) +u1D440(2u1D43F2u1D45D + 5u1D43Fu1D45D) +u1D43F3u1D45D/3 +u1D43F2u1D45D
uni007D.alt02
SW-RLS 2 ?u1D443
uni007B.alt02
u1D4403 +u1D4402(3u1D43Fu1D45D + 1) +u1D440(2u1D43F2u1D45D + 5u1D43Fu1D45D) +u1D43F3u1D45D/3
uni007D.alt02
4.3.2 Simulation Examples
Example 1
A random time- and frequency-selective Rayleigh fading MIMO channel is considered.
We assume h(u1D45B;u1D459) are zero-mean, complex Gaussian, and spatially white. We take u1D43F = 2
(3 taps) in (4.18), and u1D70E2? = u1D438{?u1D45Fu1D458(u1D45B;u1D459)??u1D45Fu1D458(u1D45B;u1D459)} = 1/(u1D43F+ 1). For different u1D459?s, h(u1D45B;u1D459)?s
are mutually independent and satisfy Jakes? model. To this end, we simulate each single tap
following [75] (with a correction in the appendix of [63]).
We consider a communication system with carrier frequency of 2GHz, data rate of
40kBd (kilo-Bauds), therefore u1D447u1D460 = 25u1D707s, and a varying Doppler spread u1D453u1D451 in the range
of 0 to 400Hz, or the normalized Doppler spread u1D453u1D451u1D447u1D460 from 0 to 0.01 (corresponding to a
maximum mobile velocity from 0 to 216km/h). The additive noise is zero-mean complex
67
Table 4.3: Block-wise least-square and subblock-wise RLS: comparative flop count for chan-
nel estimation over one block of u1D447u1D435 symbols.
Example 1: u1D447 = 400,u1D444 = 9,u1D447u1D435 = 200,u1D45Au1D44F = 20, ?u1D443 = 10,u1D43F = 2
Number of flops BA-LS EW-RLS SW-RLS
actual 51,867 286,110 557,460
relative 1.00 5.52 10.75
Example 2: u1D447 = 800,u1D444 = 5,u1D447u1D435 = 400,u1D45Au1D44F = 80, ?u1D443 = 5,u1D43F = 8
Number of flops BA-LS EW-RLS SW-RLS
actual 216,720 797,445 1,573,830
relative 1.00 3.68 7.26
white Gaussian. The (receiver) SNR refers to the average energy per symbol over one-sided
noise spectral density.
The TM training scheme of [30], which is optimal for channels satisfying critically sam-
pled CE-BEM representation, is adopted, where each subblock of equal length u1D45Au1D44F symbols
consists of an information session of u1D45Au1D451 symbols and a succeeding training session of u1D45Au1D461
symbols (u1D45Au1D44F = u1D45Au1D451 +u1D45Au1D461). We assume that each information symbol has unit power, while
at every training session, we set u1D6FE =
uni221A.alt02
u1D43E(u1D43F+ 1) +u1D43F so that the average power per symbol
at training sessions is equal to that of information sessions. In the simulations, we consider
a simple two-receiver and two-user scenario, i.e., u1D441 = 2,u1D43E = 2 with the same transmitted
power. We set u1D45Au1D44F = 20 or 40 with u1D45Au1D461 = 8 and u1D6FE = ?8 for every user following the TM
training scheme. For CE-BEM, we select the period u1D447 = 400 and hence u1D444 = 9 by (4.21).
We compared the following four schemes:
1. The block-adaptive channel estimation in [30], where the transmitted symbols are
segmented into consecutive blocks ofu1D447u1D435 symbols each. Every block consists of ?u1D443 (?u1D444)
subblocks as in Section 2.3. We use an oversampled CE-BEM with u1D447u1D435 = u1D447/2 for u1D45Au1D44F =
20 since the over-sampled CE-BEM approximates WSSUS channels much better than
critically sampled CE-BEM [14]; whereas for u1D45Au1D44F = 40, since an oversampled CE-BEM
is not possible to satisfy ?u1D443 ? u1D444, we take u1D447u1D435 = u1D447. Considering the oversampled CE-
BEM basis functions are not orthogonal, we apply a regularized least-square approach
with a regularization parameter u1D6FD = 1; so it is fair to compare with our proposed
schemes. In the figures, this scheme is denoted by ?BA-LS?.
68
2. Our subblock-wise Kalman filtering in Chapter 3, which uses the first order AR model
as a priori model for BEM coefficients. We take the AR coefficient u1D6FC = 0.9944 and
0.9667 for u1D45Au1D44F = 20 and 40 respectively. In the figures, this scheme is denoted by
?SB-KF?.
3. Proposed subblock-wise EW-RLS algorithm with u1D6FD = 1. For u1D45Au1D44F = 20 and 40, we
take the forgetting factor u1D706 = 0.65 and 0.5 respectively (those values were determined
empirically : see the Section 4.4 for the choices). In the figures, this scheme is denoted
by ?SB-EWRLS?.
4. Proposed subblock-wise SW-RLS algorithm with u1D6FD = 1. We take u1D447u1D435 = u1D447/2 = 200, so
that for u1D45Au1D44F = 20 and 40, the window size u1D44A = 10 and 5 respectively. In the figures,
this scheme is denoted by ?SB-SWRLS?.
We evaluate the performances of those schemes by considering their normalized channel
mean square error (NCMSE) and their bit error rates (BER). The NCMSE is defined as
NCMSE :=
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0
vextenddoublevextenddouble
vextenddouble?h(u1D456) (u1D45B;u1D459)?h(u1D456) (u1D45B;u1D459)
vextenddoublevextenddouble
vextenddouble2
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0?h(u1D456) (u1D45B;u1D459)?
2
where h(u1D456) (u1D45B;u1D459) is the true channel and ?h(u1D456) (u1D45B;u1D459) is the estimated channel at the u1D456-th Monte
Carlo run, among total u1D440u1D45F runs, and u1D447u1D441 is the total observation length in symbols. For each
of the above four schemes, the MMSE-DFE described in Section 2.4.2 [40] is employed at
the receiver, using the obtained channel estimates, with u1D459u1D453 = 8, u1D459u1D44F = 2, and the delay u1D451 = 5
symbols. In each run, a symbol sequence of length, u1D447u1D441 = 4000 for each user is modulated
by QPSK and the first 200 symbols are discarded in evaluations. All the simulation results
are based on 500 runs.
In Fig. 4.4, the performances of the four schemes are compared for different SNR?s
with the normalized Doppler spread u1D453u1D451u1D447u1D460 = 0.01 fixed. Since we use only training session
for channel estimation, the smaller subblock with more training (u1D45Au1D44F = 20) results in better
performance than the larger subblock (u1D45Au1D44F = 40). All the subblock-wise schemes including
Kalman tracking outperform the block-wise one in [30] and the proposed SB-EWRLS and SB-
SWRLS approach have superior performances to the subblock-wise Kalman tracking that has
an arbitrary model for BEM coefficients (e.g., AR model) and this may incur modeling errors.
The proposed subblock-wise EW-RLS and SW-RLS schemes track the channel subblock-by-
subblock using the regularized least-square solution, assuming no a priori models for the
BEM coefficients.
69
0 5 10 15 20 25 30?35
?30
?25
?20
?15
?10
?5
0
SNR(dB)
NCMSE (dB)
N2K2,L=2,T=400,Q=9,fd=400Hz,Ts=25?s,500Runs
 
 
mb=20
mb=40
BA?LS
SB?KF
SB?EWRLS
SB?SWRLS
(a) NCMSE vs SNR
0 5 10 15 20 25 30
10?4
10?3
10?2
10?1
100
SNR(dB)
BER
DFE,N2K2,L=2,T=400,Q=9,fd=400Hz,Ts=25?s,500Runs
 
 
mb=20
mb=40
BA?LS
SB?KF
SB?EWRLS
SB?SWRLS
(b) BER vs SNR, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and delay
u1D451 = 5
Figure 4.4: Subblock-wise RLS MIMO channel estimation: performance comparison for
SNR?s under u1D441 = u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.01.
In Fig. 4.5, we compare the same schemes with the subblock size u1D45Au1D44F = 20 over the wide
range of Doppler spread u1D453u1D451?s under SNR = 20dB. It is cleat that the subblock-wise RLS
approaches have more stable and better performance than BA-LS and SB-KF over all the
range of Doppler spread.
Example 2
The random Rayleigh fading MIMO channel in this example is as in Example 1, except
that we change the channel length u1D43F from 2 to 8 leading to 9 taps, and the power delay
profile is exponential satisfying u1D438{??u1D45Fu1D458(u1D45B : u1D459)?2} = u1D70E2?(u1D459) = u1D44Eu1D452?u1D459/u1D43F where the constant u1D44E is
pick to satisfy uni2211.alt01u1D43Fu1D459=0u1D70E2?(u1D459) = 1. We consider a communication system with carrier frequency
of 2GHz, data rate of 0.1MBd (mega-Bauds), therefore u1D447u1D460 = 10u1D707s, and a varying Doppler
spread u1D453u1D451 = 250Hz, or the normalized Doppler spread u1D453u1D451u1D447u1D460 = 0.0025 (corresponding to a
maximum mobile velocity of 135km/h). The training session is described by (2.24) of length
u1D45Au1D461 = u1D43E(u1D43F+ 1) + u1D43F = 26 symbols with u1D6FE =
uni221A.alt02
u1D43E(u1D43F+ 1) +u1D43F so that the average symbol
power of training sessions is equal to that of information sessions. We select the period of
the CE-BEM u1D447 = 800 symbols, and hence u1D444 = 5 by (4.21) and a block size u1D447u1D435 = 400 for
oversampled CE-BEM.
Five estimation and tracking schemes are compared: block-adaptive channel estimation
in [30] using the regularized LS approach (?BA-LS?), the subblock-wise Kalman tracking
70
0 50 100 150 200 250 300 350 400?30
?28
?26
?24
?22
?20
?18
?16
?14
?12
?10
N2K2,L=2,mb=20,T=400,Q=9,Ts=25?s,SNR=20dB,500Runs
fd(Hz)
NCMSE(dB)
 
 
BA?LS
SB?KF
SB?EWRLS
SB?SWRLS
(a) NCMSE vs u1D453u1D451u1D447u1D460
0 50 100 150 200 250 300 350 400
10?4
10?3
10?2
10?1
DFE,N2K2,L=2,mb=20,T=400,Q=9,Ts=25?s,SNR=20dB,500Runs
fd(Hz)
Bit Error Rate
 
 
BA?LS
SB?KF
SB?EWRLS
SB?SWRLS
(b) BER vs u1D453u1D451u1D447u1D460, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and delay
u1D451 = 5
Figure 4.5: Subblock-wise RLS MIMO channel estimation: performance comparison for
Doppler spread under u1D441 = u1D43E = 2,SNR = 20dB.
(?SB-KF?) introduced in Chapter 3, the proposed subblock-wise EW-RLS tracking (?SB-
EWRLS?) and SW-RLS tracking (?SB-SWRLS?). With the subblock size u1D45Au1D44F = 80, we take
the AR coefficient u1D6FC = 0.993 for SB-KF, the forgetting factor u1D706 = 0.5 for SB-EWRLS and
the window size u1D44A = u1D447u1D435/u1D45Au1D44F = 5 for SB-SWRLS. [See Section 3.4 and 4.4 for the choices of
the AR coefficient u1D6FC and the forgetting factor u1D706 respectively.] We also consider the perfect
channel estimates (?TrueCH?). The BER?s are evaluated by employing the MMSE-DFE with
u1D459u1D453 = 14, u1D459u1D44F = 8 and the delay u1D451 = 10, using the channel estimates obtained by each scheme.
In Figs. 4.6, the performances of the five schemes are compared for different SNR?s.
The NCMSE curves for u1D441 = 2 and 3 coincide with each other since we assume the chan-
nel is spatially uncorrelated for different outputs. The subblock-wise Kalman filtering and
RLS tracking schemes outperform the block-wise LS estimation in [30]. In particular, the
proposed two finite-memory RLS schemes, SB-EWRLS and SB-SWRLS, have better perfor-
mances in NCMSE and BER than the subblock-wise Kalman tracking scheme. Note that
SB-KF assumes the BEM coefficients follow a first-order AR process, which is not neces-
sarily true in the real channel environment. This AR modeling for the BEM coefficients
may introduce the modeling errors in channel estimation. The subblock-wise EW-RLS and
SW-RLS schemes update the channel estimates subblock-by-subblock using the regularized
least-square solution without any a priori models, which makes the channel tracking more
practical and effective.
71
0 5 10 15 20 25 30?35
?30
?25
?20
?15
?10
?5
0
SNR(dB)
NCMSE (dB)
K=2,L=8,mb=80,T=800,Q=5,fd=250Hz,Ts=10?s,500Runs
 
 
N=2
N=3
BA?LS
SB?KF
SB?EWRLS
SB?SWRLS
(a) NCMSE vs SNR
0 5 10 15 20 25 3010
?6
10?5
10?4
10?3
10?2
10?1
100
DFE,K=2,L=8,mb=80,T=800,Q=5,fd=250Hz,Ts=10?s,500Runs
SNR(dB)
Bit Error Rate
 
 
N=2
N=3
BA?LS
SB?KF
SB?EWRLS
SB?SWRLS
TrueCH
(b) BER vs SNR, with DFE of u1D459u1D453 = 14,u1D459u1D44F = 8 and delay
u1D451 = 10
Figure 4.6: Subblock-wise RLS MIMO channel estimation: performance comparison for
SNR?s under u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.0025, u1D45Au1D44F = 80.
4.4 Forgetting Factor u1D706 for Subblock-wise EW-RLS Tracking
Here we analyze the theoretical choices of u1D706 for subblock-wise channel estimation using
EW-RLS algorithm in this chapter. Consider only one pair of channel hu1D45Fu1D458 betweenu1D45F-th input
and u1D458-th output, since we track the channel of every pair independently for u1D441 receivers and
u1D43E users by design of time-multiplexed (TM) training in Section 2.3 [30]. The cost function
for EW-RLS algorithm is given as (4.14), which we can rewrite for ?large? (u1D45D? ?) as
u1D49E =
?uni2211.alt02
u1D456=0
u1D706u1D456??yu1D45F(u1D45D?u1D456)?u1D6FE?(u1D45D?u1D456)hu1D45Fu1D458?2. (4.34)
Let ?u1D43F the number of subblocks in which we would like to have the estimate of BEM
coefficient hu1D45Fu1D458. It is clear that ?(u1D456) is periodic with period u1D447/u1D45Au1D44F as CE-BEM is periodic
with period u1D447, that is ?(u1D45D?u1D457u1D447/u1D45Au1D44F) = ?(u1D45D) for any integer u1D457. In practice, we would like
to have the memory length (in symbols) to be less than the model period (recall that the
channel is by no means periodic) so that there are no deleterious effects due to the use of
(4.2) for all time, i.e., u1D45Au1D44F?u1D43F ? u1D447 in order to avoid this periodicity. Let us pick ?u1D43F = u1D447/u1D45Au1D44F.
What other restriction should we impose on ?u1D43F?
A least-square solution for hu1D45Fu1D458 to minimize u1D49E (with u1D45D? ?) is given by [50, p. 796]
?hu1D45Fu1D458 = A?1B, (4.35)
72
where
A =
?uni2211.alt02
u1D456=0
u1D706u1D456u1D6FE2?u1D43B(u1D45D?u1D456)?(u1D45D?u1D456), (4.36)
B =
?uni2211.alt02
u1D456=0
u1D706u1D456u1D6FE?u1D43B(u1D45D?u1D456)?yu1D45F(u1D45D?u1D456). (4.37)
In order to analyze the behavior of hu1D45Fu1D458 we need a model for ?yu1D45F(u1D456) for all subblocks, u1D456 =
u1D45D,u1D45D?1,???. To this end, for the analysis presented in this section, we assume the following
?simplified? model (recall (4.13)):
?yu1D45F(u1D45D?u1D456) = u1D6FE?(u1D45D?u1D456)h(u1D461u1D45F)u1D45Fu1D458 (u1D45D?u1D456) +vu1D45F(u1D45D?u1D456), (4.38)
where h(u1D461u1D45F)u1D45Fu1D458 (u1D45D?u1D456) is the ?true? BEM coefficient vector satisfying (u1D45A = 1,2,???)
h(u1D461u1D45F)u1D45Fu1D458 (u1D45D?u1D456) := h(u1D461u1D45F)u1D45Fu1D458,u1D45A for (u1D45A?1)?u1D43F?u1D45D?u1D456<u1D45A?u1D43F. (4.39)
That is, there exist some true BEM parameters that are ?fixed? over one BEM period of u1D447
symbols (therefore, ?u1D43F = u1D447/u1D45Au1D44F subblocks), and are allowed to change over non-overlapping
periods. In this set-up, we are trying to estimate the most recent true BEM coefficient vector
h(u1D461u1D45F)u1D45Fu1D458,1 via ?hu1D45Fu1D458 without it being unduly influenced by h(u1D461u1D45F)u1D45Fu1D458,u1D45A,u1D45A? 2.
Using the periodicity, ?(u1D45D?(u1D456+(u1D45A?1)?u1D43F)) = ?(u1D45D?u1D456) for u1D456 = 0,1,??? , ?u1D43F?1 and u1D45A =
1,2,???, we can rewrite (4.36) as (for u1D706< 1)
A = u1D6FE2
?uni2211.alt02
u1D45A=1
u1D706(u1D45A?1)?u1D43F
?
?
?u1D43F?1uni2211.alt02
u1D456=0
u1D706u1D456?u1D43B(u1D45D?u1D456)?(u1D45D?u1D456)
?
?
= u1D6FE
2
1?u1D706?u1D43F
?u1D43F?1uni2211.alt02
u1D456=0
u1D706u1D456?u1D43B(u1D45D?u1D456)?(u1D45D?u1D456). (4.40)
73
Similarly, using (4.38) and (4.39), after some manipulations, we obtain
B = u1D6FE
?uni2211.alt02
u1D456=0
u1D706u1D456?u1D43B(u1D45D?u1D456)
uni005B.alt02
u1D6FE?(u1D45D?u1D456)h(u1D461u1D45F)u1D45Fu1D458 (u1D45D?u1D456) +vu1D45F(u1D45D?u1D456)
uni005D.alt02
= u1D6FE2
?uni2211.alt02
u1D456=0
u1D706u1D456?u1D43B(u1D45D?u1D456)?(u1D45D?u1D456)h(u1D461u1D45F)u1D45Fu1D458 (u1D45D?u1D456) +u1D6FE
?uni2211.alt02
u1D456=0
u1D706u1D456?u1D43B(u1D45D?u1D456)vu1D45F(u1D45D?u1D456)
= u1D6FE2
?uni2211.alt02
u1D45A=1
u1D706(u1D45A?1)?u1D43F
?u1D43F?1uni2211.alt02
u1D456=0
u1D706u1D456?u1D43B(u1D45D?u1D456)?(u1D45D?u1D456)h(u1D461u1D45F)u1D45Fu1D458,u1D45A +u1D6FE
?uni2211.alt02
u1D456=0
u1D706u1D456?u1D43B(u1D45D?u1D456)vu1D45F(u1D45D?u1D456)
= u1D6FE2
?uni2211.alt02
u1D45A=1
u1D706(u1D45A?1)?u1D43Fh(u1D461u1D45F)u1D45Fu1D458,u1D45A
?
?
?u1D43F?1uni2211.alt02
u1D456=0
u1D706u1D456?u1D43B(u1D45D?u1D456)?(u1D45D?u1D456)
?
?
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=:D
+u1D6FE
?uni2211.alt02
u1D456=0
u1D706u1D456?u1D43B(u1D45D?u1D456)vu1D45F(u1D45D?u1D456)
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=:?w
. (4.41)
Then using (4.40) and (4.41), we can rewrite (4.35) as
?hu1D45Fu1D458 =
uni005B.alt041?u1D706?u1D43F
u1D6FE2 D
?1
uni005D.alt04uni005B.alt04
u1D6FE2D
?uni2211.alt02
u1D45A=1
u1D706(u1D45A?1)?u1D43Fh(u1D461u1D45F)u1D45Fu1D458,u1D45A + ?w
uni005D.alt04
= (1?u1D706?u1D43F)
?uni2211.alt02
u1D45A=1
u1D706(u1D45A?1)?u1D43Fh(u1D461u1D45F)u1D45Fu1D458,u1D45A + (1?u1D706?u1D43F)(u1D6FE2D)?1?w. (4.42)
For memory length considerations, we will consider only the noise?free case, and hence
we set ?w = 0 in (4.40). With this restriction together with u1D706?u1D43F ? 1, we obtain
?hu1D45Fu1D458 = (1?u1D706?u1D43F) ?uni2211.alt02
u1D45A=1
u1D706(u1D45A?1)?u1D43Fh(u1D461u1D45F)u1D45Fu1D458,u1D45A
= (1?u1D706?u1D43F)
uni005B.alt02
h(u1D461u1D45F)u1D45Fu1D458,1 +u1D706?u1D43Fh(u1D461u1D45F)u1D45Fu1D458,2 +u1D7062?u1D43Fh(u1D461u1D45F)u1D45Fu1D458,3 +???
uni005D.alt02
? h(u1D461u1D45F)u1D45Fu1D458,1 +u1D706?u1D43Fh(u1D461u1D45F)u1D45Fu1D458,2. (4.43)
For WSSUS channel, h(u1D461u1D45F)u1D45Fu1D458,1 and h(u1D461u1D45F)u1D45Fu1D458,2 have the same statistics, u1D438
uni007B.alt03vextenddoublevextenddouble
vextenddoubleh(u1D461u1D45F)u1D45Fu1D458,1
vextenddoublevextenddouble
vextenddouble2
uni007D.alt03
= u1D438
uni007B.alt03vextenddoublevextenddouble
vextenddoubleh(u1D461u1D45F)u1D45Fu1D458,2
vextenddoublevextenddouble
vextenddouble2
uni007D.alt03
.
The normalized error norm in estimating h(u1D461u1D45F)u1D45Fu1D458,1 via ?hu1D45Fu1D458 is therefore given by
u1D452u1D45Bu1D45Cu1D45Fu1D45A :=
uni221A.alt04
u1D438
uni007B.alt03vextenddoublevextenddouble
vextenddoubleu1D706?u1D43Fh(u1D461u1D45F)u1D45Fu1D458,2
vextenddoublevextenddouble
vextenddouble2
uni007D.alt03
uni221A.alt04
u1D438
uni007B.alt03vextenddoublevextenddouble
vextenddoubleh(u1D461u1D45F)u1D45Fu1D458,1
vextenddoublevextenddouble
vextenddouble2
uni007D.alt03 = u1D706
?u1D43F. (4.44)
We would like to keep u1D452u1D45Bu1D45Cu1D45Fu1D45A ? u1D6FF where u1D6FF is pre-specified for the desired performance. This
leads to
u1D706?u1D6FF1/?u1D43F = u1D6FFu1D45Au1D44F/u1D447. (4.45)
74
Given the CE-BEM period u1D447 = 400, some theoretical choices of u1D706 are shown in Table
4.4. In Figs. 4.7a and 4.7b, the performances of EW-RLS algorithm are shown for different
values of u1D706 based on 500 Monte Carlo runs with the same environment as the simulation
examples in Section 4.2.3 and 4.3.2. In our simulations, we picked the forgetting factor
u1D706 = 0.65 and 0.5 for u1D45Au1D44F = 20 and 40 respectively, which satisfy the theoretical condition in
(4.45). Indeed, u1D6FF = 0.001 yields values of u1D706 = u1D6FFu1D45Au1D44F/u1D447 that are close to the values suggested
by the minima in Fig. 4.7.
Table 4.4: Theoretical u1D706 for subblock-wise EW-RLS channel estimation.
u1D6FF u1D447 u1D45Au1D44F u1D706
0.01 400 20 ? 0.794
40 ? 0.631
0.001 400 20 ? 0.71
40 ? 0.50
0.4 0.5 0.6 0.7 0.8 0.9?30
?25
?20
?15
?10
?5
0
? for EW?RLS
NCMSE(dB)
QPSK,SB?EWRLS,T=400,Q=9,fdTs=0.01,SNR=20dB,500runs
 
 
mb=40
mb=20
(a) NCMSE vs u1D706
0.4 0.5 0.6 0.7 0.8 0.910
?4
10?3
10?2
10?1
100
DFE,QPSK,SB?EWRLS,T=400,Q=9,fdTs=0.01,SNR=20dB,500runs
? for EW?RLS
Bit Error Rate
 
 
mb=40
mb=20
(b) BER vs u1D706, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and delay
u1D451 = 5
Figure 4.7: Subblock-wise EW-RLS channel estimation: performances for forgetting factor
u1D706?s, under u1D453u1D451u1D447u1D460 = 0.01,SNR = 20dB,u1D45Au1D44F = 20 and 40.
75
4.5 Conclusions
The subblock-wise finite-memory recursive least-squares (RLS) tracking approach is
presented, exploiting the oversampled complex exponential basis expansion model (CE-
BEM) for the overall channel variations. We apply the exponentially-weighted and sliding-
window recursive least-squares (RLS) algorithms to track the BEM coefficients, using time-
multiplexed periodically transmitted training symbols. The proposed subblock-wise RLS
schemes, which have no a priori model for the BEM coefficients, outperform the subblock-
wise Kalman tracking in Chapter 3, since the AR assumption for the BEM coefficients used
in the latter scheme may introduce additional modeling errors. Simulation examples illus-
trate the superior performance of our subblock-wise scheme to the conventional block-wise
channel estimator and the subblock-wise Kalman tracking in Chapter 3.
76
Chapter 5
Turbo Equalization for Doubly-Selective Channels Using Nonliner
Kalman Filtering and Basis Expansion Models
We present a turbo equalization receiver with nonlinear Kalman filtering, exploiting the
complex exponential basis expansion model (CE-BEM) for the overall channel variations and
the autoregressive (AR) model to update the BEM coefficients. The time-varying nature of
the channel is well captured by the CE-BEM while the time-variations of the (unknown)
BEM coefficients are likely much slower than those of the channel. In the proposed receiver,
an adaptive equalizer using nonlinear Kalman filters with delay is coupled with a soft-in
soft-out (SfiSfo) module to perform the reception process iteratively. The proposed adaptive
equalizer jointly optimizes the estimation of BEM channel coefficients and data symbols,
thereby automatically accounting for the correlation between the channel estimates and
data symbols in the equalization process. Simulation examples demonstrate our CE-BEM
based approach has superior performance over an existing AR-only-based turbo equalizer.
5.1 Introduction
Among various models for channel time-variations, the AR process, particularly the first-
order AR model, is regarded as a tractable formulation to describe a time-varying channel
on a symbol-by-symbol basis [6,16,68]. In fast time-varying channel environments, however,
channel prediction using an AR model may lead to high estimation variance resulting in
erroneous symbol decisions [6]. The BEM depicts evolutions of the channel over a period
(block) of time, in which the time-varying channel taps are expressed as superpositions of
time-varying basis functions in modeling Doppler effects, weighted by time-invariant coeffi-
cients [70].
In Chapter 3, a subblock-wise tracking approach was investigated for doubly-selective
channels using time-multiplexed (TM) training. It exploits the complex exponential BEM
for the overall channel variations of each (overlapping) block, and a first-order AR model
to describe the evolutions of the BEM coefficients. The slow-varying BEM coefficients are
updated via Kalman filtering at each training session; during information sessions, chan-
nel estimates are generated by the CE-BEM using the estimated BEM coefficients. This
approach achieves better performance in fast-fading environments, than using conventional
symbol-wise AR models.
77
In this chapter, we propose BEM-based approach to coded modulation communication
systems using turbo equalization receiver. Turbo (iterative) equalization is a powerful sub-
optimal technique used in place of the computationally prohibitive but maximum-likelihood
(ML) or maximum a posteriori (MAP) sequence detection based on a super trellis. Although
originally proposed for parallel concatenated error correction codes, the turbo principle is
shown to be applicable to the detection problem for coded systems with ISI in [5]. By com-
bining a MAP equalizer and a MAP decoder, and exchanging probabilistic information about
data symbols iteratively, turbo equalization usually can achieve close-to-optimal performance
but much lower complexity [1,5]. In [71], a turbo-equalization-like system using linear equal-
izers based on soft interference cancellation and linear minimum mean-square error (MMSE)
filtering is proposed as part of a multiuser detector for CDMA. Based on this work, a variety
of SfiSfo equalizers employing linear MMSE and decision feedback equalization (DFE) are
proposed in [38,39].
For doubly-selective channels, an adaptive SfiSfo equalizer has been presented in [68],
using extended Kalman filter (EKF) to incorporate channel estimation into the equalization
process. This adaptive soft nonlinear Kalman equalizer takes the soft decisions of data
symbols from the SfiSfo decoder as its a priori information, and performs equalization process
iteratively. With such an approach, the proposed scheme jointly optimizes the estimates of
the channel and data symbols in each iteration. This avoids the common drawback in
separate channel estimation and equalization/detection approach in that the correlation
between channel estimate and data symbol decision is considered. The complexity of [68]
is comparable to that of the turbo equalizers using linear filters [36,48,57], and is usually
much lower than that of the ML/MAP based joint channel estimation and data detection
schemes.
Based on the turbo equalization approach proposed in [68] and CE-BEM, we present
an adaptive turbo equalizer with nonlinear Kalman filtering. The channel variations can be
well captured by the CE-BEM since the time-variations of the BEM coefficients are likely
much slower than those of real channel. This adaptive SfiSfo equalizer takes the decision of
data symbols provided by SfiSfo decoder as its a priori information and the performance can
be improved iteratively. The proposed adaptive equalizer jointly optimizes the estimates of
BEM channel coefficients and data symbols in each equalization process since the correlation
between the estimates of the channel and data symbols is considered. Simulation examples
demonstrate our CE-BEM based scheme has superior performance over the turbo equalizer
in [68] that relies on the AR modeling of channel. [It has been shown in [68] that their
approach has better performance than any other turbo approaches of [36,48,57]; hence we
compare our approach only with [68].]
78
5.2 Turbo Equalization using Extended Kalman Filter (EKF) and CE-BEM
Unlike the prior works [11,14,70], we will now allow the blocks ofu1D447u1D435 symbols to overlap.
By exploiting the invariance of the coefficients of the CE-BEM over each block, we consider
two overlapping blocks (each of u1D447u1D435 symbols) that differ by just one symbol: the ?past? block
beginning at time u1D45B0 and the ?present? block beginning at time u1D45B0 +1. Since the two blocks
overlap so significantly, one would expect the BEM coefficients to vary only ?a little? from
the past block to the present overlapping one. We propose to track the BEM coefficients
(rather than the channel tap gains) symbol-by-symbol using a first-order AR model for their
variations, where we will use (5.2) for all times u1D45B, not just the particular block of size u1D447u1D435
symbols, by allowing the coefficients ?u1D45E(u1D459)?s to change with time.
5.2.1 System Model and Turbo Equalization Receiver
Consider a doubly-selective (time- and frequency-selective) single-input single-output
(SISO), finite impulse response (FIR) linear channel. Let {u1D460(u1D45B)} denote a scalar sequence
that is input to the time-varying channel with discrete-time response {?(u1D45B;u1D459)} (channel
response at time u1D45B to a unit input at time u1D45B?u1D459). Then the symbol-rate noisy channel output
is given by (u1D45B = 0,1,...)
u1D466(u1D45B) =
u1D43Funi2211.alt02
u1D459=0
?(u1D45B;u1D459)u1D460(u1D45B?u1D459) +u1D463(u1D45B) (5.1)
where u1D463(u1D45B) is zero-mean white complex Gaussian noise, with variance u1D70E2u1D463. We assume that
{?(u1D45B;u1D459)} represents a wide-sense stationary uncorrelated scattering (WSSUS) channel [60].
We also assume that u1D460(u1D45B) is mutually independent and identically distributed (i.i.d.) with
zero mean and variance u1D438{u1D460(u1D45B)u1D460? (u1D45B)} = u1D70E2u1D460.
We use the oversampled CE-BEM for channel modeling comparing AR model in [68]. In
CE-BEM [11,14,70], over the u1D456-th block consisting of an observation window of u1D447u1D435 symbols,
the channel is represented as
?(u1D45B;u1D459) =
u1D444uni2211.alt02
u1D45E=1
?(u1D459)u1D45E u1D452u1D457u1D714u1D45Eu1D45B, (5.2)
79
Figure 5.1: Bit-interleaved coded modulation system model for doubly-selective fading chan-
nel
where u1D45B = ?u1D45Bu1D456,?u1D45Bu1D456 + 1,??? ,?u1D45Bu1D456 +u1D447u1D435 ?1 with ?u1D45Bu1D456 := (u1D456?1)u1D447u1D435, u1D459 = 0,1,??? ,u1D43F and one chooses
(? is an integer)
u1D447 := ?u1D447u1D435, ? ? 1, (5.3)
u1D444 := 2?u1D453u1D451u1D447u1D447u1D460?+ 1, (5.4)
u1D714u1D45E := 2u1D70Bu1D447 [u1D45E?(u1D444+ 1)/2], u1D45E = 1,2,??? ,u1D444 (5.5)
u1D43F := ?u1D70Fu1D451/u1D447u1D460? (5.6)
u1D70Fu1D451 and u1D453u1D451 are respectively the delay spread and the Doppler spread, and u1D447u1D460 is the symbol
duration. The BEM coefficients ?(u1D459)u1D45E ?s remain invariant during this block, but are allowed to
change at the next consecutive block; the Fourier basis functions {u1D452u1D457u1D714u1D45Eu1D45B} (u1D45E = 1,2,??? ,u1D444)
are common for every block. If the delay spread and the Doppler spread (or at least their
upper bounds) are known, one can infer the basis functions of the CE-BEM [70]. Treating
the basis functions as known, estimation of a time-varying process is reduced to estimating
the invariant coefficients over a block of u1D447u1D435 symbols.
Bit-Interleaved Coded Modulation (BICM)
We consider a BICM transmitter (as in [67]) for a doubly-selective fading channel as
shown in Fig. 5.1. A sequence of independent data vectoru1D44F(u1D45B?) = [u1D44F1(u1D45B?),u1D44F2(u1D45B?),??? ,u1D44Fu1D4580(u1D45B?)] ?
{1,0}u1D4580 are fed into a convolutional encoder with a code rate u1D445u1D450 = u1D4580/u1D45B0. The coded output
c(u1D45B?) = [u1D4501(u1D45B?),u1D4502(u1D45B?),??? ,u1D450u1D45B0(u1D45B?)] ? {1,0}u1D45B0 is passed through a bit-wise random interleaver
u1D70B, generating the interleaved coded bit sequence c(u1D45B) = u1D70B[c(u1D45B?)]. The binary coded bits are
then mapped to a signal sequence u1D451(u1D45B) over a 2-dimensional signal constellation u1D4B3 of cardi-
nality ? = 2u1D45A by a ?-ary modulator with an one-to-one binary map u1D707 : {1,0}u1D45A ? u1D4B3. In
this section, we only consider the case of phase-shift keying (PSK) or quadrature amplitude
modulation (QAM) with the average energy of the constellation u1D4B3 to be unity. That is,
the signal u1D451(u1D45B) drawn from u1D4B3 has mean u1D438[u1D451(u1D45B)] = 0 and variance u1D438[?u1D451(u1D45B)?2] = 1. After
modulation, we periodically insert short training sequences into the data symbol sequence.
The training symbols u1D461(u1D45B), which are known to the receiver, are randomly drawn from the
80
Figure 5.2: Turbo equalization receiver. Following [10,49,68] and contrary to the original
turbo principle, a posteriori LLR Lu1D44E{c(u1D45B)} = Lu1D440u1D452 {c(u1D45B)}+Lu1D437u1D452 {c(u1D45B)} instead of the extrinsic
LLR Lu1D437u1D452 {c(u1D45B)} can be input to the LLR-to-symbol block. Inclusion of Lu1D440u1D452 {c(u1D45B)} to create
a posteriori LLR is shown via dashed line. For our proposed approach we follow [10,49,68].
signal constellation u1D4B3 with equal probabilities. The symbol {u1D460(u1D45B)} will be used to denote
the symbol sequence after training({u1D461(u1D45B)}) insertion into data symbol sequence({u1D451(u1D45B)}).
Receiver Structure
A turbo equalization structure, as depicted in Fig. 5.2, is employed in the receiver,
as in [68] except that [68] uses symbol-wise AR models. The adaptive SfiSfo equalizer
is embedded into the iterative decoding (ID) process of the BICM transmission system
(BICM-ID) [67]. In each decoding iteration, the equalizer takes the training symbols and
the soft decision information about data symbols supplied by the SfiSfo decoder from the
previous iteration as its a priori information to perform joint adaptive channel estimation
and equalization. The equalizer produces the soft-valued extrinsic estimate of the data
symbols, which are independent of their a priori information. The output of the equalizer is
an updated sequence of soft estimates ?u1D451(u1D45B) and its error variance u1D70E2(u1D45B). Using the adaptive
SfiSfo equalizer in Section 5.2.2, we have extrinsic information for the data symbols u1D451(u1D45B).
The training symbols are removed at the SfiSfo equalizer output and the iterative process
that follows is only for data symbols. The SfiSfo equalizer based on the CE-BEM is described
in Section 5.2.2. The SfiSfo demodulator follows [67] whereas the SfiSfo decoder follows the
MAP decoding algorithm (?BCJR?) [72, Section 6.2].
The data symbol estimates ?u1D451(u1D45B) and its error variance u1D70E2(u1D45B) are passed to the SfiSfo
demodulator to generate extrinsic log-likelihood ratios (LLR?s) Lu1D440u1D452 {c(u1D45B)} for the coded bits
c(u1D45B) given u1D447u1D456 received symbols {u1D466(u1D459), 0 ?u1D459<u1D447u1D456} (only for the information symbols, not the
training), denoted by
Lu1D440u1D452 {c(u1D45B)} =
uni005B.alt02
u1D43Fu1D440u1D452
uni007B.alt02
u1D450u1D456(u1D45B)
uni007D.alt02
, u1D456 = 1,2??? ,u1D45B0
uni005D.alt02
, (5.7)
81
where u1D447u1D456 is the information block size after mapping the interleaved coded bits to the signal
sequence,
u1D43Fu1D440u1D452
uni007B.alt02
u1D450u1D456(u1D45B)
uni007D.alt02
:= ln u1D443{u1D450
u1D456(u1D45B) = 1 ?u1D466(u1D459), 0 ?u1D459<u1D447u1D456}
u1D443{u1D450u1D456(u1D45B) = 0 ?u1D466(u1D459), 0 ?u1D459<u1D447u1D456} ?ln
u1D443{u1D450u1D456(u1D45B) = 1}
u1D443{u1D450u1D456(u1D45B) = 0}bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=:u1D43F{u1D450u1D456(u1D45B)}
, (5.8)
and u1D43F{u1D450u1D456(u1D45B)} is the a priori LLR. In (5.8), u1D443{u1D450u1D456(u1D45B) = u1D44F ? u1D466(u1D459), 0 ? u1D459 < u1D447u1D456}, u1D44F ? {0,1}, is
approximated as
u1D443{u1D450u1D456(u1D45B) = u1D44F?u1D466(u1D459), 0 ?u1D459<u1D447u1D456} ? u1D443{u1D450u1D456(u1D45B) = u1D44F? ?u1D451(u1D45B)} (5.9)
by replacing the data {u1D466(u1D459), 0 ? u1D459 < u1D447u1D456} with the soft estimate ?u1D451(u1D45B). Since u1D443{u1D450u1D456(u1D45B) = u1D44F ?
?u1D451(u1D45B)} = u1D443{?u1D451(u1D45B) ?u1D450u1D456(u1D45B) = u1D44F}u1D443{u1D450u1D456(u1D45B) = u1D44F}/u1D443{?u1D451(u1D45B)}, it follows from (5.8) and (5.9) that
u1D43Fu1D440u1D452
uni007B.alt02
u1D450u1D456(u1D45B)
uni007D.alt02
= ln u1D443{
?u1D451(u1D45B) ?u1D450u1D456(u1D45B) = 1}
u1D443{?u1D451(u1D45B) ?u1D450u1D456(u1D45B) = 0}. (5.10)
The soft estimate ?u1D451(u1D45B) of u1D451(u1D45B) follows from the fixed-lag SfiSfo Kalman equalizer discussed
later in Section 5.2.2 (following [68]), and it and its variance are given by (5.51) and (5.52),
respectively. We assume that ?u1D451(u1D45B) is complex Gaussian distributed with mean u1D451 ? u1D4B3 and
variance u1D70E2(u1D45B) and follow [67] to calculate (5.10). Let u1D450u1D456?u1D451? denote the u1D456-th coded bit in
the block of coded bits [u1D4501?u1D451?,u1D4502?u1D451?,??? ,u1D450u1D45B0 ?u1D451?] that is mapped to the symbol u1D451; therefore
u1D707([u1D4501,u1D4502,??? ,u1D450u1D45B0]) = u1D451. It then follows that
u1D443
uni007B.alt02
u1D451?u1D450u1D456
uni007D.alt02
=
u1D45B0uni220F.alt02
u1D457=1,u1D457?=u1D456
u1D443{u1D450u1D457 = u1D450u1D457 ?u1D451?}. (5.11)
Furthermore, under the assumptions on ?u1D451(u1D45B), we have
u1D443
uni007B.alt02?
u1D451(u1D45B) ?u1D451
uni007D.alt02
= 1u1D70Bu1D70E2(u1D45B)exp
uni0028.alt04
??
?u1D451(u1D45B)?u1D451?2
u1D70E2(u1D45B)
uni0029.alt04
. (5.12)
Recall that we used u1D4B3 to denote the set of all possible data symbols. Let u1D4B3 (u1D456,u1D44F) =
{u1D707([u1D4501,u1D4502,??? ,u1D450u1D45B0]) ?u1D450u1D456 = u1D44F}, with u1D44F ? {1,0} and u1D456 ? {1,2,??? ,u1D45B0}, denote the collection
82
of all data symbols whose corresponding u1D456-th coded bit is ?fixed? as u1D44F. Then using (5.10)-
(5.12) one obtains
u1D43Fu1D440u1D452
uni007B.alt02
u1D450u1D456(u1D45B)
uni007D.alt02
= ln
uni2211.alt01
u1D451?u1D4B3(u1D456,1)
uni005B.alt03
exp
uni0028.alt03
???u1D451(u1D45B)?u1D451?2u1D70E2(u1D45B)
uni0029.alt03uni220F.alt01
u1D45B0
u1D457=1,u1D457?=u1D456u1D443{u1D450
u1D457(u1D45B) = u1D450u1D457 ?u1D451?}
uni005D.alt03
uni2211.alt01
u1D451?u1D4B3(u1D456,0)
uni005B.alt03
exp
uni0028.alt03
???u1D451(u1D45B)?u1D451?2u1D70E2(u1D45B)
uni0029.alt03uni220F.alt01
u1D45B0
u1D457=1,u1D457?=u1D456u1D443{u1D450u1D457(u1D45B) = u1D450u1D457 ?u1D451?}
uni005D.alt03. (5.13)
The output extrinsic bit LLR?s of the SfiSfo demodulator are bit-wise de-interleaved as
Lu1D440u1D452 {c(u1D45B?)} = u1D70B?1[Lu1D440u1D452 {c(u1D45B)}], which are then input to the SfiSfo convolutional decoder. In
SfiSfo decoder, the MAP decoding algorithm for convolutional codes (see Section 2.5.2 [72,
Section 6.2]) is applied to update the LLR?s of the coded bits {c(u1D45B)} as well as the LLR?s
of the information bits {b(u1D45B)}, based on the code constraints. The decoder computes the
extrinsic LLR for coded bits
Lu1D437u1D452 {c(u1D45B?)} =
uni005B.alt02
u1D43Fu1D437u1D452
uni007B.alt02
u1D450u1D456(u1D45B?)
uni007D.alt02
, u1D456 = 1,2??? ,u1D45B0
uni005D.alt02
, (5.14)
where
u1D43Fu1D437u1D452
uni007B.alt02
u1D450u1D456(u1D45B?)
uni007D.alt02
:= ln u1D443{u1D450
u1D456(u1D45B?) = 1 ? Lu1D440
u1D452 {c(u1D459)}, 0 ?u1D459<u1D447u1D456}
u1D443{u1D450u1D456(u1D45B?) = 0 ? Lu1D440u1D452 {c(u1D459)}, 0 ?u1D459<u1D447u1D456} ?u1D43F
u1D440
u1D452 {u1D450
u1D456(u1D45B?)}
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=:u1D43F{u1D450u1D456(u1D45B?)}
. (5.15)
The output bit LLR?s of SfiSfo decoder are bit-wise interleaved asLu1D437u1D452 {c(u1D45B)} = u1D70B[Lu1D437u1D452 {c(u1D45B?)}].
The SfiSfo demodulator performs symbol-by-symbol MAP demodulation using LLR?sLu1D437u1D452 {c(u1D45B)}
for the coded bits generated by SfiSfo decoder in the previous iteration as its a priori informa-
tion: u1D43F{u1D450u1D456(u1D45B)} = u1D43Fu1D437u1D452 {u1D450u1D456(u1D45B)} . We set u1D43Fu1D437u1D452 {u1D450u1D456(u1D45B)} = 0 for the initial step (first iteration). The
LLR for use in LLR-to-symbol block is computed via Lu1D44E{c(u1D45B)} = Lu1D440u1D452 {c(u1D45B)}+Lu1D437u1D452 {c(u1D45B)},
which is the the a posteriori LLR. [It is claimed in [10] that, unlike the original turbo principle
where one takes Lu1D44E{c(u1D45B)} = Lu1D440u1D452 {c(u1D45B)}, usage of the full SfiSfo decoder?s soft information
embodied in the a posteriori LLR Lu1D44E{c(u1D45B)} enhances performance compared to using only
Lu1D437u1D452 {c(u1D45B)}; [68] also uses this set-up. This has also been our experience in the simulations
presented; therefore, we have followed this approach.] The bit probabilities (converted from
the corresponding LLR Lu1D44E{c(u1D45B)}) at time u1D45B are used (following [67]) to compute the mean
?u1D451(u1D45B) and variance u1D6FEu1D451(u1D45B) for data symbols u1D451(u1D45B) as
?u1D451(u1D45B) = u1D438[u1D451(u1D45B)] = uni2211.alt02
u1D451?u1D4B3
u1D451u1D443 {u1D451(u1D45B) = u1D451} = uni2211.alt02
u1D451?u1D4B3
u1D451
u1D45B0uni220F.alt02
u1D457=1
u1D443u1D44E
uni007B.alt02
u1D450u1D457(u1D45B) = u1D450u1D457u1D451
uni007D.alt02
, (5.16)
u1D6FEu1D451(u1D45B) = var[u1D451(u1D45B)] = uni2211.alt02
u1D451?u1D4B3
?u1D451(u1D45B)? ?u1D451(u1D45B)?2u1D443 {u1D451(u1D45B) = u1D451} = 1? ?u1D4512(u1D45B), (5.17)
83
where
u1D443u1D44E
uni007B.alt02
u1D450u1D457(u1D45B) = 1
uni007D.alt02
= 11 + exp(?u1D43F
u1D44E{u1D450u1D457(u1D45B)})
, u1D443u1D44E
uni007B.alt02
u1D450u1D457(u1D45B) = 0
uni007D.alt02
= 11 + exp(u1D43F
u1D44E{u1D450u1D457(u1D45B)})
.
(5.18)
Then ?u1D451(u1D45B) and u1D6FEu1D451(u1D45B) are fed back to the equalizer as a priori information, along with the
training symbols.
5.2.2 Adaptive Soft-In Soft-Out Nonlinear Kalman Equalizer
Using a symbol-wise AR-model for channel variations, an adaptive SfiSfo equalizer using
fixed-lag EKF was presented in [68] for joint channel estimation and equalization where their
correlation was (implicitly) considered. In this section, we present a CE-BEM model-based
SfiSfo nonlinear Kalman equalizer for turbo equalization.
State-Space Model using CE-BEM and a Priori Information
Stack the channel coefficients in (5.2) into vectors
hu1D459 :=
uni005B.alt02
?1 (u1D459) ?2 (u1D459) ??? ?u1D444 (u1D459)
uni005D.alt02u1D447
, (5.19)
h :=
uni005B.alt02
hu1D4470 hu1D4471 ??? hu1D447u1D43F
uni005D.alt02u1D447
(5.20)
of size u1D444 and u1D444(u1D43F+ 1) respectively. We will allow h in (5.20) to change with ?time? u1D45B, in
which case it will be denoted by h(u1D45B). We assume that the channel BEM coefficients follow
an AR model. One could fit a general AR(u1D443) model with a high value of u1D443, but we seek a
?simple? AR(1) model given by
h(u1D45B) = A1h(u1D45B?1) +w(u1D45B) (5.21)
where A1 is the time-invariant AR coefficient matrix and the driving noise vector w(u1D45B) is
zero-mean white with identity covariance. Collecting all channel tap gains over one block,
further define the [(u1D43F+ 1)u1D447u1D435]?1 vector
?hu1D435(u1D45B) := uni005B.alt02 ?(u1D45B;0) ?(u1D45B?1;0) ??? ?(u1D45B?u1D447u1D435 + 1;0) ?(u1D45B;1) ?(u1D45B?1;1)
??? ?(u1D45B?u1D447u1D435 + 1;1) ??? ?(u1D45B;u1D43F) ??? ?(u1D45B?u1D447u1D435 + 1;u1D43F)
uni005D.alt02u1D447
. (5.22)
Define
uD4D4 (u1D45B) :=
uni005B.alt02
u1D452?u1D457u1D7141u1D45B u1D452?u1D457u1D7142u1D45B ??? u1D452?u1D457u1D714u1D444u1D45B
uni005D.alt02u1D447
. (5.23)
84
Using (5.23), we further define
B(u1D45B) :=
uni005B.alt02
uD4D4 (u1D45B) uD4D4 (u1D45B?1) ??? uD4D4 (u1D45B?u1D447u1D435 + 1)
uni005D.alt02u1D43B
, ? := diag
uni007B.alt02
u1D452u1D457u1D7141,u1D452u1D457u1D7142,...,u1D452u1D457u1D714u1D444
uni007D.alt02
,
(5.24)
where B(u1D45B) is u1D447u1D435 ?u1D444 and ? is u1D444?u1D444. Consider two overlapping blocks that differ by just
one symbol: g(u1D45B) and g(u1D45B+1), with h(u1D45B) and h(u1D45B+ 1), respectively, as the corresponding
BEM coefficients. It then follows that
?hu1D435(u1D45B) = ?B(u1D45B)h(u1D45B), ?hu1D435(u1D45B+ 1) = ?B(u1D45B)??h(u1D45B+ 1) (5.25)
where (?B is [(u1D43F+ 1)u1D447u1D435]?[(u1D43F+ 1)u1D444] and ?? is [(u1D43F+ 1)u1D444]?[(u1D43F+ 1)u1D444])
?B(u1D45B) := diag{B(u1D45B), B(u1D45B), ??? , B(u1D45B)}, ?? := diag{?, ?, ??? , ?}. (5.26)
If (5.21) holds, then using the Yule-Walker equation we have
A1 = u1D438
uni007B.alt02
h(u1D45B+ 1)hu1D43B(u1D45B)
uni007D.alt02uni005B.alt02
u1D438
uni007B.alt02
h(u1D45B)hu1D43B(u1D45B)
uni007D.alt02uni005D.alt02?1
(5.27)
where using (5.25) we have
u1D438
uni007B.alt02
h(u1D45B+ 1)hu1D43B(u1D45B)
uni007D.alt02
= ???1?B?(u1D45B)u1D438
uni007B.alt02?
hu1D435(u1D45B+ 1)?hu1D43Bu1D435(u1D45B)
uni007D.alt02?
B?u1D43B(u1D45B), (5.28)
u1D438
uni007B.alt02
h(u1D45B)hu1D43B(u1D45B)
uni007D.alt02
= ?B?(u1D45B)u1D438
uni007B.alt02?
hu1D435(u1D45B)?hu1D43Bu1D435(u1D45B)
uni007D.alt02?
B?u1D43B(u1D45B), (5.29)
and u1D438
uni007B.alt02?
hu1D435(u1D45B)?hu1D43Bu1D435(u1D45B)
uni007D.alt02
and u1D438
uni007B.alt02?
hu1D435(u1D45B+ 1)?hu1D43Bu1D435(u1D45B)
uni007D.alt02
can be calculated using (5.22) if we know
the channel correlation function R?(u1D70F). This procedure results in matching the correlation
function of h(u1D45B) at lags 0 and 1.
Typically R?(u1D70F) will not be available. Therefore, to simplify we will assume that A1 =
u1D6FCI (implying that all tap gains have the same Doppler spectrum), and u1D438{w(u1D45B)wu1D43B(u1D45B)} =
u1D70E2u1D464Iu1D444(u1D43F+1), leading to
h(u1D45B) = u1D6FCh(u1D45B?1) +w(u1D45B). (5.30)
If the channel is stationary (WSSUS) and coefficients ?u1D45E(u1D459)?s are independent (as assumed
in [70]), then by (5.30) and Yule-Walker equations, we can estimate u1D6FC as
u1D6FC =
?
?u1D438
uni007B.alt02
hu1D43B(u1D45B+ 1)h(u1D45B)
uni007D.alt02
u1D438{hu1D43B(u1D45B)h(u1D45B)}
?
?
?
= tr
uni007B.alt02?
??1?B?(u1D45B)u1D438
uni007B.alt02?
hu1D435(u1D45B+ 1)?hu1D43Bu1D435(u1D45B)
uni007D.alt02?
B?u1D43B(u1D45B)
uni007D.alt02
tr
uni007B.alt02?
B?(u1D45B)u1D438
uni007B.alt02?
hu1D435(u1D45B)?hu1D43Bu1D435(u1D45B)
uni007D.alt02?
B?u1D43B(u1D45B)
uni007D.alt02 , (5.31)
85
and u1D70E2u1D464 = u1D438{??(u1D45B;u1D459)?2}(1 ? ?u1D6FC?2)/u1D444 where u1D70E2? = u1D438{??(u1D45B;u1D459)?2}. Note that (5.31) requires
knowledge of R?(u1D70F). In order to avoid this, one can somewhat arbitrarily pick a value of
u1D6FC such that u1D6FC ? 1 but u1D6FC < 1; this has been done in, e.g. [28] (in a different but similar
context). Besides, for tracking, one needs u1D6FC < 1 [28]. To gain more insight, let us consider
a mutually independent channel tap ?(u1D45B;u1D459) for different u1D459?s following the Jakes? spectrum
(also used in Section 5.2.3 in simulation examples). When u1D447 = 200, u1D447u1D435 = 100, u1D444 = 5, and
u1D453u1D451u1D447u1D460 = 0.01, one gets u1D6FC =0.99989 using (5.31). We compared it with A1 obtained via (5.27)-
(5.29), yielding the normalized difference ?A1 ?u1D6FCI?u1D439/?A1?u1D439 =0.0095 where ?.?u1D439 denotes
the Frobenius norm. [As we will see later in Section 5.2.3 (Fig. 5.8 and 5.9), this value of u1D6FC
is too close to one to permit tracking; we used u1D6FC = 0.996 in Section 5.2.3.] Thus, for channel
taps following the Jakes? spectrum, A1 = u1D6FCI is an excellent choice.
Under this formulation, we do not need a ?strict? definition of the block size u1D447u1D435. A key
parameter now is the CE-BEM period u1D447, not the block size u1D447u1D435. Later we use (5.2) for all
times u1D45B, not just the particular block of size u1D447u1D435 symbols, by allowing the coefficients ?u1D45E(u1D459)?s
to change with time (symbol-wise). Note that model (5.2) is periodic with period u1D447 whereas
the channel is by no means periodic. So long as the effective ?memory? of the Kalman filter
used later is less than the model period u1D447, there are no deleterious effects due to the use of
(5.2) for all time.
We will perform equalization with a delay u1D6FF> 0. Define a parameter
?u1D6FF := max{u1D6FF+ 1,u1D43F+ 1} (5.32)
and the data vector
z(u1D45B) :=
uni005B.alt02
u1D460(u1D45B) u1D460(u1D45B?1) ??? u1D460
uni0028.alt02
u1D45B? ?u1D6FF+ 1
uni0029.alt02uni005D.alt02u1D447
. (5.33)
Consider (5.30). In order to apply (extended) Kalman filtering to joint channel estimation
and equalization, we stack h(u1D45B) and data vector z(u1D45B) together into a u1D43D ? 1 state vector
x(u1D45B) at time u1D45B as
x(u1D45B) :=
uni005B.alt02
zu1D447 (u1D45B) hu1D447 (u1D45B)
uni005D.alt02u1D447
, u1D43D := ?u1D6FF+u1D444(u1D43F+ 1). (5.34)
As in [68] (and others), we consider the symbol sequence {u1D460(u1D45B)} as a stochastic process
so as to utilize the soft decisions on the data symbols generated in the iterative decoding
process as its a priori information. We can express u1D460(u1D45B) as u1D460(u1D45B) = ?u1D460(u1D45B) + ?u1D460(u1D45B) where
?u1D460(u1D45B) = u1D438[u1D460(u1D45B)] and ?u1D460(u1D45B) is approximated as a zero-mean uncorrelated sequence such that
u1D438[?u1D460(u1D45B)?u1D460?(u1D45B + u1D457)] = u1D6FE(u1D45B)u1D6FF(u1D457), assuming an ideal interleaver. Note that ?u1D460(u1D45B) and u1D6FE(u1D45B) are
86
provided via the a priori information. We have ?u1D460(u1D45B) = ?u1D451(u1D45B) and u1D6FE(u1D45B) = u1D6FEu1D451(u1D45B) for a data
symbol u1D451(u1D45B) (where ?u1D451(u1D45B) and u1D6FEu1D451(u1D45B) are specified in (5.16) and (5.17), respectively), while
?u1D460(u1D45B) = u1D461(u1D45B) and u1D6FE(u1D45B) = 0 for a training symbol u1D461(u1D45B).
Using x(u1D45B), the state equation turns out to be
x(u1D45B) = u1D4AFx(u1D45B?1) +e0?u1D460(u1D45B) +u(u1D45B), (5.35)
where
u1D4AF =
?
? ? 0?u1D6FF?u1D444(u1D43F+1)
0u1D444(u1D43F+1)??u1D6FF F
?
?
u1D43D?u1D43D
, F = u1D6FCIu1D444(u1D43F+1), (5.36)
? =
?
?01?(?u1D6FF?1) 01?1
I(?u1D6FF?1) 0(?u1D6FF?1)?1
?
?
?u1D6FF??u1D6FF
, e0 =
uni005B.alt02
1 01?(u1D43D?1)
uni005D.alt02u1D447
, (5.37)
the vector
u(u1D45B) :=
uni005B.alt02
eu1D447?u1D6FF ?u1D460(u1D45B) wu1D447(u1D45B)
uni005D.alt02u1D447
(5.38)
is zero-mean uncorrelated process noise where e?u1D6FF = [1 01?(?u1D6FF?1)]u1D447, w(u1D45B) is given in (5.21) and
Q(u1D45B) := u1D438[u(u1D45B)uu1D43B(u1D45B)] = ?Q+u1D6FE(u1D45B)e0eu1D4470, ?Q :=
?
? 0?u1D6FF??u1D6FF 0?u1D6FF?u1D444(u1D43F+1)
0u1D444(u1D43F+1)??u1D6FF u1D70E2u1D464Iu1D444(u1D43F+1)
?
?
u1D43D?u1D43D
. (5.39)
The channel output u1D466(u1D45B) in (5.1) can be rewritten by CE-BEM given in (5.2) as
u1D466(u1D45B) = su1D447(u1D45B)
uni005B.alt02
I(u1D43F+1) ?uD4D4(u1D45B)
uni005D.alt02u1D43B
h(u1D45B) +u1D463(u1D45B), (5.40)
where s(u1D45B) = [u1D460(u1D45B) u1D460(u1D45B?1) ??? u1D460(u1D45B?u1D43F)]u1D447 (and uD4D4 (u1D45B) is as defined in (5.23)). Using the
state vector that comprises the information symbols and channel coefficients, the measure-
ment equation can be given as
u1D466(u1D45B) = u1D453[x(u1D45B)] +u1D463(u1D45B), (5.41)
where
u1D453[x(u1D45B)] := xu1D447(u1D45B)
uni005B.alt02
I(u1D43F+1) 0(u1D43F+1)?(u1D43D?u1D43F?1)
uni005D.alt02u1D447 uni005B.alt02
I(u1D43F+1) ?uD4D4(u1D45B)
uni005D.alt02u1D43B uni005B.alt02
0[u1D444(u1D43F+1)]??u1D6FF Iu1D444(u1D43F+1)
uni005D.alt02
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=:D
x(u1D45B).
(5.42)
With (5.35) and (5.41) as the state and measurement equations, respectively, nonlinear
Kalman filtering is applied to track x(u1D45B) for joint channel estimation and equalization.
87
Fixed-Lag Soft Input Extended Kalman Filtering
In EKF, the state transition and observation models need not be linear functions of the
state but may instead be (differentiable) functions. The state function can be used to com-
pute the predicted state from the previous estimate and similarly the measurement function
can be used to compute the predicted measurement from the predicted state. However, these
functions cannot be applied to the covariance directly. Instead a matrix of partial derivatives
(Jacobian) is computed. At each time-step the Jacobian is evaluated with current predicted
states. Those matrices can be used in the Kalman filter equations. This process essentially
linearizes the non-linear function around the current estimate.
For the nonlinear system represented by (5.35) and (5.41), EKF is applied to track the
channel BEM coefficients and to decode data symbols jointly. The EKF is initialized with
?x(?1 ? ?1) = 0 and P(?1 ? ?1) = ?Q (5.43)
where?x(u1D45D?u1D45A) denotes the estimate ofx(u1D45D) given the observations{y(0),y(1),??? ,y(u1D45A)},
and P(u1D45D?u1D45A) denotes the error covariance matrix of ?x(u1D45D?u1D45A), defined as
P(u1D45D?u1D45A) := u1D438{[?x(u1D45D?u1D45A)?x(u1D45D)][?x(u1D45D?u1D45A)?x(u1D45D)]u1D43B}. (5.44)
Extended Kalman recursive filtering (for u1D45B = 0,1,2,???) is applied as in [68] but with a
different state and measurement equations, to generate ?x(u1D45B?u1D45B) and P(u1D45B?u1D45B). The following
steps are executed:
1. Time update:
?x(u1D45B?u1D45B?1) = u1D4AF ?x(u1D45B?1 ?u1D45B?1) +e0?u1D460(u1D45B), (5.45)
P(u1D45B?u1D45B?1) = u1D4AFP(u1D45B?1 ?u1D45B?1)u1D4AF u1D447 + ?Q+u1D6FE(u1D45B)e0eu1D4470. (5.46)
2. Kalman gain:
u1D702(u1D45B) = ?u1D453 [x]?x
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglex=?x(u1D45B?u1D45B?1) = ?x
u1D447(u1D45B?u1D45B?1)uni0028.alt02D+Du1D447uni0029.alt02 ... Jacobian matrix, (5.47)
K(u1D45B) = P(u1D45B?u1D45B?1)u1D702u1D43B(u1D45B)/
uni005B.alt02
u1D70E2u1D463 +u1D702(u1D45B)P(u1D45B?u1D45B?1)u1D702u1D43B(u1D45B)
uni005D.alt02
. (5.48)
88
Figure 5.3: Structure of the adaptive SfiSfo equalizer proposed in [68]
3. Measurement update:
?x(u1D45B?u1D45B) = ?x(u1D45B?u1D45B?1) +K(u1D45B)(u1D466(u1D45B)?u1D453 [?x(u1D45B?u1D45B?1)]), (5.49)
P(u1D45B?u1D45B) = [Iu1D43D ?K(u1D45B)u1D702(u1D45B)]P(u1D45B?u1D45B?1). (5.50)
The a priori information {?u1D460(u1D45B),u1D6FE(u1D45B)} is the soft input at time u1D45B acquired via (5.16) and
(5.17), while u1D6FF-th element of the estimate ?x(u1D45B+u1D6FF ?u1D45B+u1D6FF) is the delayed a posteriori estimate
of data symbol. [Note that, in order to compensate the estimation errors, we take the noise
variance in (5.48) to be u1D70E2u1D463 + 0.01u1D70E2u1D460 instead of u1D70E2u1D463 in our simulations, which is similar as
Kalman detector in Section 2.4.1 and MMSE-DFF in Section 2.4.2. One can also take this
increase as the stabilizing noise that is useful to solve a problem of EKF underestimating
the true covariance matrix.]
Structure of Adaptive soft-in soft-out Equalizer
The fixed-lag EKF takes soft inputs and generates a delayed a posteriori estimate
for u1D460(u1D45B). In order to generate extrinsic estimate independent of the a priori information
{?u1D460(u1D45B),u1D6FE(u1D45B)}, a ?comb? structure in conjunction with the EKF in Fig. 5.3 is used for SfiSfo
equalization, just as in [68]. At each time u1D45B, the vertical branch composed of (u1D6FF+1) EKF?s
produce the extrinsic estimate ?u1D460(u1D45B), while the horizontal branch keeps updating the a pos-
teriori estimate ?x(u1D45B ? u1D45B) and its error covariance P(u1D45B ? u1D45B). The first vertical EKF has an
input {0,1} in place of {?u1D460(u1D45B),u1D6FE(u1D45B)} to exclude the effect of the a priori information. Let
?xu1D452(u1D45B + u1D456 ? u1D45B + u1D456) and Pu1D452(u1D45B + u1D456 ? u1D45B + u1D456) denote the state estimate and its error covariance
matrix, respectively, generated by the (u1D456+1)-th vertical filtering branch. Then the extrinsic
89
estimate ?u1D460(u1D45B) of u1D460(u1D45B) and its error variance u1D70E2(u1D45B) are given by
?u1D460(u1D45B) = u1D6FF-th component of vector ?xu1D452(u1D45B+u1D6FF ?u1D45B+u1D6FF), (5.51)
u1D70E2(u1D45B) = (u1D6FF,u1D6FF)-th component of matrix Pu1D452(u1D45B+u1D6FF ?u1D45B+u1D6FF). (5.52)
Note that the extrinsic outputs ?u1D460(u1D45B) and u1D70E2(u1D45B) are computed for data symbol u1D451(u1D45B), not for
training symbol u1D461(u1D45B), and then used in the later parts of the turbo-equalization receiver (see
Fig. 5.2). Further details regarding generation of extrinsic estimates can be found in [68].
Computational Complexity
Here we consider computational complexity using the floating point operation (flop)
counting for turbo equalization using EKF and CE-BEM. The computational complexity
of the approach of [68] is u1D4AA((u1D6FF + 2)[?u1D6FF +u1D443(u1D43F+ 1)]2) where u1D6FF is the equalization delay, ?u1D6FF is
given by (5.32) and an AR(P) channel model is used. Note that it is independent of the
constellation size ?. As we follow [68] with the difference that we use CE-BEM instead
of AR modeling of the channel, the computational complexity of our proposed approach
readily follows as u1D4AA((u1D6FF+ 2)[?u1D6FF+u1D444(u1D43F+ 1)]2) = u1D4AA((u1D6FF+ 2)u1D43D2) where u1D444 is the number of basis
functions in the CE-BEM. Therefore, the proposed approach and the approach of [68] have
comparable computational complexity if one takes u1D443 = u1D444. As in [68], the computational
complexity of our proposed approach is independent of the constellation size ?. In the
simulations presented in Section 5.2.3, we have u1D6FF = 5, u1D43F = 2 and ?u1D6FF = 6. For CE-BEM,
we take u1D444 = 5 or u1D444 = 9, therefore, corresponding values of the AR model order u1D443 in the
approach of [68] were picked as 5 or 9 to attain comparable computational requirements for
a fair performance comparison.
5.2.3 Simulation Examples
A random time- and frequency-selective Rayleigh fading channel is considered. We
assume ?(u1D45B;u1D459) are zero-mean, complex Gaussian, and spatially white with autocorrelation
u1D70E2?. We take u1D43F = 2 (3 taps) in (5.1), and u1D70E2? = 1/(u1D43F+ 1). For different u1D459?s, ?(u1D45B;u1D459)?s
are mutually independent and satisfy Jakes? model. To this end, we simulate each single
tap following [75](with a correction in the appendix of [63]). We consider a communication
system with carrier frequency of 2GHz, data rate of 40kBd (kilo-Bauds), thereforeu1D447u1D460 = 25u1D707s,
and a varying Doppler spread u1D453u1D451 in the range of 40 to 400Hz, or the normalized Doppler
spread u1D453u1D451u1D447u1D460 from 0.001 to 0.01. The additive noise is zero-mean complex white Gaussian.
The (receiver) SNR refers to the average energy per symbol over one-sided noise spectral
density.
90
4 6 8 10 12 14 16?20
?18
?16
?14
?12
?10
?8
?6
?4
?2
0
SNR(dB)
NCMSE (dB)
QPSK,L=2,d=5,lp=5,ls=20,fd=400Hz,1000runs
 
 
1st iteration
5th iteration
TE?LE
TE?AR5
TE?AR9
TE?BEM(200)
TE?BEM(400)
(a) NCMSE vs SNR
4 6 8 10 12 14 1610
?6
10?5
10?4
10?3
10?2
10?1
100
SNR(dB)
BER
QPSK,L=2,d=5,lp=5,ls=20,fd=400Hz,1000runs
 
 
1st iteration
2nd iteration
5th iteration
TE?LE
TE?AR5
TE?AR9
TE?BEM(200)
TE?BEM(400)
TrueCH
Opt?MAP?TrueCH
(b) BER vs SNR, with equalization delay u1D451 = 5
Figure 5.4: Turbo equalization: performance comparison for SNR?s under u1D453u1D451u1D447u1D460 = 0.01,u1D459u1D45D =
5,u1D459u1D460 = 20 (20% training overhead).
In the simulations, we use a 4-state convolutional code of rate u1D445u1D450 = 1/2 with octal
generators (5,7). The information block size is set to 3000 bits (u1D447u1D456=3000) leading to a coded
block size of 6000 bits, and the interleaver size is equal to the coded block size. In the
modulator, the QPSK constellation with Gray mapping is used, which gives ? = 4 and a
block size of 3000 symbols. After modulation, training symbol sequences of length u1D459u1D45D are
inserted in front of every u1D459u1D460 data symbols, leading to a sequence of length u1D447u1D45F = 3750 when
u1D459u1D45D = 5 and u1D459u1D460 = 20 (20% training overhead).
We compared the following schemes:
1. The approach of [48] that uses the linear MMSE equalizer (e.g. [38]) coupled with
modified RLS channel estimation, where we set the linear filter length = 3 (6 pre-
cursor taps and 3 post-cursor taps are used). This scheme is denoted by ?TE-LE?.
2. The AR(P) model-based scheme in [68]. The AR(P) model is as described in Sec-
tion 2.2.1 and is fitted using [27] to Jakes? spectrum with u1D453u1D451u1D447u1D460=0.01 (the maximum
anticipated normalized Doppler spread), denoted by ?TE-AR5? for AR(5) model and
?TE-AR9? for AR(9) model.
3. The proposed BEM-based turbo equalization schemes, where we consider BEM period
u1D447 = 200 and 400 respectively, so that u1D444 = 5 and 9, respectively, by (5.2). For the
channel BEM coefficients, we take the AR-coefficient in (5.30) asu1D6FC = 0.996 foru1D447 = 200
and u1D6FC = 0.998 for u1D447 = 400. This scheme is denoted by ?TE-BEM(200)? for u1D447 = 200
and ?TE-BEM(400)? for u1D447 = 400.
91
1 2 3 4 5 6 7 8 9 10
x 10?3
?20
?18
?16
?14
?12
?10
?8
?6
?4
?2
0
Normalized Doppler spread (fdTs)
NCMSE (dB)
QPSK,L=2,d=5,lp=5,ls=20,SNR=10dB,1000runs
 
 
1st iteration
5th iteration
TE?LE
TE?AR5
TE?AR9
TE?BEM(200)
TE?BEM(400)
(a) NCMSE vs u1D453u1D451u1D447u1D460
1 2 3 4 5 6 7 8 9 10
x 10?3
10?6
10?5
10?4
10?3
10?2
10?1
100
Normalized Doppler spread (fdTs)
BER
QPSK,L=2,d=5,lp=5,ls=20,SNR=10dB,1000runs
 
 
1st iteration
5th iteration
TE?LE
TE?AR5
TE?AR9
TE?BEM(200)
TE?BEM(400)
(b) BER vs u1D453u1D451u1D447u1D460, with equalization delay u1D451 = 5
Figure 5.5: Turbo equalization: performance comparison for normalized Doppler
spread(u1D453u1D451u1D447u1D460)?s under SNR = 10dB,u1D459u1D45D = 5,u1D459u1D460 = 20 (20% training overhead).
4. The turbo equalizer based on the fixed-lag Kalman filter with perfect knowledge of the
true channel, denoted by ?TrueCH?.
5. The turbo equalizer based on the optimum trellis-based MAP (BJCR) method [46]
with perfect knowledge of the true channel, denoted by ?Opt-MAP-TrueCH?.
We evaluate the performances of various schemes by considering their normalized chan-
nel mean square error (NCMSE) and their bit error rates (BER). The NCMSE is defined
as
NCMSE :=
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0
vextenddoublevextenddouble
vextenddouble??(u1D456) (u1D45B;u1D459)??(u1D456) (u1D45B;u1D459)
vextenddoublevextenddouble
vextenddouble2
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0??(u1D456) (u1D45B;u1D459)?
2
where ?(u1D456) (u1D45B;u1D459) is the true channel and ??(u1D456) (u1D45B;u1D459) is the estimated channel at the u1D456-th Monte
Carlo run, among total u1D440u1D45F runs. The BER?s are evaluated by employing the equalization
delay u1D6FF = 5, using the decoded information symbol sequences at the turbo-equalization
receiver. All the simulation results are based on 1000 runs.
In Fig. 5.4, the performance of all the above schemes, under normalized Doppler spread
u1D453u1D451u1D447u1D460 = 0.01, are compared for different SNR?s. In Fig. 5.5, those schemes are compared over
varying Doppler spread u1D453u1D451?s, under SNR = 10dB. Other settings of the simulation are same
as Fig. 5.4, including the fact that u1D444 = 5 (for u1D447 = 200) or u1D444 = 9 (for u1D447 = 400), regardless
of the actual u1D453u1D451. It is clear that since the channel variations are well captured by the BEM
coefficients, our proposed TE-BEM approach yields good performance even for ?low? SNR?s.
92
4 6 8 10 12 14 16?20
?18
?16
?14
?12
?10
?8
?6
?4
?2
0
SNR(dB)
NCMSE (dB)
QPSK,L=2,d=5,lp=5,ls=20,fd=160Hz,1000runs
 
 
1st iteration
4th iteration
TE?AR3
TE?AR5
TE?AR9
TE?BEM(100)
TE?BEM(200)
TE?BEM(400)
(a) NCMSE vs SNR
4 6 8 10 12 14 16
10?4
10?3
10?2
10?1
100
SNR(dB)
BER
QPSK,L=2,d=5,lp=5,ls=20,fd=160Hz,1000runs
 
 
1st iteration
4th iteration
TE?AR3
TE?AR5
TE?AR9
TE?BEM(100)
TE?BEM(200)
TE?BEM(400)
(b) BER vs SNR, with equalization delay u1D451 = 5
Figure 5.6: Turbo equalization: performance comparison for SNR?s under u1D453u1D451u1D447u1D460 = 0.004,u1D459u1D45D =
5,u1D459u1D460 = 20 (20% training overhead).
We can expect to save the signal power to get the same BER performance. Note that TE-
BEM with larger block size (u1D447 = 400) has a little better performance than with smaller
one (u1D447 = 200), since CE-BEM with u1D447 = 400 has the more basis functions in (5.2) and
Kalman filtering utilize the past data implicitly(see Section 2.2.2). The NCMSE and BER
for TE-BEM (especially, withu1D447 = 400) vary ?slightly? along the range of normalized Doppler
spread, so that given the CE-BEM representation used in channel tracking, its performances
are not sensitive to the actual Doppler spread. Therefore, we do not have to know the exact
Doppler spread of the channel ? an upper bound of that is sufficient in practice.
The performance of TE-AR5 is significantly worse than that of TE-BEM(200) (the two
approaches have comparable computational complexity) in Fig. 5.4 with increasing SNR for
a fixed u1D453u1D451u1D447u1D460 = 0.01, and is slightly worse in Fig. 5.5 for a fixed SNR of 10dB and varying
Doppler spreads. On the other hand, while the performance of TE-AR9 is slightly better
than that of TE-BEM(400) (the two approaches have comparable computational complexity)
in Fig. 5.4 with increasing SNR for a fixed u1D453u1D451u1D447u1D460 = 0.01, it is significantly worse in Fig. 5.5
for a fixed SNR of 10dB and varying Doppler spreads. While increasing the BEM period
u1D447 improves performance, increasing the AR model order does not necessarily do so: we
get inconsistent performance. A possible reason is that, as noted in [27], AR model fitting
to a given correlation function can be numerically ill-conditioned for ?large? model orders;
it turned out to be so for AR(9) model and we followed the recommendations of [27] in
choosing the regularization parameter for the matrix inversion involved. Such inconsistent
behavior is also seen in Fig. 5.6 where we compare performance of various schemes (including
93
4 5 6 7 8 9 10 11 12?15
?10
?5
0
SNR(dB)
NCMSE (dB)
QPSK,L=2,d=5,lp=5,ls=20,T=200,Q=5,fd=400Hz,1000runs
 
 
1st iteration
3rd iteration
5th iteration
Ti=1000
Ti=3000
(a) NCMSE vs SNR
4 5 6 7 8 9 10 11 1210
?6
10?5
10?4
10?3
10?2
10?1
100
SNR(dB)
BER
QPSK,L=2,d=5,lp=5,ls=20,T=200,Q=5,fd=400Hz,1000runs
 
 
1st iteration
3rd iteration
5th iteration
Ti=1000
Ti=3000
(b) BER vs SNR, with equalization delay u1D451 = 5
Figure 5.7: Turbo equalization: performance comparison of TE-BEM(200) with different
interleaver lengths for SNR?s under u1D453u1D451u1D447u1D460 = 0.01,u1D459u1D45D = 5,u1D459u1D460 = 20 (20% training overhead).
TE-BEM(100) with u1D447 = 100 and u1D444 = 3, and AR3 with order u1D443 = 3) for different SNR?s
under normalized Doppler spread u1D453u1D451u1D447u1D460 = 0.004. It is seen that increasing the BEM period
u1D447 improves performance but increasing the AR model order does not necessarily do so.
Moreover, for the same computational complexity, BEM models outperform AR models.
In Figs. 5.4 and 5.5, it is seen that the linear MMSE equalizer coupled with modified RLS
channel estimation (TE-LE) only works for normalized Doppler spread values of ? 0.002.
Note that turbo equalization using linear MMSE equalizer with separate channel estimation
only works well for slow fading channels, no matter what value of the filter length to choose
[68]. In Fig. 5.4, we present the performance of the turbo equalizer based on the fixed-
lag Kalman filter with knowledge of the true channel (TrueCH) in order to illustrate the
effectiveness of the proposed channel estimation approach; as there was little improvement
beyond the second iteration, we only show the second iterative result with dotted curve. It is
seen that there is a slightly more than 2dB SNR penalty due to channel estimation. As has
been noted in the literature, the Kalman filter based equalization is a sub-optimum equalizer
compared to the trellis-based MAP (BCJR) equalizer [46]. We also present the performance
of the turbo equalizer based on the optimum BCJR method with knowledge of the true
channel (Opt-MAP-TrueCH) in order to illustrate loss in performance due to suboptimality
of the Kalman equalizer; as there was little improvement beyond the second iteration, we
only show the second iterative result with dotted curve. It is seen that while there is a
large difference in performance initially (see 1st iteration results for TrueCH and Opt-MAP-
TrueCH), just one turbo iteration yields very close performance (see the two dotted curves).
94
0.993 0.994 0.995 0.996 0.997 0.998 0.999 1?20
?18
?16
?14
?12
?10
?8
?6
?4
?2
0
?
NCMSE (dB)
TE?BEM,T=200,Q=5,QPSK,L=2,d=5,lp=5,ls=20,SNR=10dB,fd=400Hz,1000runs
 
 
iter1
iter2
iter3
(a) NCMSE vs u1D6FC
0.993 0.994 0.995 0.996 0.997 0.998 0.999 110
?6
10?5
10?4
10?3
10?2
10?1
100
?
BER
TE?BEM,T=200,Q=5,QPSK,L=2,d=5,lp=5,ls=20,SNR=10dB,fd=400Hz,1000runs
 
 
iter1
iter2
iter3
(b) BER vs u1D6FC with equalization delay u1D451 = 5
Figure 5.8: Turbo equalization: performances of TE-BEM(200) for AR(1) coefficient u1D6FC?s
under u1D453u1D451u1D447u1D460 = 0.01,SNR = 10dB,u1D459u1D45D = 5,u1D459u1D460 = 20 (20% training overhead).
That is, at least for this example, performance loss in using Kalman equalizer instead of the
BCJR equalizer is quite negligible.
In Fig. 5.7, a smaller information block size (u1D447u1D456 = 1000) in BICM transmitter is consid-
ered leading to a coded block size 2000 bits and interleaver length of 2000 bits also. Thus,
we have a smaller interleaver size compared to 6000 bits in our earlier setting, designed to
reduce the overall delay at turbo equalization receiver output. We compare the performance
of TE-BEM with u1D447 = 200,u1D444 = 5 in CE-BEM for different SNR?s, under normalized Doppler
spread u1D453u1D451u1D447u1D460 = 0.01 with two different interleaver size (equivalently different information
block size). It is seen that a smaller interleaver length results in a ?small? deterioration in
NCMSE and BER (when five iterations are considered).
In Fig. 5.8 and 5.9 we show the BER performance of schemes TE-BEM(200) and TE-
BEM(400) for different values of u1D6FC respectively. It is seen that while the performance is
not sensitive to the choice u1D6FC over a relatively wide range of values, it does deteriorate as
u1D6FC approaches one. Note that u1D6FC = 1 in (5.30) implies time-invariance and u1D6FC < 1 permits
tracking by discounting older values of the channel BEM coefficients ? smaller the value of u1D6FC
higher this discounting effect but discrepancy with the value of u1D6FC obtained from (5.31) also
increases.
5.2.4 EXIT Chart Analysis
The extrinsic information transfer (EXIT) chart is a useful semi-analytic tool [37,47,59]
to analyze the exchange of mutual information between the equalizer and the decoder and
95
0.993 0.994 0.995 0.996 0.997 0.998 0.999 1?20
?18
?16
?14
?12
?10
?8
?6
?4
?2
0
?
NCMSE (dB)
TE?BEM,T=400,Q=9,QPSK,L=2,d=5,lp=5,ls=20,SNR=10dB,fd=400Hz,1000runs
 
 
iter1
iter2
iter3
(a) NCMSE vs u1D6FC
0.993 0.994 0.995 0.996 0.997 0.998 0.999 110
?6
10?5
10?4
10?3
10?2
10?1
100
?
BER
TE?BEM,T=400,Q=9,QPSK,L=2,d=5,lp=5,ls=20,SNR=10dB,fd=400Hz,1000runs
 
 
iter1
iter2
iter3
(b) BER vs u1D6FC, with equalization delay u1D451 = 5
Figure 5.9: Turbo equalization: performances of TE-BEM(400) for AR(1) coefficient u1D6FC?s
under u1D453u1D451u1D447u1D460 = 0.01,SNR = 10dB,u1D459u1D45D = 5,u1D459u1D460 = 20 (20% training overhead).
to describe the convergence of the iterative receiver algorithm. The EXIT chart makes it
possible to predict the system trajectory from extrinsic mutual information transfer functions
without performing simulations on the complete iterative receiver. The (extrinsic) mutual
information u1D43C(u1D43F;u1D450) between the equally likely u1D450 ? {+1,?1} and the symmetric LLR u1D43F
simplifies to [37,59]
u1D43C(u1D43F;u1D450) = 1?u1D438
uni005B.alt02
log2(1 +u1D452?u1D43F) ?u1D450 = +1
uni005D.alt02
. (5.53)
Under ergodicity, for a large sample of size u1D447u1D45F, we have [37]
u1D43C(u1D43F;u1D450) ? 1?u1D447?1u1D45F
u1D447u1D45Funi2211.alt02
u1D461=1
log2(1 +u1D452?u1D450(u1D461)u1D43F{u1D450(u1D461)}). (5.54)
We observe the mutual information u1D43Cu1D440u1D452 = u1D43C(u1D43Fu1D440u1D452 {u1D450(u1D45B)};u1D450(u1D45B)) at the equalizer output and
u1D43Cu1D437u1D452 = u1D43C(u1D43Fu1D437u1D452 {u1D450(u1D45B?)};u1D450(u1D45B?)) at the decoder output. The EXIT chart combines the equalizer
transfer function and the decoder transfer function. Since the output LLR?s from the equal-
izer are input to the decoder and vice versa, both transfer functions are drawn in the same
plot with the axes being flipped for the decoder transfer function. The system trajectory
of the turbo equalization receiver forms a ?zigzag-path? between the two transfer functions
where each equalization (or decoding) task is represented as a vertical (or horizontal) arrow.
The simulation setup to generate the extrinsic information transfer function is shown in
Fig. 5.10. Following [59] (and others),u1D43Fu1D440u1D452 {u1D450u1D456(u1D45B?)}(input to the SfiSfo decoder) is modeled as
independent and identically distributed (i.i.d.) Gaussian with mean u1D450u1D456(u1D45B?)u1D70E2u1D43F/2 and variance
96
(a) Decoder
(b) Equalizer
Figure 5.10: Simulation setup for generating extrinsic information transfer functions
u1D70E2u1D43F; then mutual informationu1D43Cu1D440u1D452 andu1D43Cu1D437u1D452 at the input and output, respectively, of the decoder
are functions of a single parameter u1D70Eu1D43F. For a range of values of u1D70Eu1D43F and randomly generated
u1D43Fu1D440u1D452 {u1D450u1D456(u1D45B?)}, we can estimate u1D43Cu1D440u1D452 and u1D43Cu1D437u1D452 (the same for all channel models) via simulations
using (5.54). The interleaved random extrinsic LLR?s u1D43Fu1D437u1D452 {u1D450u1D456(u1D45B)} are input to the ?LLR to
symbol? block in Fig. 5.10b together with the corresponding a priori LLR u1D43Fu1D440u1D452 {u1D450u1D456(u1D45B)}, the
input LLR?s of the decoder, in order to obtain the a posteriori LLR?s u1D43Fu1D44E{u1D450u1D456(u1D45B)}. Then we
can estimate input-output mutual information u1D43Cu1D437u1D452 and u1D43Cu1D440u1D452 of the equalizer (dependent upon
the channel model) using the input LLR?s u1D43Fu1D437u1D452 {u1D450u1D456(u1D45B)} (not u1D43Fu1D44E{u1D450u1D456(u1D45B)}) and output LLR?s
u1D43Fu1D440u1D452 {u1D450u1D456(u1D45B)}, respectively, of a given SfiSfo equalizer. For a given equalizer we plot curves
(transfer function) with input u1D43Cu1D437u1D452 along horizontal axis and output u1D43Cu1D440u1D452 along the vertical
axis; the axes are ?flipped? for the decoder. The iteration process between equalizer and
decoder can be visualized by using a trajectory trace where each vertical trace represents
equalization task and each horizontal trace represents decoding task and the trajectory starts
at the (0,0) point (see Fig. 5.11 for instance).
Using the set-up and parameters of Fig. 5.5 but with the information block size set to
30000 bits (the coded block size u1D447u1D45F = 60000), the normally distributed LLR?s were generated
with values of u1D70E2u1D43F ? [10?2,102], and then u1D43Cu1D440u1D452 and u1D43Cu1D437u1D452 were calculated. We analyze the EXIT
charts of our CE-BEM based approach withu1D447 = 200 andu1D444 = 5 (TE-BEM(200) scheme) and
the symbol-wise AR-model-based approach in [68] using AR(5) model (TE-AR5 scheme). In
97
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA for Equalizer (IE for Decoder)
I E for Equalizer (I
A for Decoder)
EXIT charts,TE?BEM,T=200,Q=5,lp=5,ls=20,fdTs=0.008
 
 
SNR=8dB
SNR=10dB
SNR=12dB
SNR=14dB
3rd iteration
2nd iteration
1st iteration
Figure 5.11: EXIT charts of the proposed turbo equalizer using CE-BEM with u1D447 = 200,u1D444 =
5 for different SNR?s under u1D453u1D451u1D447u1D460 = 0.008,u1D459u1D45D = 5,u1D459u1D460 = 20 (20% training overhead).
Fig. 5.11, EXIT charts for TE-BEM(200) are shown under a fixed normalized Doppler spread
u1D453u1D451u1D447u1D460 = 0.008 for different SNR?s. We show the trajectory trace for SNR=10 dB where the
first iteration is not visible as it is ?cramped? in the lower left corner. Note that, as SNR
increases, the output mutual information of the SfiSfo equalizer increases. In Fig. 5.12,
EXIT charts for the symbol-wise AR(P)-based turbo equalizations (TE-AR5, TE-AR9) and
the proposed ones using CE-BEM (TE-BEM(200), TE-BEM(400)) are compared under fixed
u1D453u1D451u1D447u1D460 = 0.004 and 0.008, SNR = 10dB. In Fig. 5.13, EXIT charts for TE-BEM(200) and
TE-AR5 schemes are depicted under SNR = 10dB for different normalized Doppler spreads.
One can compare EXIT charts with the actual simulation performances for every scheme
in Section 5.2.3. Table 5.1 compares the BER?s obtained via full Monte Carlo simulations
(as in Section 5.2.3, Fig. 5.13, with u1D459u1D45D = 5, u1D459u1D460 = 20 and SNR=10 dB) and predicted by
EXIT chart analysis (shown in parentheses). It is seen that while the two sets of BER?s are
?close,? there are discrepancies. One reason for this is that while EXIT charts are based on
the assumption of infinite interleaver length, simulation results are based on finite length
interleaver. Furthermore, drawing of trajectory traces is subject to ?manual? errors.
98
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA for Equalizer (IE for Decoder)
I E for Equalizer (I
A for Decoder)
EXIT charts,lp=5,ls=20,fdTs=0.004,SNR=10dB
 
 
TE?AR5
TE?AR9
TE?BEM(200)
TE?BEM(400)
(a) u1D453u1D451u1D447u1D460 = 0.004
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA for Equalizer (IE for Decoder)
I E for Equalizer (I
A for Decoder)
EXIT charts,lp=5,ls=20,fdTs=0.008,SNR=10dB
 
 
TE?AR5
TE?AR9
TE?BEM(200)
TE?BEM(400)
(b) u1D453u1D451u1D447u1D460 = 0.008
Figure 5.12: EXIT charts for TE-AR5, TE-AR9, TE-BEM(200) and TE-BEM(400) under
fixed u1D453u1D451u1D447u1D460, SNR = 10dB,u1D459u1D45D = 5,u1D459u1D460 = 20 (20% training overhead).
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA for Equalizer (IE for Decoder)
I E for Equalizer (I
A for Decoder)
EXIT charts,TE?AR5,lp=5,ls=20,SNR=10dB
 
 
fdTs=0.002
fdTs=0.004
fdTs=0.006
fdTs=0.008
(a) TE-AR5: turbo equalization of [68] using symbol-
wise AR(5) channel model
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA for Equalizer (IE for Decoder)
I E for Equalizer (I
A for Decoder)
EXIT charts,TE?BEM,T=200,Q=5,lp=5,ls=20,SNR=10dB
 
 
fdTs=0.002
fdTs=0.004
fdTs=0.006
fdTs=0.008
(b) TE-BEM(200): turbo equalization using CE-BEM
with u1D447 = 200,u1D444 = 5
Figure 5.13: EXIT charts for different u1D453u1D451u1D447u1D460?s under SNR = 10dB,u1D459u1D45D = 5,u1D459u1D460 = 20 (20%
training overhead).
99
Table 5.1: Turbo equalization: comparison between actual BER (via simulations) and pre-
dicted BER (via EXIT charts)
Via Monte Carlo runs (predicted by EXIT charts): SNR=10dB
u1D453u1D451 (u1D453u1D451u1D447u1D460) iteration TE-BEM(200) TE-AR5
80u1D43Bu1D467 (0.002)
1st 1.2?10?1 (1.4?10?1) 0.8?10?1 (1.2?10?1)
2nd 2.3?10?2 (2.4?10?2) 8.5?10?3 (1.5?10?3)
3rd 1.9?10?3 (1.5?10?5) 6.3?10?4 (< 10?5)
160u1D43Bu1D467 (0.004)
1st 1.3?10?1 (1.5?10?1) 1.5?10?1 (1.7?10?1)
2nd 2.7?10?2 (5.5?10?2) 4.8?10?2 (9.0?10?2)
3rd 2.1?10?3 (2.0?10?4) 7.0?10?3 (2.4?10?3)
240u1D43Bu1D467 (0.006)
1st 8.5?10?2 (1.1?10?1) 6.6?10?2 (8.5?10?2)
2nd 6.9?10?3 (2.2?10?3) 3.3?10?3 (4.0?10?4)
3rd 1.9?10?4 (< 10?5) 7.1?10?5 (< 10?5)
320u1D43Bu1D467 (0.008)
1st 1.4?10?1 (1.7?10?1) 1.1?10?1 (1.6?10?1)
2nd 3.9?10?2 (7.0?10?2) 1.8?10?2 (4.5?10?2)
3rd 2.9?10?3 (1.3?10?3) 7.2?10?4 (2.5?10?4)
100
5.3 MIMO Turbo Equalization using EKF and CE-BEM
In this section, we extend the approach of Section 5.2 to multi-input multi-output
(MIMO) systems. Several variants of DFE and linear transversal equalizer (LTE), such
as the delayed decision feedback sequence estimator along with schemes like ordered suc-
cessive interference cancellation has been proposed in [3,73]. Considering a coded system,
Partial response equalizers, which effectively shorten the original MIMO channel so as to
follow it by a trellis based equalizers and the SfiSfo iterative Kalman equalizer pursuing
low complexity for MIMO frequency selective fading channels have been proposed in [12]
and [56] respectively. In contrast to [6] based on AR model only, our adaptive turbo equal-
izer takes advantage of BEM in addition to the conventional turbo processing and results in
superior performance, even with smaller SNR?s and over a wide range of Doppler spreads as
demonstrated in simulation examples.
5.3.1 MIMO System Model and Turbo Equalization Receiver
Consider a doubly-selective (time- and frequency-selective) MIMO, finite impulse re-
sponse (FIR) linear channel with u1D43E inputs and u1D441 outputs. Let {su1D458 (u1D45B)} denote u1D458-th user?s
information sequence that is input to the time-varying channel with discrete-time response
{hu1D458 (u1D45B;u1D459)} (channel response for the u1D458-th user at time instance u1D45B to a unit input at time
instance u1D45B?u1D459). Then the symbol-rate noisy u1D441-column channel output vector is given by
(u1D45B = 0,1,...)
y(u1D45B) =
u1D43Euni2211.alt02
u1D458=1
u1D43Funi2211.alt02
u1D459=0
hu1D458 (u1D45B;u1D459)u1D460u1D458 (u1D45B?u1D459) +v(u1D45B) (5.55)
where the u1D441-column vector v(u1D45B) is zero-mean, white, uncorrelated with u1D460u1D458 (u1D45B), complex
Gaussian noise, with the autocorrelation u1D438{v(u1D45B+u1D70F)vu1D43B (u1D45B)} = u1D70E2u1D463Iu1D441u1D6FF(u1D70F). We assume that
{hu1D458 (u1D45B;u1D459)} represents a wide-sense stationary uncorrelated scattering (WSSUS) channel [60],
independent for different u1D458?s. We assume that u1D460u1D458 (u1D45B) is mutually independent and iden-
tically distributed (i.i.d.) with zero mean and variance u1D438{u1D460u1D458 (u1D45B)u1D460?u1D458 (u1D45B)} = u1D70E2u1D460u1D458 = u1D70E2u1D460 for
u1D458 = 1,2,??? ,u1D43E. Define
s(u1D45B) :=
uni005B.alt02
u1D4601(u1D45B) u1D4602(u1D45B) ??? u1D460u1D43E(u1D45B)
uni005D.alt02u1D447
h(u1D45B;u1D459) :=
uni005B.alt02
h1(u1D45B;u1D459) h2(u1D45B;u1D459) ??? hu1D43E(u1D45B;u1D459)
uni005D.alt02
and then we may rewrite (5.55) as
y(u1D45B) =
u1D43Funi2211.alt02
u1D459=0
h(u1D45B;u1D459)s(u1D45B?u1D459) +v(u1D45B). (5.56)
101
In our approach, we use an oversampled CE-BEM for the overall channel variations
and first-order AR model for the BEM coefficients. These two models are described in this
subsection. In CE-BEM [11,14,70], over the u1D456-th block consisting of an observation window
of u1D447u1D435 symbols, the channel is represented as (u1D45B = (u1D456?1)u1D447u1D435,(u1D456?1)u1D447u1D435 + 1,??? ,u1D456u1D447u1D435 ?1 and
u1D459 = 0,1,??? ,u1D43F )
hu1D458(u1D45B;u1D459) =
u1D444uni2211.alt02
u1D45E=1
hu1D458,u1D45E(u1D459)u1D452u1D457u1D714u1D45Eu1D45B, (5.57)
where h(u1D459)u1D458,u1D45E is the u1D441-column time-invariant BEM coefficient vector for u1D458-th user and one
chooses (? is an integer)
u1D447 := ?u1D447u1D435, ? ? 1, (5.58)
u1D444 := 2?u1D453u1D451u1D447u1D447u1D460?+ 1, (5.59)
u1D714u1D45E := 2u1D70Bu1D447 [u1D45E?(u1D444+ 1)/2], u1D45E = 1,2,??? ,u1D444 (5.60)
u1D43F := ?u1D70Fu1D451/u1D447u1D460?, (5.61)
u1D70Fu1D451 andu1D453u1D451 are respectively the delay spread and the Doppler spread, andu1D447u1D460 is the symbol
duration. The BEM coefficients hu1D458,u1D45E(u1D459)?s remain invariant during this block, but are allowed
to change at the next consecutive block; the Fourier basis functions {u1D452u1D457u1D714u1D45Eu1D45B} (u1D45E = 1,2,??? ,u1D444)
are common for every block. If the delay spread and the Doppler spread (or at least their
upper bounds) are known, one can infer the basis functions of the CE-BEM [70]. Treating
the basis functions as known, estimation of a time-varying process is reduced to estimating
the invariant coefficients over a block of u1D447u1D435 symbols.
Bit-Interleaved Coded Modulation (BICM) for multiple users
Figure 5.14: Bit-interleaved coded modulation system model for doubly-selective fading
MIMO channels
102
We consider a BICM transmitter (as in [67]) for a doubly-selective MIMO channel as
shown in Fig. 5.14. At the Transmitter, the input bit stream {b(u1D45B)} for u1D43E users are
processed independently one another. A sequence of information data vector bu1D458(u1D45B?) =
[u1D44F1u1D458(u1D45B?),u1D44F2u1D458(u1D45B?),??? ,u1D44Fu1D4580u1D458 (u1D45B?)] ? {1,0}u1D4580 for u1D458-th input are fed into a convolutional encoder with
a code rate u1D445u1D450 = u1D4580/u1D45B0. The coded output cu1D458(u1D45B?) = [u1D4501u1D458(u1D45B?),u1D4502u1D458(u1D45B?),??? ,u1D450u1D45B0u1D458 (u1D45B?)] ? {1,0}u1D45B0
is passed through a bit-wise random interleaver u1D70B, generating the interleaved coded bit se-
quence cu1D458(u1D45B) = u1D70B[cu1D458(u1D45B?)]. The binary coded bits are then mapped to a signal sequence u1D451u1D458(u1D45B)
over a 2-dimensional signal constellation u1D4B3 of cardinality ? = 2u1D45A by a ?-ary modulator
with an one-to-one binary map u1D707 : {1,0}u1D45A ? u1D4B3. In this section, we only consider the case
of phase-shift keying (PSK) or quadrature amplitude modulation (QAM) with the average
energy of the constellation u1D4B3 to be unity. That is, the signal u1D451u1D458(u1D45B) drawn from u1D4B3 has mean
u1D438{u1D451u1D458(u1D45B)} = 0 and variance u1D438
uni007B.alt02
?u1D451u1D458(u1D45B)?2
uni007D.alt02
= 1. After modulation, we periodically insert short
training sequences into the data symbol sequence. The training symbols u1D461u1D458(u1D45B), which are
known to the receiver, are randomly drawn from the signal constellation u1D4B3 with equal prob-
abilities. The {u1D460u1D458(u1D45B)} will be used to denote the symbol sequence after training({u1D461u1D458(u1D45B)})
insertion into data symbol sequence({u1D451u1D458(u1D45B)}).
MIMO Turbo Equalization Receiver
Figure 5.15: MIMO turbo equalization receiver. Following [10,49,68], a posteriori LLR
Lu1D44E{cu1D458(u1D45B)} = Lu1D440u1D452 {cu1D458(u1D45B)} + Lu1D437u1D452 {cu1D458(u1D45B)} is input to the LLR-to-symbol block. Inclusion of
Lu1D440u1D452 {c(u1D45B)} to create a posteriori LLR is shown via dashed line.
A turbo equalization structure, as depicted in Fig. 5.15, is employed in the receiver
as used in [68] except that [68] uses symbol-wise AR models. The adaptive SfiSfo equalizer
is embedded into the iterative decoding (ID) process of the BICM transmission system
103
(BICM-ID) [67]. In each decoding iteration, the equalizer takes the training symbols and
the soft decision information about data symbols supplied by the SfiSfo decoder from the
previous iteration as its a priori information to perform joint adaptive channel estimation
and equalization. The equalizer produces the soft-valued extrinsic estimate of the data
symbols, which are independent of their a priori information.
The output of the equalizer is an updated sequence of soft estimates ?d(u1D45B) and its error
covariance u1D70Eu1D70E2(u1D45B) for u1D43E users. Using the adaptive SfiSfo equalizer in Section 5.3.2, we have
an extrinsic information for the data symbols d(u1D45B). The training symbols in the received
signal are removed at the SfiSfo equalizer and the following iterative process is only for
data symbols since training symbols are known to the receiver. Note that the extrinsic
output of SfiSfo equalizer
uni007B.alt02?
u1D451u1D458(u1D45B),u1D70E2u1D458(u1D45B)
uni007D.alt02
for u1D458-th user is processed independently in SfiSfo
module set, where SfiSfo demodulation and SfiSfo decoding are implemented. The SfiSfo
demodulator follows [67] whereas the SfiSfo decoder follows the MAP decoding algorithm
(?BCJR?) [72, Section 6.2]. [See the details for the SfiSfo demodulator and the SfiSfo decoder
for u1D458-th user in Section 5.2.1.]
5.3.2 Adaptive Soft-In Soft-Out Nonlinear Kalman Equalizer for Doubly-Selective
MIMO Channels
In this section, we extend a CE-BEM model-based SfiSfo nonlinear Kalman equalizer
in Section 5.2.2 to MIMO systems, implementing joint channel estimation and equalization
where their correlation was (implicitly) considered.
State-Space Model using CE-BEM and a Priori Information
Stack the channel coefficients in (5.57) into into vectors
h(u1D459)u1D45E :=
uni005B.alt02
? (u1D459)11,u1D45E ???? (u1D459)u1D4411,u1D45E ??? ? (u1D459)1u1D43E,u1D45E ???? (u1D459)u1D441u1D43E,u1D45E
uni005D.alt02u1D447
, (5.62)
h(u1D459) :=
uni005B.alt02
h(u1D459)u1D4471 h(u1D459)u1D4472 ??? h(u1D459)u1D447u1D444
uni005D.alt02u1D447
, (5.63)
h :=
uni005B.alt02
h(0)u1D447 h(1)u1D447 ??? h(u1D43F)u1D447
uni005D.alt02u1D447
(5.64)
of size u1D441u1D43E, u1D441u1D43Eu1D444 and u1D43D1 := u1D441u1D43Eu1D444(u1D43F+ 1) respectively. We allow the coefficient vector in
(5.64) to change with ?time? u1D45B, in which case it will be denoted by h(u1D45B). We assume that
the channel BEM coefficients follow an AR model. One could fit a general AR(u1D443) model
with a high value of u1D443, but we seek a ?simple? AR(1) model given by
h(u1D45B) = A1h(u1D45B?1) +w(u1D45B) (5.65)
104
whereA1 = u1D6FCIu1D43D1 is the AR coefficient matrix, and the driving noise vectorw(u1D45B) is zero-mean
complex Gaussian with variance u1D70E2u1D464Iu1D43D1 and statistically independent of h(u1D45B?1). Assuming
the channel is wide-sense stationary (WSS) and coefficients ? (u1D459)u1D45Fu1D458,u1D45E ?s are independent, we have
u1D70E2u1D464 = u1D70E2?(1??u1D6FC?2)/u1D444 (5.66)
where u1D70E2?Iu1D43D1 := u1D438
uni007B.alt02
h(u1D45B;u1D459)hu1D43B (u1D45B;u1D459)
uni007D.alt02
. [Typically u1D6FC < 1 but close to one. Strictly speaking
one should try a ?full? matrix A1 in (5.64); however, it can only be calculated if the MIMO
channel statistics are known ? we do not assume such knowledge here, just an upperbound
on the Doppler and multipath delay spreads. See details in Section 5.2.2]
Define with the equalization with a delay u1D6FF> 0 and u1D43E-column vector s(u1D45B),
?u1D6FF := max{u1D6FF+ 1,u1D43F+ 1} (5.67)
z(u1D45B) :=
uni005B.alt02
su1D447 (u1D45B) su1D447 (u1D45B?1) ??? su1D447
uni0028.alt02
u1D45B? ?u1D6FF+ 1
uni0029.alt02uni005D.alt02u1D447
. (5.68)
In order to apply (extended) Kalman filtering to joint channel estimation and equalization,
we stack h(u1D45B) and data vector z(u1D45B) together into a u1D43D ?1 state vector x(u1D45B) at time u1D45B as
x(u1D45B) :=
uni005B.alt02
zu1D447 (u1D45B) hu1D447 (u1D45B)
uni005D.alt02u1D447
, u1D43D := u1D43E?u1D6FF+u1D441u1D43Eu1D444(u1D43F+ 1) = u1D43E?u1D6FF+u1D43D1. (5.69)
As in [68] (and others), we consider the symbol sequence {s(u1D45B)} as a stochastic process
so as to utilize the soft decisions on the data symbols generated in the iterative decoding
process as its a priori information. We can express u1D458-th user?s information sequence u1D460u1D458(u1D45B)
as
u1D460u1D458(u1D45B) = ?u1D460u1D458(u1D45B) + ?u1D460u1D458(u1D45B), (5.70)
where ?u1D460u1D458(u1D45B) = u1D438[u1D460u1D458(u1D45B)] is a deterministic sequence and ?u1D460u1D458(u1D45B) is approximated as a zero-mean
uncorrelated stochastic process such that u1D438[?u1D460u1D458(u1D45B)?u1D460?u1D458(u1D45B+u1D457)] = u1D6FEu1D458(u1D45B)u1D6FF[u1D457] assuming an ideal
interleaver. Note that ?u1D460u1D458(u1D45B) and u1D6FEu1D458(u1D45B), the statistical characteristics of u1D460u1D458(u1D45B), are provided
via the a priori information. We have ?u1D460u1D458(u1D45B) = ?u1D451u1D458(u1D45B) and u1D6FEu1D458(u1D45B) = u1D6FEu1D451u1D458(u1D45B) for a data symbol
u1D451u1D458(u1D45B), where ?u1D451u1D458(u1D45B) and u1D6FEu1D451u1D458(u1D45B) are given in (5.16) and (5.17) respectively, while ?u1D460u1D458(u1D45B) = u1D461u1D458(u1D45B)
and u1D6FEu1D458(u1D45B) = 0 for a training symbol u1D461u1D458(u1D45B). Using x(u1D45B), the state equation turns out to be
Using the state vector, x(u1D45B) in (5.69), the state equation can be written as
x(u1D45B) = u1D4AFx(u1D45B?1) +??u1D460(u1D45B) +u(u1D45B), (5.71)
105
with the following definitions :
u1D4AF =
?
? ? 0u1D43E?u1D6FF?u1D43D1,
0u1D43D1?u1D43E?u1D6FF F
?
?
u1D43D?u1D43D
, F = u1D6FCIu1D43D1, (5.72)
? =
?
?01?(?u1D6FF?1) 01?1
I(?u1D6FF?1) 0(?u1D6FF?1)?1
?
??Iu1D43E, ? = uni005B.alt02Iu1D43E 0u1D43E?(u1D43D?u1D43E)uni005D.alt02u1D447 . (5.73)
The vector u(u1D45B) is zero-mean uncorrelated process noise, defined as
u(u1D45B) =
uni005B.alt02
?u1D447?u1D6FF ?u1D460(u1D45B) wu1D447(u1D45B)
uni005D.alt02u1D447
, (5.74)
where ??u1D6FF = [1 01?(?u1D6FF?1)]u1D447 ?Iu1D43E, and w(u1D45B) is given in (5.65). The covariance matrix of u(u1D45B) is
given by
Q(u1D45B) = u1D438[u(u1D45B)uu1D43B(u1D45B)] = ?Q+?u1D451u1D456u1D44Eu1D454{u1D6FEu1D6FE(u1D45B)}?u1D447, (5.75)
where u1D451u1D456u1D44Eu1D454{u1D6FEu1D6FE(u1D45B)} is a diagonal matrix composed of u1D6FEu1D458(u1D45B) and matrix ?Q is defined as
?Q =
?
?0u1D43E?u1D6FF?u1D43E?u1D6FF 0u1D43E?u1D6FF?u1D43D1
0u1D43D1?u1D43E?u1D6FF u1D70E2u1D464Iu1D43D1
?
?
u1D43D?u1D43D
. (5.76)
Meanwhile, based on CE-BEM given in (5.57), the channel output y(u1D45B) in (5.56) can
be rewritten as
y(u1D45B) = [S(u1D45B)?Iu1D441]u1D447
uni005B.alt02
I(u1D43F+1) ?(uD4D4(u1D45B)?Iu1D441u1D43E)
uni005D.alt02u1D43B
h(u1D45B) +v(u1D45B), (5.77)
where
S(u1D45B) :=
uni005B.alt02
su1D447(u1D45B) su1D447(u1D45B?1) ??? su1D447(u1D45B?u1D43F)
uni005D.alt02u1D447
, (5.78)
uD4D4(u1D45B) :=
uni005B.alt02
u1D452?u1D457u1D7141u1D45B u1D452?u1D457u1D7142u1D45B ??? u1D452?u1D457u1D714u1D444u1D45B
uni005D.alt02u1D447
. (5.79)
where s(u1D45B) = [u1D4601 (u1D45B) u1D4602 (u1D45B) ??? u1D460u1D43E (u1D45B)]u1D447. Note that the state vector x(u1D45B) comprises S(u1D45B)
and h(u1D45B) in (5.69). Hence, the measurement equation using x(u1D45B) can be given as
y(u1D45B) = f[x(u1D45B)] +v(u1D45B), (5.80)
where the nonlinear function for f[x(u1D45B)] is defined as
f[x(u1D45B)] := [S(u1D45B)?Iu1D441]u1D447
uni005B.alt02
I(u1D43F+1) ?(uD4D4(u1D45B)?Iu1D441u1D43E)
uni005D.alt02u1D43B
h(u1D45B), (5.81)
106
Treating (5.71) and (5.80) respectively as the state and the measurement equations,
nonlinear Kalman filtering can be applied to track the vector x(u1D45B) at each time for the joint
channel estimation and equalization.
Fixed-Lag Soft Input Extended Kalman Filtering
The EKF is applied to (5.71) and (5.80) to track the channel BEM coefficients and to
decode data symbols jointly. Kalman tracking is initialized with
?x(?1 ? ?1) = 0 and P(?1 ? ?1) = ?Q,
where?x(u1D45D?u1D45A) denotes the estimate ofx(u1D45D) given the observations{y(0),y(1),??? ,y(u1D45A)},
and P(u1D45D?u1D45A) denotes the error covariance matrix of ?x(u1D45D?u1D45A), defined as
P(u1D45D?u1D45A) := u1D438{[?x(u1D45D?u1D45A)?x(u1D45D)][?x(u1D45D?u1D45A)?x(u1D45D)]u1D43B}.
Extended Kalman recursive filtering (for u1D45B = 0,1,2,???) is applied one by one via the
following steps:
1. Time update:
?x(u1D45B?u1D45B?1) = u1D4AF ?x(u1D45B?1 ?u1D45B?1) +??s(u1D45B), (5.82)
P(u1D45B?u1D45B?1) = u1D4AFP(u1D45B?1 ?u1D45B?1)u1D4AF u1D447 + ?Q+?u1D451u1D456u1D44Eu1D454{u1D6FEu1D6FE(u1D45B)}?u1D447; (5.83)
2. Kalman gain:
u1D702u1D702u1D702(u1D45B) = ?f [x]?x
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglex=?x(u1D45B?u1D45B?1)(Jacobian matrix), (5.84)
K(u1D45B) = P(u1D45B?u1D45B?1)u1D702u1D702u1D702u1D43B(u1D45B)/
uni005B.alt02
u1D70E2u1D463Iu1D441 +u1D702u1D702u1D702(u1D45B)P(u1D45B?u1D45B?1)u1D702u1D702u1D702u1D43B(u1D45B)
uni005D.alt02
(u1D45B). (5.85)
3. Measurement update:
?x(u1D45B?u1D45B) = ?x(u1D45B?u1D45B?1) +K(u1D45B){y(u1D45B)?f [?x(u1D45B?u1D45B?1)]}, (5.86)
P(u1D45B?u1D45B) = [Iu1D43D ?K(u1D45B)u1D702u1D702u1D702(u1D45B)]P(u1D45B?u1D45B?1). (5.87)
The a priori information {?s(u1D45B),u1D6FEu1D6FE(u1D45B)} is the soft input at time u1D45B, a a priori information
acquired from LLR-to-symbol block in Fig. 5.15, while (u1D43E(u1D6FF ? 1) + u1D458)-th elements (for
u1D458 = 1,2,??? ,u1D43E) of the estimate ?x(u1D45B + u1D6FF ? u1D45B + u1D6FF) is the delayed a posteriori estimate of
107
data symbol for the u1D458-th user. [Note that we increase the noise variance in (5.85) from u1D70E2u1D463
to u1D70E2u1D463 + 0.1u1D70E2u1D460 in our simulations in Section 5.3.3 to compensate the estimation errors.]
Structure of Adaptive soft-in soft-out MIMO Equalizer
The fixed-lag EKF takes soft inputs and generates a delayed a posteriori estimate for
s(u1D45B). However, we need an extrinsic estimate as an output of SfiSfo equalizer at turbo-
equalization receiver in Fig. 5.15. In order to generate extrinsic estimate independent of
the a priori information {?s(u1D45B),u1D6FEu1D6FE(u1D45B)}, we use a ?comb? structure in conjunction with the
EKF in Fig. 5.3, where we have u1D43E-column vectors ?s(u1D45B),u1D6FEu1D6FE(u1D45B) and u1D441-column vector y(u1D45B) for
MIMO systems. At each time u1D45B, the vertical branch composed of (u1D6FF+1) EKFs produce the
extrinsic estimate ?s(u1D45B), while the horizontal branch keeps updating the a posteriori estimate
?x(u1D45B?u1D45B) and its error covariance P(u1D45B?u1D45B). The first vertical EKF has an input {0u1D43E?1,1u1D43E?1}
in place of {?s(u1D45B),u1D6FEu1D6FE(u1D45B)} to exclude the effect of the a priori information. Let ?xu1D452(u1D45B+u1D456?u1D45B+u1D456)
and Pu1D452(u1D45B+u1D456?u1D45B+u1D456) denote the state estimate and its error covariance matrix, respectively,
generated by the (u1D456+ 1)-th vertical filtering branch. Then the extrinsic estimate ?u1D460u1D458(u1D45B) and
its error covariance u1D70E2u1D458(u1D45B) for u1D460u1D458(u1D45B) are given by
?u1D460u1D458(u1D45B) = (u1D43E(u1D6FF?1) +u1D458)-th component of vector ?xu1D452(u1D45B+u1D6FF ?u1D45B+u1D6FF), (5.88)
u1D70E2u1D458(u1D45B) = (u1D43E(u1D6FF?1) +u1D458,u1D43E(u1D6FF?1) +u1D458)-th component of matrix Pu1D452(u1D45B+u1D6FF ?u1D45B+u1D6FF). (5.89)
Computational Complexity
Here we consider computational complexity using the floating point operation (flop)
counting for MIMO turbo equalization in this section. Since the adaptive SfiSfo equalizer
has (u1D6FF+ 2) EKFs at time u1D45B as shown in Fig. 5.3, the computational complexity is given by
u1D4AA((u1D6FF + 2)u1D43D2), where u1D6FF is the equalization delay and u1D43D is the size of state vector x(u1D45B); the
computational complexity of a Kalman filtering is u1D4AA(u1D43D2). When based on AR(P) model [68],
the computational complexity is u1D4AA((u1D6FF+2)
uni005B.alt02
u1D43E?u1D6FF+u1D441u1D43Eu1D443(u1D43F+ 1)
uni005D.alt022
) where ?u1D6FF is given by (5.67).
Note that it is independent of the constellation size ?. As we follow [68] with the difference
that we use CE-BEM instead of AR modeling of the channel, the computational complexity
of our proposed approach readily follows as u1D4AA((u1D6FF+2)
uni005B.alt02
u1D43E?u1D6FF+u1D441u1D43Eu1D444(u1D43F+ 1)
uni005D.alt022
) where u1D444 is the
number of basis functions in CE-BEM. Therefore, the proposed approach and the approach
of [68] have comparable computational complexity if one takes u1D443 = u1D444. In addition, since
the size of state vector using CE-BEM is given by (5.69), the computational complexity of
our proposed approach is also independent of the constellation size ?. In the simulations
presented in Section 5.3.3, we have u1D6FF = 5, u1D43F = 2 and ?u1D6FF = 6. For BEMs we take u1D444 = 5 or
108
u1D444 = 9, therefore, corresponding values of the AR model order u1D443 in the approach of [68] were
picked as 5 or 9 to attain comparable computational requirements for a fair performance
comparison.
5.3.3 Simulation Examples
4 6 8 10 12 14 16?20
?18
?16
?14
?12
?10
?8
?6
?4
?2
0
SNR(dB)
NCMSE (dB)
QPSK,N=2,K=2,L=2,d=5,lp=6,ls=14,fd=400Hz,500runs
 
 
1st iteration
3rd iteration
TE?LE
TE?AR5
TE?AR9
TE?BEM(200)
TE?BEM(400)
(a) NCMSE vs SNR
4 6 8 10 12 14 16
10?4
10?3
10?2
10?1
100
SNR(dB)
BER
QPSK,N=2,K=2,L=2,d=5,lp=6,ls=14,fd=400Hz,500runs
 
 
1st iteration
2nd iteration
3rd iteration
TE?LE
TE?AR5
TE?AR9
TE?BEM(200)
TE?BEM(400)
TrueCH
(b) BER vs SNR, with equalization delay u1D451 = 5
Figure 5.16: MIMO turbo equalization: performance comparison for SNR?s under u1D441 = u1D43E =
2,u1D453u1D451u1D447u1D460 = 0.01,u1D459u1D45D = 6,u1D459u1D460 = 14 (30% training overhead).
A random time- and frequency-selective multi-input multi-output (MIMO) Rayleigh
fading channel is considered. We assume h(u1D45B;u1D459) are zero-mean, complex Gaussian, and
spatially white. We take u1D43F = 2 (3 taps) in (5.56), and u1D70E2? = 1/(u1D43F+ 1). For different
u1D459?s, h(u1D45B;u1D459)?s are mutually independent and satisfy Jakes? model. To this end, we simulate
each single tap following [75] (with a correction in the appendix of [63]). We consider a
communication system with carrier frequency of 2GHz, data rate of 40kBd (kilo-Bauds),
therefore u1D447u1D460 = 25u1D707s, and a varying Doppler spread u1D453u1D451 in the range of 40 to 400Hz, or the
normalized Doppler spreadu1D453u1D451u1D447u1D460 from 0.001 to 0.01. The additive noise is zero-mean complex
white Gaussian. The (receiver) SNR refers to the average energy per symbol over one-sided
noise spectral density.
In the simulations, we consider a simple two-receiver and two-user scenario, i.e., u1D441 =
2,u1D43E = 2 with the same transmitted power. We use a 4-state convolutional code of rate
u1D445u1D450 = 1/2 with octal generators (5,7). The information block size is set to 3000 bits (u1D447u1D456=3000)
leading to a coded block size of 6000 bits, and the interleaver size is equal to the coded block
size. In the modulator, the QPSK constellation with Gray mapping is used, which gives
109
1 2 3 4 5 6 7 8 9 10
x 10?3
?20
?18
?16
?14
?12
?10
?8
?6
?4
?2
0
Normalized Doppler spread (fdTs)
NCMSE (dB)
QPSK,N=2,K=2,L=2,d=5,lp=6,ls=14,SNR=10dB,500runs
 
 
1st iteration
3rd iteration
TE?LE
TE?AR5
TE?AR9
TE?BEM(200)
TE?BEM(400)
(a) NCMSE vs u1D453u1D451u1D447u1D460
1 2 3 4 5 6 7 8 9 10
x 10?3
10?6
10?5
10?4
10?3
10?2
10?1
100
Normalized Doppler spread (fdTs)
BER
QPSK,N=2,K=2,L=2,d=5,lp=6,ls=14,SNR=10dB,500runs
 
 
1st iteration
3rd iteration
TE?LE
TE?AR5
TE?AR9
TE?BEM(200)
TE?BEM(400)
(b) BER vs u1D453u1D451u1D447u1D460, with equalization delay u1D451 = 5
Figure 5.17: MIMO turbo equalization: performance comparison for normalized Doppler
spread(u1D453u1D451u1D447u1D460)?s under u1D441 = u1D43E = 2,SNR = 10dB,u1D459u1D45D = 6,u1D459u1D460 = 14 (30% training overhead).
? = 4 and a block size of 3000 symbols. After modulation, training symbol sequences of
length u1D459u1D45D are inserted in front of every u1D459u1D460 data symbols. We set u1D459u1D45D = 6 and u1D459u1D460 = 14 leading to
30% training overhead.
We compared the following schemes:
1. The approach of [48] that uses the linear MMSE equalizer (e.g. [38]) coupled with
modified RLS channel estimation, where we set the linear filter length = 3 (6 pre-
cursor taps and 3 post-cursor taps are used). This scheme is denoted by ?TE-LE?.
2. The AR(P) model-based scheme in [68]. The AR(P) model is as described in Sec-
tion 2.2.1 and is fitted using [27] to Jakes? spectrum with u1D453u1D451u1D447u1D460=0.01 (the maximum
anticipated normalized Doppler spread), denoted by ?TE-AR5? for AR(5) model and
?TE-AR9? for AR(9) model.
3. The proposed BEM-based turbo equalization schemes, where we consider BEM period
u1D447 = 200 and 400 respectively, so that u1D444 = 5 and 9, respectively, by (5.57). For the
channel BEM coefficients, we take the AR-coefficient in (5.30) asu1D6FC = 0.994 foru1D447 = 200
and u1D6FC = 0.996 for u1D447 = 400. This scheme is denoted by ?TE-BEM(200)? for u1D447 = 200
and ?TE-BEM(400)? for u1D447 = 400.
4. The turbo equalizer based on the fixed-lag Kalman filter with perfect knowledge of the
true channel, denoted by ?TrueCH?.
110
4 6 8 10 12 14 16?16
?14
?12
?10
?8
?6
?4
?2
0
SNR(dB)
NCMSE (dB)
QPSK,N=2,K=2,L=2,d=5,lp=6,ls=14,fd=160Hz,500runs
 
 
1st iteration
3rd iteration
TE?AR3
TE?AR5
TE?AR9
TE?BEM(100)
TE?BEM(200)
TE?BEM(400)
(a) NCMSE vs SNR
4 6 8 10 12 14 16
10?4
10?3
10?2
10?1
100
SNR(dB)
BER
QPSK,N=2,K=2,L=2,d=5,lp=6,ls=14,fd=160Hz,500runs
 
 
1st iteration
3rd iteration
TE?AR3
TE?AR5
TE?AR9
TE?BEM(100)
TE?BEM(200)
TE?BEM(400)
(b) BER vs SNR, with equalization delay u1D451 = 5
Figure 5.18: MIMO turbo equalization: performance comparison for SNR?s under u1D441 = u1D43E =
2,u1D453u1D451u1D447u1D460 = 0.004,u1D459u1D45D = 6,u1D459u1D460 = 14 (30% training overhead).
For the MIMO systems, unlike SISO system in Section 5.2.3, we do not have the turbo
equalizer based on the optimum trellis-based MAP (BJCR) method [46] with perfect knowl-
edge of the true channel due to the complexity.
We evaluate the performances of various schemes by considering their normalized chan-
nel mean square error (NCMSE) and their bit error rates (BER). The NCMSE is defined
as
NCMSE :=
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0
vextenddoublevextenddouble
vextenddouble?h(u1D456) (u1D45B;u1D459)?h(u1D456) (u1D45B;u1D459)
vextenddoublevextenddouble
vextenddouble2
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0?h(u1D456) (u1D45B;u1D459)?
2
where h(u1D456) (u1D45B;u1D459) is the true channel and ?h(u1D456) (u1D45B;u1D459) is the estimated channel at the u1D456-th Monte
Carlo run, among total u1D440u1D45F runs. The BER?s are evaluated by employing the equalization
delay u1D6FF = 5, using the decoded information symbol sequences at the turbo-equalization
receiver. All the simulation results are based on 500 runs.
In Fig. 5.16, the performance of all the above schemes, under normalized Doppler spread
u1D453u1D451u1D447u1D460 = 0.01, are compared for different SNR?s. In Fig. 5.17, they are compared over varying
Doppler spread u1D453u1D451?s, under SNR = 10dB. Other settings of the simulation are same as Fig.
5.16, including the fact that u1D444 = 5 (for u1D447 = 200) or u1D444 = 9 (for u1D447 = 400), regardless of
the actual u1D453u1D451. Since the channel variations are well captured by the BEM coefficients, our
proposed TE-BEM approach yields good performance even for ?low? SNR?s, which makes it
possible to save the signal power to get the same BER performance. Note that TE-BEM
with larger block size (u1D447 = 400) has a better performance than with smaller one (u1D447 = 200),
111
4 5 6 7 8 9 10 11 12?15
?10
?5
0
SNR(dB)
NCMSE (dB)
QPSK,N=2,K=2,L=2,d=5,lp=6,ls=14,T=200,Q=5,fd=400Hz,500runs
 
 
1st iteration
2nd iteration
3rd iteration
Ti=1000
Ti=3000
(a) NCMSE vs SNR
4 5 6 7 8 9 10 11 12
10?5
10?4
10?3
10?2
10?1
100
SNR(dB)
BER
QPSK,N=2,K=2,L=2,d=5,lp=6,ls=14,T=200,Q=5,fd=400Hz,500runs
 
 
1st iteration
2nd iteration
3rd iteration
Ti=1000
Ti=3000
(b) BER vs SNR, with equalization delay u1D451 = 5
Figure 5.19: MIMO turbo equalization: performance comparison of TE-BEM(200) with
different interleaver lengths for SNR?s under u1D441 = u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.01,u1D459u1D45D = 6,u1D459u1D460 = 14 (30%
training overhead).
since CE-BEM with u1D447 = 400 has the more basis functions in (5.57) and Kalman filtering
utilize the past data implicitly (see Section 2.2.2). It is clear that the NCMSE and BER of
turbo equalization using the CE-BEM for the channel tracking (TE-BEM) are not sensitive
to the actual Doppler spread. Especially, TE-BEM with u1D447 = 400,u1D444 = 9 seems so realizable
over all the range of normalized Doppler spread. Therefore, we can neglect the actual
Doppler spread of the channel for TE-BEM, esp., TE-BEM(400). With the comparable
computational complexity, the performance of the symbol-wise AR(u1D443)-based schemes are
significantly worse than that of the proposed turbo equalization using CE-BEM. TE-AR5
is than that of TE-BEM(200) in Fig. 5.16 with increasing SNR for a fixed u1D453u1D451u1D447u1D460 = 0.01,
and is slightly worse in Fig. 5.17 for a fixed SNR of 10dB and varying Doppler spreads.
On the other hand, TE-AR9 is significantly worse than that of TE-BEM(400) in Fig. 5.16
with increasing SNR for a fixed u1D453u1D451u1D447u1D460 = 0.01 and in Fig. 5.17 for a fixed SNR of 10dB and
varying Doppler spreads. For the same computational complexity, BEM models outperform
AR models.
Note that while increasing the BEM period u1D447 (and corresponding u1D444) in CE-BEM im-
proves performance, increasing the AR model order does not necessarily do so: we get
inconsistent performance. A possible reason is that, as noted in [27], AR model fitting to
a given correlation function can be numerically ill-conditioned for ?large? model orders; it
turned out to be so for AR(9) model and we followed the recommendations of [27] in choosing
the regularization parameter for the matrix inversion involved. Such inconsistent behavior
112
is also seen in Fig. 5.18 where we compare performance of various schemes (including TE-
BEM(100) with u1D447 = 100 and u1D444 = 3, and AR3 with order u1D443 = 3) for different SNR?s
under normalized Doppler spread u1D453u1D451u1D447u1D460 = 0.004. It is seen that increasing the BEM period
u1D447 improves performance but increasing the AR model order does not necessarily do so.
For the linear MMSE equalizer coupled with modified RLS channel estimation (TE-LE),
it performs well only for the small Doppler spread (u1D453u1D451u1D447u1D460 ? 0.002 in Fig. 5.17), since it uses
the linear MMSE equalizer with separate channel estimation (not joint channel estimation
and equalization). In Fig. 5.16, we present the performance of the turbo equalizer based
on the fixed-lag Kalman filter with knowledge of the true channel (TrueCH) in order to
illustrate the effectiveness of the proposed channel estimation approach; as there was little
improvement beyond the second iteration, we only show the second iterative result with
dotted curve. It is seen that there is a slightly more than 2dB SNR penalty due to channel
estimation. As has been noted in the literature, the Kalman filter based equalization is a
sub-optimum equalizer compared to the trellis-based MAP (BCJR) equalizer [46].
In Fig. 5.19, we compare the performance of TE-BEM(200) for different SNR?s, under
normalized Doppler spread u1D453u1D451u1D447u1D460 = 0.01 with two different interleaver size (equivalently
different information block size). A smaller information block size (u1D447u1D456 = 1000) in BICM
transmitter is considered leading to a coded block size 2000 bits and interleaver length of
2000 bits also. Thus, we have a smaller interleaver size compared to 6000 bits in our earlier
setting, designed to reduce the overall delay at turbo equalization receiver output. It is seen
that a smaller interleaver length results in a ?small? deterioration in NCMSE and BER.
5.3.4 EXIT Chart Analysis
Figure 5.20: Simulation setup for generating extrinsic information transfer functions for
MIMO system
113
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA for Equalizer (IE for Decoder)
I E for Equalizer (I
A for Decoder)
EXIT charts,TE?BEM,T=200,Q=5,N=2,K=2,lp=6,ls=14,fdTs=0.008
 
 
SNR=8dB
SNR=10dB
SNR=12dB
SNR=14dB
1st iteration
2nd iteration
3rd iteration
Figure 5.21: EXIT charts for different SNR?s under u1D441 = u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.008,u1D459u1D45D = 6,u1D459u1D460 = 14
(30% training overhead).
The simulation setup to generate the extrinsic information transfer function is shown
in Fig. 5.20. Note that the data symbol estimate and its error covariance for the u1D458-th
user,
uni007B.alt02?
u1D451u1D458(u1D45B),u1D70E2u1D458(u1D45B)
uni007D.alt02
is processed independently for the SfiSfo demodulation and the SfiSfo
decoding at the receiver, as the input bit stream for u1D458-th user, {bu1D458(u1D45B)} is processed in-
dependently of other users at the transmitter. For EXIT chart for the SfiSfo decoder, we
observe u1D43Fu1D440u1D452 {u1D450u1D456u1D458(u1D45B?)} (input LLR?s to the SfiSfo decoder) and u1D43Fu1D437u1D452 {u1D450u1D456u1D458(u1D45B?)} (output LLR?s to
the SfiSfo decoder). Following [59] (and others), u1D43Fu1D440u1D452 {u1D450u1D456u1D458(u1D45B?)} is modeled as independent and
identically distributed (i.i.d.) Gaussian with mean u1D450u1D458(u1D45B?)u1D70E2u1D43F/2 and variance u1D70E2u1D43F; then mutual
information u1D43Cu1D440u1D452 and u1D43Cu1D437u1D452 at the input and output, respectively, of the decoder are functions of
a single parameter u1D70Eu1D43F. For a range of values of u1D70Eu1D43F and randomly generated u1D43Fu1D440u1D452 {u1D450u1D456u1D458(u1D45B?)}, we
can estimate u1D43Cu1D440u1D452 and u1D43Cu1D437u1D452 (the same for all channel models) via simulations. The interleaved
random extrinsic LLR?s u1D43Fu1D437u1D452 {u1D450u1D456u1D458(u1D45B)} are input to the ?LLR to symbol? block in Fig. 5.20
together with the corresponding a priori LLR u1D43Fu1D440u1D452 {u1D450u1D456u1D458(u1D45B)}, the input LLR?s of the decoder,
in order to obtain the a posteriori LLR?s u1D43Fu1D44E{u1D450u1D456u1D458(u1D45B)}. Then we can estimate input-output
mutual information u1D43Cu1D437u1D452 and u1D43Cu1D440u1D452 of the equalizer (dependent upon the channel model) using
the input LLR?s u1D43Fu1D437u1D452 {u1D450u1D456u1D458(u1D45B)} (not u1D43Fu1D44E{u1D450u1D456u1D458(u1D45B)}) and output LLR?s u1D43Fu1D440u1D452 {u1D450u1D456u1D458(u1D45B)}, respectively, of
a given SfiSfo equalizer. For a given equalizer we plot curves (transfer function) with input
u1D43Cu1D437u1D452 along horizontal axis and output u1D43Cu1D440u1D452 along the vertical axis; the axes are ?flipped? for the
decoder. The iteration process between equalizer and decoder can be visualized by using a
trajectory trace where each vertical trace represents equalization task and each horizontal
114
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA for Equalizer (IE for Decoder)
I E for Equalizer (I
A for Decoder)
EXIT charts,N=2,K=2,lp=6,ls=14,fdTs=0.004,SNR=10dB
 
 
TE?AR5
TE?AR9
TE?BEM(200)
TE?BEM(400)
(a) u1D453u1D451u1D447u1D460 = 0.004
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA for Equalizer (IE for Decoder)
I E for Equalizer (I
A for Decoder)
EXIT charts,N=2,K=2,lp=6,ls=14,fdTs=0.008,SNR=10dB
 
 
TE?AR5
TE?AR9
TE?BEM(200)
TE?BEM(400)
(b) u1D453u1D451u1D447u1D460 = 0.008
Figure 5.22: EXIT charts for TE-AR5, TE-AR9, TE-BEM(200) and TE-BEM(400) under
u1D441 = u1D43E = 2, fixed u1D453u1D451u1D447u1D460, SNR = 10dB,u1D459u1D45D = 6,u1D459u1D460 = 14 (30% training overhead).
trace represents decoding task and the trajectory starts at the (0,0) point (see Fig. 5.21 for
instance).
Using the set-up and parameters of Fig. 5.5 but with the information block size set to
30000 bits (the coded block size u1D447u1D45F = 60000), the normally distributed LLR?s were generated
with values of u1D70E2u1D43F ? [10?2,102], and then u1D43Cu1D440u1D452 and u1D43Cu1D437u1D452 were calculated. We analyze the EXIT
charts of our CE-BEM based approach withu1D447 = 200 andu1D444 = 5 (TE-BEM(200) scheme) and
the symbol-wise AR-model-based approach in [68] using AR(5) model (TE-AR5 scheme). In
Fig. 5.21, EXIT charts for TE-BEM(200) are shown under a fixed normalized Doppler spread
u1D453u1D451u1D447u1D460 = 0.008 for different SNR?s, where the output mutual information of the SfiSfo equalizer
increases as SNR increases. We also show the trajectory trace for SNR=10 dB. In Fig. 5.22,
EXIT charts for the symbol-wise AR(P)-based turbo equalizations (TE-AR5, TE-AR9) and
the proposed ones using CE-BEM (TE-BEM(200), TE-BEM(400)) are compared under fixed
u1D453u1D451u1D447u1D460 = 0.004 and 0.008, SNR = 10dB. In Fig. 5.23, EXIT charts for TE-BEM(200) and
TE-AR5 schemes are depicted under SNR = 10dB for different normalized Doppler spreads.
One can estimate the actual simulation performances in Section 5.3.3 using the corresponding
EXIT charts [Compare Fig. 5.23 and 5.22a with Fig. 5.17 and 5.18 respectively].
5.4 Conclusions
We proposed a turbo equalization receiver based on complex exponential basis expan-
sion model (CE-BEM) for doubly selective fading channels, extending the single-user turbo
115
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA for Equalizer (IE for Decoder)
I E for Equalizer (I
A for Decoder)
EXIT charts,TE?AR5,N=2,K=2,lp=6,ls=14,SNR=10dB
 
 
fdTs=0.002
fdTs=0.004
fdTs=0.006
fdTs=0.008
(a) TE-AR5: Turbo Equalization of [68] using AR(5)
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA for Equalizer (IE for Decoder)
I E for Equalizer (I
A for Decoder)
EXIT charts,TE?BEM,T=200,Q=5,N=2,K=2,lp=6,ls=14,SNR=10dB
 
 
fdTs=0.002
fdTs=0.004
fdTs=0.006
fdTs=0.008
(b) TE-BEM(200): Turbo Equalization using CE-BEM
with u1D447 = 200,u1D444 = 5
Figure 5.23: EXIT charts for different u1D453u1D451u1D447u1D460?s under u1D441 = u1D43E = 2,SNR = 10dB,u1D459u1D45D = 6,u1D459u1D460 = 14
(30% training overhead).
equalization approach of [68] based on symbol-wise AR modeling of channels to channels
based on CE-BEMs where the adaptive equalizer using nonlinear Kalman filters is coupled
with an SfiSfo decoder to iteratively perform equalization and decoding using soft infor-
mation feedback. The proposed adaptive equalizer jointly optimizes the estimation of BEM
channel coefficients and data symbol decoding in each iteration with the assistance of a priori
information for the data symbols given by the SfiSfo decoder. While the BEM coefficients
are updated via the autoregressive (AR) model, the time-varying nature of channels can be
well captured by the CE-BEM, since the time-variations of the (unknown) BEM coefficients
are likely much slower than those of the real channels. Unlike [68], an EXIT chart analysis
of the proposed approach was also provided. Simulation examples demonstrated that our
CE-BEM-based approach had significantly superior performance over the symbol-wise AR
model-based turbo equalizer of [68] for comparable computational complexity.
116
Chapter 6
Decision-Directed Tracking for Doubly-Selective Channel using Basis
Expansion Models
We present a decision-directed tracking approach to doubly-selective channel estimation,
exploiting the complex exponential basis expansion model (CE-BEM) for the overall channel
variations, where we track the BEM coefficients rather than the channel tap gains, aided by
the symbol decisions from a decision-feedback equalizer (DFE). The time-varying nature of
the channel is well captured by the CE-BEM while the time-variations of the (unknown)
BEM coefficients are likely much slower than those of the channel. Two different schemes
are investigated for BEM coefficient tracking based on the first-order autoregressive (AR)
model for the BEM coefficients, including the Kalman filtering scheme and the extended
exponentially-weighted (EW) recursive least-squares (RLS) algorithm. Simulation examples
illustrate the superior performance of our approaches.
6.1 Introduction
Exploiting the detected information symbols as virtual training, decision-directed track-
ing is often utilized in both training-based and blind channel estimation approaches [6,9,
21,53]. Wireless channels, due to multipath propagation and Doppler spread, are character-
ized by frequency- and time-selectivity [60]. Accurate modeling of temporal evolution of the
channel plays a crucial role for estimation and tracking purposes. Among various models
for channel time-variations, the autoregressive (AR) process, particularly the first-order AR
model, is regarded as a tractable formulation to describe a time-varying channel [16].
The AR model, however, does not perform well in predicting a (fast-varying) channel,
which is often required in decision-directed schemes. In fast-fading environments, channel
prediction using an AR model may lead to high estimation variance resulting in erroneous
symbol decisions [6]. With these incorrect decisions, channel estimation error can be fur-
ther worsened, inducing error propagation and even a breakdown in symbol detection [53].
Since decision-directed approaches are sensitive to error propagation, more accurate channel
modeling becomes necessary for tracking fast-fading channels.
In contrast to AR models that describe channel variations on a symbol-by-symbol update
basis, a basis expansion model (BEM) depicts evolutions of the channel over a period (block)
of time, where the time-varying channel taps are expressed as superpositions of time-varying
117
basis functions in modeling Doppler effects, weighted by time-invariant coefficients [11].
Candidate basis functions include complex exponential (Fourier) functions [11,70] leading to
CE-BEM. Since the time-varying nature of the channel can be well captured in a BEM by
the known basis functions, the time-variations of the (unknown) BEM coefficients are likely
much slower than that of the channel, and thus more convenient to track in a fast-fading
environment.
In Chapter 3, a subblock-wise tracking approach was proposed for doubly-selective chan-
nels using time-multiplexed (TM) training. It exploits the CE-BEM for the overall channel
variations, and a first-order AR model to describe the evolutions of the BEM coefficients.
The slow-varying BEM coefficients, rather than the fast-varying channel, are tracked and up-
dated at each training session; during information sessions, channel estimates are generated
by the CE-BEM using the estimated BEM coefficients. The BEM coefficients are updated
via Kalman filtering at each training session.
Based on the subblock approach (as opposed to blockwise approaches of [8,26,41,44]),
decision-directed tracking of doubly-selective channels is investigated. Acting as virtual
training symbols, the information symbol decisions, provided by a DFE (with delay u1D451 ? 0)
utilizing the estimated channel, are used to enhance the estimation of the BEM coefficients,
so that much of the spectrum resource allocated to training can be saved. Although a time
gap still exists between the available symbol decisions and the channel estimates required by
the DFE, it can be successfully bridged by the CE-BEM-based channel prediction, without
incurring much estimation variance.
Decision-directed channel tracking using a polynomial BEM has been investigated in [8],
where the BEM coefficients are updated via the recursive least-squares (RLS) algorithm
within a sliding window; channel estimation using Kalman filtering and polynomial or com-
plex exponential BEM for OFDM systems has been explored in [26,41,44], among which
decision-directed tracking is considered in [26,41]. Our decision-directed scheme updates the
BEM coefficients of much smaller size of (sub)block than BEM period. The periodic training
symbols are still necessary to recover the channel tracking from possible phase ambiguity
due to the error propagation.
Two different decision-directed schemes are investigated exploiting CE-BEM for the
channel variations: the Kalman filtering and the extended EW-RLS algorithm. We assume
that the BEM coefficients (rather than the time-varying channel tap gains) follow a first-order
AR model. The BEM coefficients are tracked and used to generate the channel estimates
by the CE-BEM. The RLS filter is only a special case of a Kalman filter and an extended
RLS scheme is developed with enhanced tracking abilities. These approaches have superior
performances in both channel estimation variance and bit error rate (BER) over the scheme
118
proposed by [6] that relies solely on the AR model. Compared with the approaches in
Chapters 3 and 4, the decision-directed tracking scheme in this chapter achieves better BER
performances with much less training overhead.
6.2 Decision-Directed Tracking using CE-BEM
Figure 6.1: Decision-directed tracking: overlapping blocks that differ by a small step size
u1D45Au1D460.
Consider two overlapping blocks (each consisting of u1D447u1D435 symbols) that differ by only u1D45Au1D460
(1 ?u1D45Au1D460 ?u1D447u1D435) symbols as Fig. 6.1: the ?past? block beginning at time u1D45B0, and the ?present?
block beginning at time u1D45B0 +u1D45Au1D460. Thanks to the significant overlapping of two blocks, one
can expect the BEM coefficients hu1D45E (u1D459)?s in CE-BEM (6.2) to vary only a little from the past
block to the current overlapping one. Therefore, we can estimate channel by updating the
BEM coefficients every u1D45Au1D460 symbols, rather than the anew at every non-overlapping block
as in [70]. Intuitively, with a smaller step size u1D45Au1D460, the BEM coefficients should vary more
slowly. At the receiver, we employ a MMSE-DFE in Section 2.4.2 [40] with delay u1D451 ? 0 to
equalize the estimated channel. In decision-directed tracking, its output symbol decisions
are fed-back to the channel estimator as pseudo-trainings. Note that we need to predict the
channel up to time u1D45B to detect symbol for ?u1D460(u1D45B?u1D451) at the DFE. In CE-BEM (6.2), we can
use the BEM coefficients hu1D45E (u1D459)?s provided by a channel estimator for the current step size u1D45Au1D460
to predict the channel ?h(u1D45B;u1D459) for the following u1D45Au1D460+u1D451 symbols since they are within the same
u1D45D-th overlapping block; the estimated BEM coefficients are shared for u1D447u1D435 symbols starting
u1D45B = u1D45Du1D45Au1D460 in CE-BEM. In this chapter, we propose to update the BEM coefficients in CE-
BEM every u1D45Au1D460 symbols via Kalman filtering or EW-RLS algorithm for the decision-directed
tracking.
Consider a doubly-selective (time- and frequency-selective) single-input multi-output
(SIMO), finite impulse response (FIR) linear channel with u1D441 outputs. Let {u1D460(u1D45B)} denote
a scalar information sequence that is input to the time-varying channel with discrete-time
response {h(u1D45B;u1D459)} (u1D441-column vector channel response at time instance u1D45B to a unit input at
119
time instance u1D45B?u1D459). We assume {u1D460(u1D45B)} is independent and identically distributed (i.i.d.)
with zero mean and variance u1D438{?u1D460(u1D45B)?2} = u1D70E2u1D460. Then the symbol-rate noisy u1D441-column
channel output vector is given by (u1D45B = 0,1,...)
y(u1D45B) =
u1D43Funi2211.alt02
u1D459=0
h(u1D45B;u1D459)u1D460(u1D45B?u1D459) +v(u1D45B) (6.1)
where the u1D441-column vector v(u1D45B) is zero-mean, white, uncorrelated with u1D460(u1D45B), complex
Gaussian noise, with the autocorrelation u1D438{v(u1D45B+u1D70F)vu1D43B (u1D45B)} = u1D70E2u1D463Iu1D441u1D6FF(u1D70F). We assume that
{h(u1D45B;u1D459)} represents a wide-sense stationary uncorrelated scattering (WSSUS) channel [60].
In CE-BEM [11,14,70], over the u1D456-th block consisting of an observation window of
u1D447u1D435 symbols, the channel is represented as (u1D45B = (u1D456? 1)u1D447u1D435,(u1D456? 1)u1D447u1D435 + 1,??? ,u1D456u1D447u1D435 ? 1 and
u1D459 = 0,1,??? ,u1D43F )
h(u1D45B;u1D459) =
u1D444uni2211.alt02
u1D45E=1
hu1D45E(u1D459)u1D452u1D457u1D714u1D45Eu1D45B, (6.2)
where hu1D45E(u1D459) is the u1D441-column time-invariant BEM coefficient vector and one chooses (u1D43E is
an integer)
u1D447 := ?u1D447u1D435, ? ? 1, (6.3)
u1D444? 2?u1D453u1D451u1D447u1D447u1D460?+ 1, (6.4)
u1D714u1D45E := 2u1D70Bu1D447 [u1D45E?(u1D444+ 1)/2], u1D45E = 1,2,??? ,u1D444 (6.5)
u1D43F := ?u1D70Fu1D451/u1D447u1D460?, (6.6)
u1D70Fu1D451 and u1D453u1D451 are respectively the delay spread and the Doppler spread, and u1D447u1D460 is the symbol
duration. The BEM coefficients hu1D45E(u1D459)?s remain invariant during this block, but are allowed
to change at the next consecutive block; the Fourier basis functions {u1D452u1D457u1D714u1D45Eu1D45B} (u1D45E = 1,2,??? ,u1D444)
are common for every block. If the delay spread and the Doppler spread (or at least their
upper bounds) are known, one can infer the basis functions of the CE-BEM [70]. Treating
the basis functions as known, estimation of a time-varying process is reduced to estimating
the invariant coefficients over a block of u1D447u1D435 symbols.
Stack the BEM coefficients in (6.2) into ?tall? vectors as
h(u1D459) :=
uni005B.alt02
h(u1D459)u1D4471 h(u1D459)u1D4472 ??? h(u1D459)u1D447u1D444
uni005D.alt02u1D447
(6.7)
h :=
uni005B.alt02
h(0)u1D447 h(1)u1D447 ??? h(u1D43F)u1D447
uni005D.alt02u1D447
(6.8)
120
of size u1D441u1D444 and u1D440 := u1D441u1D444(u1D43F+ 1) respectively. The coefficient vectors in (6.7) and (6.8) of
the u1D45D-th overlapping block will be denoted by h(u1D459) (u1D45D), and h(u1D45D). Again, we emphasize that
the u1D45D-th block and the (u1D45D+ 1)-st block differ by just u1D45Au1D460 symbols. Since a fading channel
can follow well a Markov model [16], we further assume that the BEM coefficients over each
overlapping block are also Markovian. A simplified formulation is given by the first-order
AR model, i.e.,
h(u1D45D) = A1h(u1D45D?1) +w(u1D45D), (6.9)
where A1 = u1D6FCIu1D440 is the first-order AR coefficient, and the driving noise vector w(u1D45D) is zero-
mean complex Gaussian with variance u1D70E2u1D464Iu1D440 and statistically independent of h(u1D45D?1). [We
have already investigated the theoretical choice for the first-order AR coefficient in Section
3.4, whose results apply here after replacing the subblock size u1D45Au1D44F with the smaller step size
u1D45Au1D460.] Assuming the channel is wide-sense stationary (WSS) and the BEM coefficients ?u1D45E(u1D459)?s
are independent, we have
u1D70E2u1D464 = u1D70E2?(1??u1D6FC?2)/u1D444 (6.10)
where u1D70E2?Iu1D440 := u1D438
uni007B.alt02
h(u1D45B;u1D459)hu1D43B (u1D45B;u1D459)
uni007D.alt02
. [Typically u1D6FC < 1 but close to one. Strictly speaking,
one should try a ?full? matrix A1 in (6.9); however, it can only be calculated if all the channel
statistics are known ? we do not assume such knowledge here, just an upperbound on the
Doppler and multipath delay spreads.]
Define
uD4D4 (u1D45B) :=
uni005B.alt02
u1D452?u1D457u1D7141u1D45B u1D452?u1D457u1D7142u1D45B ??? u1D452?u1D457u1D714u1D444u1D45B
uni005D.alt02u1D447
,
s(u1D45B) :=
uni005B.alt02
u1D460(u1D45B) u1D460(u1D45B?1) ??? u1D460(u1D45B?u1D43F)
uni005D.alt02u1D447
. (6.11)
For u1D45Du1D45Au1D460 ?u1D45B< (u1D45D+ 1)u1D45Au1D460, by (6.1), (6.2), (6.7) and (6.8), the received signal at time u1D45B can
be written as
y(u1D45B) = [s(u1D45B)?Iu1D441]u1D447 [Iu1D43F+1 ?(uD4D4 (u1D45B)?Iu1D441)]u1D43B h(u1D45D) +v(u1D45B).
Further defining
Cu1D456 (u1D45D) := [s(u1D45Du1D45Au1D460 +u1D456)?Iu1D441]u1D447 [Iu1D43F+1 ?(uD4D4 (u1D45Du1D45Au1D460 +u1D456)?Iu1D441)]u1D43B , (6.12)
C(u1D45D) :=
uni005B.alt02
Cu1D4470 (u1D45D) Cu1D4471 (u1D45D) ??? Cu1D447u1D45Au1D460?1 (u1D45D)
uni005D.alt02u1D447
, (6.13)
we have
yu1D45Au1D460 (u1D45D) = C(u1D45D)h(u1D45D) +vu1D45Au1D460 (u1D45D) (6.14)
121
where
yu1D45Au1D460 (u1D45D) :=
uni005B.alt02
yu1D447 (u1D45Du1D45Au1D460) yu1D447 (u1D45Du1D45Au1D460 + 1) ??? yu1D447 ((u1D45D+ 1)u1D45Au1D460 ?1)
uni005D.alt02u1D447
and vu1D45Au1D460 (u1D45D) is defined likewise.
6.2.1 Decision-Directed Kalman Tracking
The dynamical state-space system model for Kalman filtering is represented by (6.9)
and (6.14), where yu1D45Au1D460(u1D45D) is the measurement of the state of BEM coefficients h(u1D45D). Kalman
filtering can be applied to track the coefficient vector h(u1D45D) at a step size of u1D45Au1D460 symbols.
Since s(u1D45B) is unknown during information sessions, the receiver then switches to a decision-
directed mode, in which the past symbol decisions are assumed to be correct and used in
Kalman tracking. Treating (6.9) and (6.14) as the state and the measurement equations
respectively, the Kalman tracking in the training mode is initialized with
?h(?1 ? ?1) = 0u1D440 and R? (?1 ? ?1) = u1D70E2u1D464Iu1D440,
where ?h(u1D45D?u1D45A) denotes the estimate ofh(u1D45D) given the observations{yu1D45Au1D460 (0),yu1D45Au1D460 (1),??? ,yu1D45Au1D460 (u1D45A)},
and R? (u1D45D?u1D45A) denotes the error covariance matrix of ?h(u1D45D?u1D45A), defined as
R? (u1D45D?u1D45A) := u1D438{[?h(u1D45D?u1D45A)?h(u1D45D)][?h(u1D45D?u1D45A)?h(u1D45D)]u1D43B}.
Kalman recursive filtering (for u1D45D = 0,1,???) is applied via the following steps [32]:
1. Time update:
?h(u1D45D?u1D45D?1) = u1D6FC?h(u1D45D?1 ?u1D45D?1),
R? (u1D45D?u1D45D?1) = ?u1D6FC?2R? (u1D45D?1 ?u1D45D?1) +u1D70E2u1D464Iu1D440;
2. Kalman gain:
Ru1D702 (u1D45D) = C(u1D45D)R? (u1D45D?u1D45D?1)Cu1D43B (u1D45D) +u1D70E2u1D463Iu1D441u1D45Au1D460,
G(u1D45D) = R? (u1D45D?u1D45D?1)Cu1D43B (u1D45D)R?1u1D702 (u1D45D);
3. Measurement update:
?h(u1D45D?u1D45D) = ?h(u1D45D?u1D45D?1) +G(u1D45D)uni005B.alt02yu1D45Au1D460 (u1D45D)?C(u1D45D)?h(u1D45D?u1D45D?1)uni005D.alt02,
R? (u1D45D?u1D45D) = [Iu1D440 ?G(u1D45D)C(u1D45D)]R? (u1D45D?u1D45D?1).
122
In our simulations, we take the noise variance to be u1D70E2u1D463 + 0.01u1D70E2u1D460 instead of u1D70E2u1D463 to com-
pensate the symbol decision errors.
6.2.2 Decision-Directed EW-RLS Tracking
Since the CE-BEM (6.2) is periodic with a period u1D447, the memory of the algorithm
should be less than u1D447 to avoid periodicity of the model influencing the estimation results
as the actual channel is not periodic. Therefore, adaptive algorithms with finite memory
are preferred. A ?finite-memory? algorithm is considered in decision-directed tracking: the
extended exponentially-weighted recursive least-square (EW-RLS) algorithm. Assuming the
BEM coefficients follow the first-order AR model, a state-space model for the extended EW-
RLS algorithm is given by (6.9) and (6.14). [As in Chapter 4, one can use the ?normal? EW-
RLS algorithm with no a priori model for BEM coefficients. However, we adopt the extended
EW-RLS algorithm because it turns out AR(1) modeling for the BEM coefficients limits the
error propagation caused by the incorrect decision-feedbacks which are more severe than the
modeling errors (especially in low SNR?s). Note that subblock-wise channel estimation in
Chapter 4 uses only the training symbols for the channel tracking.]
Based on (6.9) and (6.14), we can apply the extended exponentially-weighted regularized
RLS (EW-RLS) algorithm [2, Section 12.B.(12.B.18)] to track an unknown BEM coefficient
vector h(u1D45D). Choose h(u1D45D) to minimize the cost function
u1D706u1D45D+1u1D6FD?h?2 +
u1D45Duni2211.alt02
u1D456=0
u1D706u1D45D?u1D456?yu1D45Au1D460(u1D456)?C(u1D456)h?2, (6.15)
where u1D6FD > 0 is a regularization parameter and 0 < u1D706 < 1 is the forgetting factor. [This
is the general cost function of the exponentially-weighted regularized RLS. See the details
in [2, Section 12.B] about the equivalent cost function and its solution for the extended
exponentially-weighted regularized RLS that is subject to the state equation in (6.9).] Note
that the forgetting factor is used to give more weight to recent data and less weight to past
data. We take u1D706 to be close to one for updating the small step size u1D45Au1D460.
Mimicking [2, Algorithm 12.B.1], the extended EW-RLS tracking is applied via the
following steps :
1. Initialization:
?h(?1) = 0u1D440?1 and P(?1) = u1D6FD?1Iu1D440
123
2. RLS recursion: For u1D45D = 0,1,???
?(u1D45D) = u1D706Iu1D441u1D45Au1D460 +C(u1D45D)P(u1D45D?1)Cu1D43B(u1D45D),
G(u1D45D) = P(u1D45D?1)Cu1D43B(u1D45D)??1(u1D45D),
?h(u1D45D) = u1D6FC?h(u1D45D?1) +u1D6FCG(u1D45D)uni005B.alt02yu1D45A
u1D460(u1D45D)?C(u1D45D)
?h(u1D45D?1)uni005D.alt02,
P(u1D45D) = u1D706?1?u1D6FC?2 [Iu1D440 ?G(u1D45D)C(u1D45D)]P(u1D45D?1) +u1D70E2u1D464Iu1D440
where ?h(u1D45D) denotes the estimate ofh(u1D45D) given the observations{yu1D45Au1D460 (0),yu1D45Au1D460 (1),??? ,yu1D45Au1D460 (u1D45D)}.
After channel estimation for every u1D45D using Kalman filtering or RLS algorithm, we can
generate the channel by the estimated ?h(u1D45D) via the CE-BEM (6.2). In order to bridge the
time gap between the symbol decisions and the channel estimates required by a MMSE-DFE
with delay u1D451 ? 0, we estimate the channel for the current u1D45Au1D460 symbols as well as following
u1D45Au1D460 +u1D451 symbols, i.e., the channel estimates in our receiver are given by
?h(u1D45B;u1D459) = uD4D4u1D43B(u1D45B)?h(u1D459)(u1D45D), (6.16)
for u1D45B = u1D45Du1D45Au1D460,u1D45Du1D45Au1D460 +1,??? ,(u1D45D+2)u1D45Au1D460 +u1D451?1. The definition of ?h(u1D459) (u1D45D) is similar to (6.7) and
?h(u1D459) (u1D45D) is based on observations up to time u1D45B = (u1D45D+ 1)u1D45Au1D460 ?1.
Using the channel estimates up to time (u1D45D+ 2)u1D45Au1D460+u1D451?1, the symbol detections are made
by DFE for the time u1D45B = u1D45Du1D45Au1D460,u1D45Du1D45Au1D460 + 1,??? ,(u1D45D+ 2)u1D45Au1D460 ?1. Note that the detected symbols
for the time u1D45B = (u1D45D+1)u1D45Au1D460,(u1D45D+1)u1D45Au1D460+1,??? ,(u1D45D+2)u1D45Au1D460?1 are used for the following channel
tracking, whereas the ?current? decisions for the time u1D45B = u1D45Du1D45Au1D460,u1D45Du1D45Au1D460 + 1,??? ,(u1D45D+ 1)u1D45Au1D460 ?1
are updated for improved performance.
Computational Complexity
We compare the computational complexity using the floating point operation (flop)
count for the decision-directed schemes. The detailed flops counts for one iteration of Kalman
filtering are already shown in Section 3.3.1 (EW-RLS algorithm in Section 4.3.1) and DFE
in Table 6.1 respectively. We consider three different channel estimation schemes: ?SB-KF-
BEM? denoting BEM-based subblock-wise Kalman filtering in Section 3, ?DD-KF-ARu1D443?
denoting AR(P) model-based symbol-by-symbol decision-directed Kalman filtering [6], ?DD-
KF-BEM? denoting BEM-based decision-directed Kalman filtering with different step sizes
u1D45Au1D460 = 1,2 and 4. All the above schemes have the same subblock (u1D45Au1D44F = 100) and hence
same training overhead. We take u1D447 = 400 and u1D444 = 9 for CE-BEM. [Since the complexities
of Kalman filtering and EW-RLS algorithm are (almost) the same, we consider Kalman
124
filter only to compare the proposed decision-directed approach with others under the same
environment.]
The number of flops for one cycle of subblock-wise Kalman filtering, decision-directed
Kalman filtering and DFE turn out to be
u1D4531(SB-KF) =u1D4403u1D460 +u1D4402u1D460(3u1D43Fu1D45D + 3) +u1D440u1D460(2u1D43F2u1D45D + 5u1D43Fu1D45D + 1) +u1D43F3u1D45D/3 +u1D43F2u1D45D
u1D4531(DD-KF) =u1D4403 +u1D4402(3u1D441u1D45Au1D460 + 2) +u1D440[2(u1D441u1D45Au1D460)2 + 3u1D441u1D45Au1D460 + 1] + (u1D441u1D45Au1D460)3/3 + (u1D441u1D45Au1D460)2
u1D4531(DFE) =2(u1D441u1D459u1D453)3 + (u1D441u1D459u1D453)2(?u1D459u1D453 + ?u1D459u1D44F + 2) + (u1D441u1D459u1D453)(?u1D459u1D453?u1D459u1D44F + ?u1D4592u1D44F + ?u1D459u1D44F + 1) + 2(?u1D4593u1D44F + ?u1D4592u1D44F + ?u1D459u1D44F) +u1D459u1D44F
where u1D440u1D460 = u1D444(u1D43F + 1) for SB-KF-BEM,u1D440 = u1D441u1D443(u1D43F + 1) for DD-KF-ARu1D443 or u1D441u1D444(u1D43F +
1) for DD-KF-BEM, and u1D43Fu1D45D = u1D43F + 1,?u1D459u1D453 = u1D459u1D453 + u1D43F,?u1D459u1D44F = u1D459u1D44F + 1. The overall flop counts
over one block of u1D447u1D435 symbols for the subblock-wise and decision-directed Kalman tracking
schemes are then given by
u1D453u1D450(SB-KF-BEM) = u1D441(u1D447u1D435/u1D45Au1D44F)u1D4531(SB-KF) + [u1D447u1D435 ?u1D45Au1D461(u1D447u1D435/u1D45Au1D44F)]u1D4531(DFE)
u1D453u1D450(DD-KF-ARu1D443) = u1D447u1D435u1D4531(DD-KF) + 2[u1D447u1D435 ?u1D45Au1D461(u1D447u1D435/u1D45Au1D44F)]u1D4531(DFE),u1D45Au1D460 = 1
u1D453u1D450(DD-KF-BEM) = (u1D447u1D435/u1D45Au1D460)u1D4531(DD-KF) + 2[u1D447u1D435 ?u1D45Au1D461(u1D447u1D435/u1D45Au1D44F)]u1D4531(DFE).
The comparative flop counts for each scheme with one receiver (u1D441 = 1) are compared in
Table 6.2 over one block (u1D447u1D435 = 200). Note that DD-KF-BEM with u1D45Au1D460 = 1 has the same
complexity as DD-KF-ARu1D443 since we select u1D443 = u1D444 = 9.
Table 6.1: DFE: flop count for one cycle.
Operation flops (with u1D70E2u1D460 = 1)
Ru1D466u1D466 = H(u1D45B)Hu1D43B(u1D45B) +u1D70E2u1D463Iu1D441u1D459u1D453 (u1D441u1D459u1D453)2(?u1D459u1D453 + 1)
Ru1D460u1D466 = ?Hu1D43B(u1D45B) (u1D441u1D459u1D453)?u1D459u1D453?u1D459u1D44F
Ru1D6FF = I?u1D459u1D44F ?Ru1D460u1D466(u1D45B)R?1u1D466u1D466 (u1D45B)Ru1D43Bu1D460u1D466(u1D45B) (u1D441u1D459u1D453)3 + (u1D441u1D459u1D453)2?u1D459u1D44F + (u1D441u1D459u1D453)?u1D4592u1D44F
bMMSE(u1D45B) = R?1u1D6FF e0
uni0028.alt02
eu1D4470R?1u1D6FF e0
uni0029.alt02?1
2(?u1D4593u1D44F + ?u1D4592u1D44F + ?u1D459u1D44F)
fMMSE(u1D45B) = R?1u1D466u1D466 (u1D45B)Ru1D43Bu1D460u1D466(u1D45B)bMMSE(u1D45B) (u1D441u1D459u1D453)3 + (u1D441u1D459u1D453)2 + (u1D441u1D459u1D453)?u1D459u1D44F
?u1D460(u1D45B?u1D451) = uni2211.alt01u1D459u1D453?1u1D45A=0fu1D447u1D45A(u1D45B)y(u1D45B?u1D45A)?uni2211.alt01u1D459u1D44Fu1D458=1u1D44Fu1D458(u1D45B)?u1D460(u1D45B?u1D451?u1D458) u1D441u1D459u1D453 +u1D459u1D44F
Total : 2(u1D441u1D459u1D453)3 + (u1D441u1D459u1D453)2(?u1D459u1D453 + ?u1D459u1D44F + 2) + (u1D441u1D459u1D453)(?u1D459u1D453?u1D459u1D44F + ?u1D4592u1D44F + ?u1D459u1D44F + 1) + 2(?u1D4593u1D44F + ?u1D4592u1D44F + ?u1D459u1D44F) +u1D459u1D44F
125
Table 6.2: Subblock-wise and decision-directed tracking with DFE: comparative flop count
over one block of u1D447u1D435 symbols.
tracking schemes comparative flops
SB-KF-BEM 1
DD-KF-BEM, u1D45Au1D460 = 4 4.78
DD-KF-BEM, u1D45Au1D460 = 2 6.78
DD-KF-BEM, u1D45Au1D460 = 1 10.86
DD-KF-AR9 10.86
6.2.3 Simulation Examples
A random time- and frequency-selective Rayleigh fading channel is considered. We
assume h(u1D45B;u1D459) are zero-mean, complex Gaussian, and spatially white with autocorrelation
u1D438
uni007B.alt02
h(u1D45B;u1D459)hu1D43B (u1D45B;u1D459)
uni007D.alt02
= u1D70E2?Iu1D441 for eachu1D459. We takeu1D43F = 2 (3 taps) in (6.1), andu1D70E2? = 1/(u1D43F+ 1).
For different u1D459?s, h(u1D45B;u1D459)?s are mutually independent and satisfy Jakes? model. To this end,
we simulate each single tap following [75] (with a correction in the appendix of [63]).
We consider a communication system with carrier frequency of 2GHz, data rate of
40kBd (kilo-Bauds), therefore u1D447u1D460 = 25u1D707s, and a varying Doppler spread u1D453u1D451 = 400Hz, or
the normalized Doppler spread u1D453u1D451u1D447u1D460 = 0.01 (corresponding to a maximum mobile velocity
216km/h). The additive noise is zero-mean complex white Gaussian. The (receiver) SNR
refers to the average energy per symbol over one-sided noise spectral density.
We evaluate the performances of various schemes by considering their normalized chan-
nel mean square error (NCMSE) and their bit error rates (BER). The NCMSE is defined
as
NCMSE :=
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0
vextenddoublevextenddouble
vextenddouble?h(u1D456) (u1D45B;u1D459)?h(u1D456) (u1D45B;u1D459)
vextenddoublevextenddouble
vextenddouble2
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0?h(u1D456) (u1D45B;u1D459)?
2
where h(u1D456) (u1D45B;u1D459) is the true channel and ?h(u1D456) (u1D45B;u1D459) is the estimated channel at the u1D456-th Monte
Carlo run, among total u1D440u1D45F runs. In each run, a training mode of 200 binary phase-shift
keying (BPSK) symbols is followed by a decision-directed mode of quadrature phase-shift
keying (QPSK) 4000 symbols (u1D447u1D441 = 4000). All the simulation results are based on 500 Monte
Carlo runs, and we consider the performances during the decision-directed mode only.
In the decision-directed mode, training sessions are also periodically sent to facilitate
the EW-RLS tracking. The TM training scheme of [70] is adopted, where each subblock of
equal length u1D45Au1D44F symbols consists of an information session of u1D45Au1D451 symbols and a succeeding
training session of u1D45Au1D461 symbols (u1D45Au1D44F = u1D45Au1D451 +u1D45Au1D461). The training session for each user contains
126
an impulse guarded by zeros (silent periods), which has the structure (u1D6FE > 0)
cu1D45D :=
uni005B.alt02
01?u1D43F u1D6FE 01?u1D43F
uni005D.alt02
. (6.17)
Therefore, u1D45Au1D461 = 2u1D43F+1 = 5, which have to be devoted for training and the remaining, u1D45Au1D451 =
u1D45Au1D44F ?u1D45Au1D461 are available for information symbols. We assume that each information symbol
has unit power, while at every training session given by (6.17), we set u1D6FE = ?2u1D43F+ 1 = ?5
so that the average power per symbol at training sessions is equal to that of information
sessions. We consider a large subblock size u1D45Au1D44F = 100 with less frequent training symbols,
comparing a small subblock size u1D45Au1D44F = 40 with frequent training symbols. For the CE-BEM,
we take u1D447 = 400 and u1D444 = 9 with u1D453u1D451u1D447u1D460 = 0.01.
0 5 10 15 20 25 30?30
?25
?20
?15
?10
?5
0
5
10
SNR(dB)
NCMSE(dB)
QPSK,N=1,K=1,L=2,mb=100,T=400,Q=9,ms=2,fdTs=0.01,500Runs
 
 
SB?RLS?BEM
DD?KF?AR9
DD?RLS?BEM
DD?KF?BEM
PD?RLS?BEM
(a) NCMSE vs SNR
0 5 10 15 20 25 3010
?4
10?3
10?2
10?1
100
SNR(dB)
Bit Error Rate
QPSK,N=1,K=1,L=2,mb=100,T=400,Q=9,ms=2,fdTs=0.01,500Runs
 
 
SB?RLS?BEM
DD?KF?AR9
DD?RLS?BEM
DD?KF?BEM
PD?RLS?BEM
(b) BER vs SNR, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and equal-
ization delay u1D451 = 5
Figure 6.2: Decision-directed tracking: performance comparison for SNR?s, under u1D441 =
1,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 100.
We compared the following schemes:
1. The subblock-wise EW-RLS algorithm with u1D6FD = 1, which is one of the best subblock-
wise approaches in Chapter 4. [One can pick a different subblock-wise scheme using
SW-RLS algorithm or Kalman filter that has similar (or slightly worse) performances.]
In the simulations, we take the forgetting factor u1D706 = 0.5 for u1D45Au1D44F = 100 in EW-RLS
algorithm. This scheme is denoted by ?SB-RLS-BEM?.
2. The decision-directed channel tracking scheme in [6] using the u1D443-th order AR model,
i.e., the time-varying channel follows (assuming each independent channel tap has same
127
0 5 10 15 20 25 30?30
?25
?20
?15
?10
?5
0
5
10
SNR(dB)
NCMSE(dB)
QPSK,N=2,K=1,L=2,mb=100,T=400,Q=9,ms=2,fdTs=0.01,500Runs
 
 
SB?RLS?BEM
DD?KF?AR9
DD?RLS?BEM
DD?KF?BEM
PD?RLS?BEM
(a) NCMSE vs SNR
0 5 10 15 20 25 3010
?6
10?5
10?4
10?3
10?2
10?1
100
SNR(dB)
Bit Error Rate
QPSK,N=2,K=1,L=2,mb=100,T=400,Q=9,ms=2,fdTs=0.01,500Runs
 
 
SB?RLS?BEM
DD?KF?AR9
DD?RLS?BEM
DD?KF?BEM
PD?RLS?BEM
(b) BER vs SNR, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and equal-
ization delay u1D451 = 5
Figure 6.3: Decision-directed tracking: performance comparison for SNR?s, under u1D441 =
2,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 100.
Doppler spread)
h(u1D45B;u1D459) =
u1D443uni2211.alt02
u1D456=1
Au1D456h(u1D45B?u1D456;u1D459) +w(u1D45B), (6.18)
where Au1D456 = u1D6FCu1D456I is the AR coefficients matrix and driving noise w(u1D45B) is zero-mean
complex Gaussian with autocorrelation,u1D438
uni007B.alt02
w(u1D45B)wu1D43B (u1D45B)
uni007D.alt02
= u1D70E2u1D464Iu1D441. Using Yule-Walker
equations, we ?fit? the AR coefficients and noise variance for u1D453u1D451u1D447u1D460 = 0.01. The channel
prediction stage is conducted by using (6.18) omitting the driving noise, i.e.,
?h(u1D45B;u1D459) = u1D443uni2211.alt02
u1D456=1
Au1D456?h(u1D45B?u1D456;u1D459). (6.19)
We take u1D443 = 9 to compare CE-BEM with u1D447 = 400,u1D444 = 9. This scheme is denoted by
?DD-KF-AR9?.
3. The proposed decision-directed tracking scheme with different step sizes in the Kalman
tracking, denoting by ?DD-KF-BEM?. For the CE-BEM, we take the first-order AR
coefficient for the BEM coefficients, u1D6FC = 0.992,0.996 and 0.998 for u1D45Au1D460 = 4,2 and 1
respectively. [We select the u1D6FC values empirically via simulations to get the best per-
formances; the corresponding theoretical values are u1D6FC = 0.9992,0.9998 and 0.9999 in
(6.9) for u1D45Au1D460 = 4,2 and 1 respectively. See the details in Section 3.4 about theoretical
128
0 0.002 0.004 0.006 0.008 0.01?25
?20
?15
?10
?5
0
fdTs
NCMSE(dB)
QPSK,N=2,K=1,L=2,mb=100,T=400,Q=9,ms=2,SNR=14dB,500Runs
 
 
SB?RLS?BEM
DD?KF?AR9
DD?RLS?BEM
DD?KF?BEM
(a) NCMSE vs u1D453u1D451u1D447u1D460
0 0.002 0.004 0.006 0.008 0.0110
?5
10?4
10?3
10?2
10?1
100
fdTs
Bit Erm<r Rate
QPSK,N=2,K=1,L=2,mb=100,T=400,Q=9,ms=2,SNR=14dB,500Runs
 
 
SB?RLS?BEM
DD?KF?AR9
DD?RLS?BEM
DD?KF?BEM
(b) BER vs u1D453u1D451u1D447u1D460, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and equal-
ization delay u1D451 = 5
Figure 6.4: Decision-directed tracking: performance comparison for u1D453u1D451u1D447u1D460?s, under u1D441 =
2,SNR = 14dB,u1D45Au1D44F = 100.
choice for AR(1) coefficient, but replacing the subblock size u1D45Au1D44F with the smaller step
size u1D45Au1D460.]
4. The proposed decision-directed tracking scheme with different step sizes in the extended
EW-RLS tracking with u1D6FD = 1, denoting by ?DD-RLS-BEM?. We take the same u1D6FC value
for each step size as the decision-directed Kalman tracking. We also take the forgetting
factor u1D706 = 0.93,0.96 and 0.98 for u1D45Au1D460 = 4,2 and 1 respectively. [The theoretical choice
for the forgetting factor in EW-RLS tracking is analyzed later in Section 6.4.]
5. Perfect symbol decisions are used as training for the extended EW-RLS channel track-
ing with step size u1D45Au1D460 = 2 and the same other setups as decision-directed RLS tracking.
First, the channel is estimated using exact knowledge of the information symbols. Then
this estimated channel is used to detect the ?unknown? information symbols. There-
fore the ?performance-gap? between ?decision-directed tracking? and ?perfect decision-
directed? in figures shows the performance deterioration by decision-directed approach.
[One can also use Kalman tracking with the perfect symbol decisions which has almost
the same performance.] This scheme provides the baseline for decision-directed track-
ing, denoted by ?PD-RLS-BEM?.
For each of the above schemes, an MMSE-DFE described in Section 2.4.2 is employed at
the receiver, using the obtained channel estimates, with u1D459u1D453 = 8, u1D459u1D44F = 2, and the equalization
delay u1D451 = 5 symbols.
129
0 5 10 15 20 25 30?30
?25
?20
?15
?10
?5
0
5
10
SNR(dB)
NCMSE(dB)
QPSK,N=1,K=1,L=2,mb=100,T=400,Q=9,ms=2,fdTs=0.004,500Runs
 
 
SB?RLS?BEM
DD?KF?AR9
DD?RLS?BEM
DD?KF?BEM
PD?RLS?BEM
(a) NCMSE vs SNR
0 5 10 15 20 25 3010
?4
10?3
10?2
10?1
100
SNR(dB)
Bit Error Rate
QPSK,N=1,K=1,L=2,mb=100,T=400,Q=9,ms=2,fdTs=0.004,500Runs
 
 
SB?RLS?BEM
DD?KF?AR9
DD?RLS?BEM
DD?KF?BEM
PD?RLS?BEM
(b) BER vs SNR, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and equal-
ization delay u1D451 = 5
Figure 6.5: Decision-directed tracking: performance comparison for SNR?s, under u1D441 =
1,u1D453u1D451u1D447u1D460 = 0.004,u1D45Au1D44F = 100.
In Figs. 6.2 and 6.3, the performances of the schemes are compared for different SNR?s,
under normalized Doppler spread u1D453u1D451u1D447u1D460 = 0.01. In Fig. 6.4, the performances are compared
over different normalized Doppler spread u1D453u1D451u1D447u1D460?s for fixed SNR = 14dB with two receivers
(u1D441 = 2); we keep the AR(9) coefficients and u1D444 = 9 for CE-BEM with u1D447 = 400, regardless of
the actual u1D453u1D451 (we do not know it in practice). For the subblock-wise RLS tracking scheme,
frequent training sessions with a small subblock size (for instance u1D45Au1D44F = 40) are required
in order to track the rapid channel variations; SB with a large subblock size u1D45Au1D44F = 100 and
hence less training overhead, does not work. Note that subblock-wise approach depends only
on the training session of size u1D45Au1D461 to estimate channel for one subblock of size u1D45Au1D44F. On the
other hand, the decision-directed tracking performs well with low training overhead for the
fast-varying channel. Our decision-directed tracking has the training symbols to reduce the
error propagation while subblock tracking uses them for channel estimation. Exploiting the
detected information symbols as virtual training, our decision-directed tracking approach
requires less training overhead to achieve satisfactory performance, so that much of the
spectrum resource can be saved. Since the DFE is also a decision-directed device and thus
prone to error propagation, periodic training sessions are still necessary to recover channel
tracking from possible ambiguity. It is clear that since the channel variations are well cap-
tured by the BEM coefficients, our proposed BEM-based decision-directed approaches yield
better performance than the symbol-wise AR-model-based decision-directed approach [6].
The performance deteriorations caused by ?not-a-perfect? decision-directed are also shown
130
0 5 10 15 20 25 30?30
?25
?20
?15
?10
?5
0
5
10
SNR(dB)
NCMSE(dB)
QPSK,N=2,K=1,L=2,mb=100,T=400,Q=9,ms=2,fdTs=0.004,500Runs
 
 
SB?RLS?BEM
DD?KF?AR9
DD?RLS?BEM
DD?KF?BEM
PD?RLS?BEM
(a) NCMSE vs SNR
0 5 10 15 20 25 3010
?6
10?5
10?4
10?3
10?2
10?1
100
SNR(dB)
Bit Error Rate
QPSK,N=2,K=1,L=2,mb=100,T=400,Q=9,ms=2,fdTs=0.004,500Runs
 
 
SB?RLS?BEM
DD?KF?AR9
DD?RLS?BEM
DD?KF?BEM
PD?RLS?BEM
(b) BER vs SNR, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and equal-
ization delay u1D451 = 5
Figure 6.6: Decision-directed tracking: performance comparison for SNR?s, under u1D441 =
2,u1D453u1D451u1D447u1D460 = 0.004,u1D45Au1D44F = 100.
in Figs. 6.2 and 6.3, where better performance in symbol detection (with u1D441 = 2) leads
to less deterioration. In Fig. 6.4, the NCMSE and BER for BEM-based decision-directed
approaches vary only ?slightly? with increasing normalized Doppler spread implying that
its performance is not sensitive to the actual Doppler spread. Therefore, we do not have
to know the exact Doppler spread of the channel ? an upper bound on it is sufficient in
practice. In Figs. 6.5 and 6.6, the performances are compared for the actual u1D453u1D451u1D447u1D460 = 0.004
(slower fading), with the same AR coefficients and CE-BEM (u1D447 = 400,u1D444 = 9) for the upper
bound u1D453u1D451u1D447u1D460 = 0.01; similar results are shown.
Note that, in Figs. 6.2, 6.3, 6.4, 6.6 and 6.5, the proposed BEM-based decision-directed
tracking schemes update the BEM coefficients every step size (u1D45Au1D460 = 2 symbols), which have
less computational complexity than the AR-model-based one; see the flop counts in Table
6.2. As shown in Fig. 6.7, a ?finer? channel tracking can be obtained by reducing step size
u1D45Au1D460 although the computational complexity increases. Fig. 6.7 also shows that the advantage
of multiple-receiver system with u1D441 = 2. At high SNR?s with two receivers, the NCMSE for
a large step size (u1D45Au1D460 = 4 symbols) is better than for a small one (u1D45Au1D460 = 1 symbol) since the
former has more measurements with the symbol decisions close to perfect decisions.
It is hard to say which one is better between Kalman filtering and extended EW-RLS
algorithm for BEM-based decision-directed tracking. For both decision-directed schemes, we
selected an arbitrary value for the first-order AR coefficient u1D6FC via simulations, although the
theoretical choice is given in Section 3.4. In Fig. 6.8, the performances of the BEM-based
131
0 5 10 15 20 25 30?30
?25
?20
?15
?10
?5
0
5
SNR(dB)
NCMSE (dB)
DD?KF?BEM,QPSK,L=2,mb=100,T=400,Q=9,fdTs=0.01,500runs
 
 
N = 1
N = 2
ms = 1
ms = 2
ms = 4
(a) NCMSE vs SNR
0 5 10 15 20 25 3010
?6
10?5
10?4
10?3
10?2
10?1
100
SNR(dB)
BER
DD?KF?BEM,QPSK,L=2,mb=100,T=400,Q=9,fdTs=0.01,500runs
 
 
N = 1
N = 2
ms = 1
ms = 2
ms = 4
(b) BER vs SNR, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and equal-
ization delay u1D451 = 5
Figure 6.7: Decision-directed Kalman tracking: performances with different step size u1D45Au1D460?s
for SNR?s, under u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 100.
decision-directed Kalman tracking is shown for different values of u1D6FC. It is seen that while the
performance is not sensitive to the value of u1D6FC over a relatively wide range of values, it does
deteriorate as u1D6FC approaches one. Typically u1D6FC is close to one (u1D6FC? 1) but not quite one. Note
that u1D6FC = 1 in (6.9) implies time-invariance and u1D6FC< 1 permits tracking by discounting older
values of the channel BEM coefficients ? smaller the value of u1D6FC higher this discounting effect
but discrepancy with the theoretical choice ofu1D6FCin Section 3.4 also increases. For the extended
EW-RLS algorithm, the sensitivity to u1D6FC is similar but a little more due to the forgetting
factor u1D706. In Fig. 6.9, the performances of the BEM-based decision-directed extended EW-
RLS tracking are shown for different values of u1D706 under u1D453u1D451u1D447u1D460 = 0.01, SNR = 14dB. As in Fig.
6.13 where we show the theoretical MSE analysis for u1D706 values, the minima of NCMSE and
BER are similar for the different step sizes. It is also shown that the larger step size leads to
more convenience to select u1D706 for the reasonable performance. One can choose the step size
u1D45Au1D460 considering performance, complexity or convenience. In the following section (Section
6.4), we analyze the mean square error (MSE) of decision-directed EW-RLS tracking and
hence obtain the baseline for the theoretical value for the forgetting factor u1D706 in EW-RLS
algorithm.
132
0.99 0.992 0.994 0.996 0.998 1?25
?20
?15
?10
?5
0
?
NCMSE(dB)
DD?KF?BEM,QPSK,N2K1,L=2,mb=100,T=400,Q=9,fdTs=0.01,SNR=14dB,500runs
 
 
ms=1
ms=2
ms=4
(a) NCMSE vs u1D6FC
0.99 0.992 0.994 0.996 0.998 110
?4
10?3
10?2
10?1
100
DD?KF?BEM,QPSK,N2K1,L=2,mb=100,T=400,Q=9,fdTs=0.01,SNR=14dB,500runs
?
Bit Error Rate
 
 
ms=1
ms=2
ms=4
(b) BER vs u1D6FC with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and equaliza-
tion delay u1D451 = 5
Figure 6.8: Decision-directed Kalman tracking: performances for AR(1) coefficientu1D6FC?s, under
u1D441 = 2,u1D453u1D451u1D447u1D460 = 0.01,SNR = 14dB,u1D45Au1D44F = 100.
6.3 Decision-Directed MIMO Tracking using CE-BEM
Consider a doubly-selective (time- and frequency-selective) multi-input multi-output
(MIMO), finite impulse response (FIR) linear channel with u1D43E inputs and u1D441 outputs. Let
{u1D460u1D458 (u1D45B)} denote u1D458-th user?s information sequence that is input to the time-varying channel
with discrete-time response {hu1D458 (u1D45B;u1D459)} (channel response for the u1D458-th user at time instance
u1D45B to a unit input at time instance u1D45B?u1D459). Then the symbol-rate noisy u1D441-column channel
output vector is given by (u1D45B = 0,1,...)
y(u1D45B) =
u1D43Euni2211.alt02
u1D458=1
u1D43Funi2211.alt02
u1D459=0
hu1D458 (u1D45B;u1D459)u1D460u1D458 (u1D45B?u1D459) +v(u1D45B) (6.20)
where the u1D441-column vector v(u1D45B) is zero-mean, white, uncorrelated with u1D460u1D458 (u1D45B), complex
Gaussian noise, with the autocorrelation u1D438{v(u1D45B+u1D70F)vu1D43B (u1D45B)} = u1D70E2u1D463Iu1D441u1D6FF(u1D70F). We assume that
{hu1D458 (u1D45B;u1D459)} represents a wide-sense stationary uncorrelated scattering (WSSUS) channel [60],
independent for different u1D458?s. We assume that u1D460u1D458 (u1D45B)?s are mutually independent and iden-
tically distributed (i.i.d.), with zero mean and variance u1D438{u1D460u1D458 (u1D45B)u1D460?u1D458 (u1D45B)} = u1D70E2u1D460u1D458 = u1D70E2u1D460 for
u1D458 = 1,2,??? ,u1D43E. Define
s(u1D45B) :=
uni005B.alt02
u1D4601(u1D45B) u1D4602(u1D45B) ??? u1D460u1D43E(u1D45B)
uni005D.alt02u1D447
h(u1D45B;u1D459) :=
uni005B.alt02
h1(u1D45B;u1D459) h2(u1D45B;u1D459) ??? hu1D43E(u1D45B;u1D459)
uni005D.alt02
.
133
0.9 0.92 0.94 0.96 0.98 1?25
?20
?15
?10
?5
0
5
10
DD/EW?RLS,N=2,K=1,L=2,mb=100,T=400,Q=9,fdTs=0.01,SNR=14dB,500runs
? of EW?RLS
NCMSE(dB)
 
 
ms=1
ms=2
ms=4
(a) NCMSE vs u1D706
0.9 0.92 0.94 0.96 0.98 110
?4
10?3
10?2
10?1
100
DD/EW?RLS,N=2,K=1,L=2,mb=100,T=400,Q=9,fdTs=0.01,SNR=14dB,500runs
? of EW?RLS
BER
 
 
ms=1
ms=2
ms=4
(b) BER vs u1D706 with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and equalization
delay u1D451 = 5
Figure 6.9: Decision-directed EW-RLS tracking: performances for forgetting factor u1D706?s,
under u1D441 = 2,u1D453u1D451u1D447u1D460 = 0.01,SNR = 14dB,u1D45Au1D44F = 100.
Then we may rewrite (6.20) as
y(u1D45B) =
u1D43Funi2211.alt02
u1D459=0
h(u1D45B;u1D459)s(u1D45B?u1D459) +v(u1D45B). (6.21)
In CE-BEM [11,14,70], over the u1D456-th block consisting of an observation window of
u1D447u1D435 symbols, the channel is represented as (u1D45B = (u1D456? 1)u1D447u1D435,(u1D456? 1)u1D447u1D435 + 1,??? ,u1D456u1D447u1D435 ? 1 and
u1D459 = 0,1,??? ,u1D43F )
hu1D458(u1D45B;u1D459) =
u1D444uni2211.alt02
u1D45E=1
hu1D458,u1D45E(u1D459)u1D452u1D457u1D714u1D45Eu1D45B, (6.22)
where hu1D458,u1D45E(u1D459) is the u1D441-column time-invariant BEM coefficient vector for u1D458-th user and one
chooses (? is an integer)
u1D447 := ?u1D447u1D435, ? ? 1, (6.23)
u1D444? 2?u1D453u1D451u1D447u1D447u1D460?+ 1, (6.24)
u1D714u1D45E := 2u1D70Bu1D447 [u1D45E?(u1D444+ 1)/2], u1D45E = 1,2,??? ,u1D444 (6.25)
u1D43F := ?u1D70Fu1D451/u1D447u1D460?, (6.26)
u1D70Fu1D451 and u1D453u1D451 are respectively the delay spread and the Doppler spread, and u1D447u1D460 is the symbol
duration. The BEM coefficients hu1D458,u1D45E(u1D459)?s remain invariant during this block, but are allowed
134
to change at the next consecutive block; the Fourier basis functions {u1D452u1D457u1D714u1D45Eu1D45B} (u1D45E = 1,2,??? ,u1D444)
are common for every block. If the delay spread and the Doppler spread (or at least their
upper bounds) are known, one can infer the basis functions of the CE-BEM [70]. Treating
the basis functions as known, estimation of a time-varying process is reduced to estimating
the invariant coefficients over a block of u1D447u1D435 symbols.
Now extend the decision-directed tracking using CE-BEM in Section 6.2 to MIMO
systems. Consider two overlapping blocks (each consistingu1D43E-users? sequences ofu1D447u1D435 symbols)
that differ by only u1D45Au1D460 (1 ? u1D45Au1D460 ? u1D447u1D435) symbols. Since the past block starting at time u1D45B0
and the present block starting at time u1D45B0 +u1D45Au1D460 overlap significantly, the BEM coefficients
representing a block of u1D447u1D435 symbols in CE-BEM vary only a little from the past block to
the present one. We wish to estimate hu1D458,u1D45E (u1D459) by updating the BEM coefficients every u1D45Au1D460
symbols using TM training or detected symbols, rather than estimating them anew at every
non-overlapping block as in [70].
Let ? (u1D459)u1D45Fu1D458,u1D45E denote the u1D45F-th component of the column h (u1D459)u1D458,u1D45E . Stack the BEM coefficients
in (6.22) into ?tall? vectors as
h(u1D459)u1D45E :=
uni005B.alt02
? (u1D459)11,u1D45E ???? (u1D459)u1D4411,u1D45E ??? ? (u1D459)1u1D43E,u1D45E ???? (u1D459)u1D441u1D43E,u1D45E
uni005D.alt02u1D447
(6.27)
h(u1D459) :=
uni005B.alt02
h(u1D459)u1D4471 h(u1D459)u1D4472 ??? h(u1D459)u1D447u1D444
uni005D.alt02u1D447
(6.28)
h :=
uni005B.alt02
h(0)u1D447 h(1)u1D447 ??? h(u1D43F)u1D447
uni005D.alt02u1D447
(6.29)
of size u1D441u1D43E, u1D441u1D43Eu1D444 and u1D440 := u1D441u1D43Eu1D444(u1D43F+ 1) respectively. The coefficient vectors in (6.27),
(6.28) and (6.29) of the u1D45D-th overlapping block will be denoted by h(u1D459)u1D45E (u1D45D), h(u1D459) (u1D45D), and h(u1D45D)
respectively. Again, we emphasize that the u1D45D-th block and the (u1D45D+ 1)-st block differ by just
u1D45Au1D460 symbols. Since a fading channel can follow well a Markov model [16], we further assume
that the BEM coefficients over each overlapping block are also Markovian. A simplified
formulation is given by the first-order AR model, i.e.,
h(u1D45D) = A1h(u1D45D?1) +w(u1D45D), (6.30)
where A1 = u1D6FCIu1D440 is the first-order AR coefficient matrix (u1D6FC < 1 but close to one), and the
driving noise vector w(u1D45D) is zero-mean complex Gaussian with variance u1D70E2u1D464Iu1D440 and statisti-
cally independent of h(u1D45D?1). Assuming the channel is wide-sense stationary (WSS) and
the BEM coefficients ?u1D45E(u1D459)?s are independent, we have
u1D70E2u1D464 = u1D70E2?(1??u1D6FC?2)/u1D444 (6.31)
135
where u1D70E2?Iu1D440 := u1D438
uni007B.alt02
h(u1D45B;u1D459)hu1D43B (u1D45B;u1D459)
uni007D.alt02
.
Define
uD4D4 (u1D45B) :=
uni005B.alt02
u1D452?u1D457u1D7141u1D45B u1D452?u1D457u1D7142u1D45B ??? u1D452?u1D457u1D714u1D444u1D45B
uni005D.alt02u1D447
,
S(u1D45B) :=
uni005B.alt02
su1D447 (u1D45B) su1D447 (u1D45B?1) ??? su1D447 (u1D45B?u1D43F)
uni005D.alt02u1D447
, (6.32)
where s(u1D45B) is u1D43E-column vector. For u1D45Du1D45Au1D460 ? u1D45B < (u1D45D+ 1)u1D45Au1D460, by (6.21), (6.22), and (6.27)?
(6.29), the received signal at time u1D45B can be written as
y(u1D45B) = [S(u1D45B)?Iu1D441]u1D447 [Iu1D43F+1 ?(uD4D4 (u1D45B)?Iu1D441u1D43E)]u1D43B h(u1D45D) +v(u1D45B).
Further defining
Cu1D456 (u1D45D) := [S(u1D45Du1D45Au1D460 +u1D456)?Iu1D441]u1D447 [Iu1D43F+1 ?(uD4D4 (u1D45Du1D45Au1D460 +u1D456)?Iu1D441u1D43E)]u1D43B , (6.33)
C(u1D45D) :=
uni005B.alt02
Cu1D4470 (u1D45D) Cu1D4471 (u1D45D) ??? Cu1D447u1D45Au1D460?1 (u1D45D)
uni005D.alt02u1D447
, (6.34)
we have
yu1D45Au1D460 (u1D45D) = C(u1D45D)h(u1D45D) +vu1D45Au1D460 (u1D45D) (6.35)
where
yu1D45Au1D460 (u1D45D) :=
uni005B.alt02
yu1D447 (u1D45Du1D45Au1D460) yu1D447 (u1D45Du1D45Au1D460 + 1) ??? yu1D447 ((u1D45D+ 1)u1D45Au1D460 ?1)
uni005D.alt02u1D447
(6.36)
and vu1D45Au1D460 (u1D45D) is defined likewise.
6.3.1 Decision-Directed MIMO Tracking via KF or EW-RLS
The dynamical state-space MIMO system model is represented by (6.30) and (6.35),
where yu1D45Au1D460(u1D45D) is the measurement of the state of BEM coefficients h(u1D45D). Kalman filtering
can be applied to track the coefficient vector h(u1D45D) at a step size of u1D45Au1D460 symbols. Since s(u1D45B)
is unknown during information sessions, the receiver then switches to a decision-directed
mode, in which the symbol decision is assumed to be correct (?s(u1D45B) = s(u1D45B)) and used in
Kalman tracking. Treating (6.30) and (6.35) as the state and the measurement equations
respectively, the Kalman tracking is applied recursively every step size. [See the details in
Section 6.2.1.]
136
In order to avoid the periodicity of CE-BEM with a period u1D447 that has an impact on
estimating ?non-periodic? channels, a ?finite-memory? algorithm is considered in decision-
directed tracking via the extended exponentially-weighted RLS (EW-RLS) algorithm. As-
suming the BEM coefficients follow the first-order AR model, a state-space model for the
extended EW-RLS algorithm is given by (6.30) and (6.35).
Similarly as Kalman filtering, the extended exponentially-weighted regularized RLS
(EW-RLS) algorithm [2, Section 12.B.(12.B.18)] is based on (6.30) and (6.35) to track the
unknown BEM coefficient vector h(u1D45D) recursively every step size. [See the details in Section
6.2.2.]
After Kalman or RLS recursion for everyu1D45D, we can generate the channel by the estimated
?h(u1D45D) via the CE-BEM (6.22). Note that we also need the channel estimate ?h(u1D45B;u1D459) for the
following u1D45Au1D460 +u1D451 symbols to make up the time gap between channel estimation and symbol
detection in DFE. Since the estimated BEM coefficients are same for a large block size
(u1D447u1D435 >>u1D45Au1D460), the channel predictions for the current u1D45Au1D460 symbols as well as following u1D45Au1D460 +u1D451
symbols are given by
?h(u1D45B;u1D459) = uD4D4u1D43B(u1D45B)?h(u1D459)(u1D45D), (6.37)
for u1D45B = u1D45Du1D45Au1D460,u1D45Du1D45Au1D460 + 1,??? ,(u1D45D+ 2)u1D45Au1D460 +u1D451? 1. The definition of ?h(u1D459) (u1D45D) is similar to (6.28)
and ?h(u1D459) (u1D45D) is based on observations up to time u1D45B = (u1D45D+ 1)u1D45Au1D460 ?1.
Using the channel estimates
uni007B.alt02?
h(u1D45B;u1D459)
uni007D.alt02
in (6.37), the symbol decisions are made by an
FIR MMSE-DFE in Section 2.4.2 [40] for the time u1D45B = u1D45Du1D45Au1D460,u1D45Du1D45Au1D460 + 1,??? ,(u1D45D + 2)u1D45Au1D460 ? 1.
Note that the detected symbols for the time u1D45B = (u1D45D+1)u1D45Au1D460,(u1D45D+1)u1D45Au1D460 +1,??? ,(u1D45D+2)u1D45Au1D460?1
are used for the following channel tracking, whereas the ?current? decisions for the time
u1D45B = u1D45Du1D45Au1D460,u1D45Du1D45Au1D460 + 1,??? ,(u1D45D+ 1)u1D45Au1D460 ?1 are updated for improved performance.
Computational Complexity
We compare computational complexity using the floating point operation (flop) count
for the decision-directed MIMO channel tracking schemes. Here we consider Kalman filtering
as the common basis for all the simulation schemes to fairly compare the decision-directed
approach with subblock-wise one. We consider three different channel estimation schemes:
?SB-KF(BEMu1D447)? denoting BEM-based subblock-wise Kalman filtering with a period u1D447 in
Chapter 3, ?DD-ARu1D443? denoting AR(P) model-based symbol-by-symbol decision-directed
Kalman filtering [6], ?DD-KF(BEMu1D447)? denoting BEM-based decision-directed Kalman fil-
tering with a period u1D447. All the above schemes have the same subblock (u1D45Au1D44F = 100) and
hence same training overhead. We take u1D447 = 400 and u1D444 = 9 for CE-BEM. [The complexity
137
of EW-RLS algorithm is almost the same as Kalman filtering. See the detailed flop counts
of Kalman filtering in Section 3.3.1 and EW-RLS algorithm in Section 4.3.1.]
The number of flops for one cycle of of subblock-wise Kalman filtering, decision-directed
Kalman filtering and DFE turn out to be
u1D453u1D450(SB-KF) =u1D4403u1D460 +u1D4402u1D460(3u1D43Fu1D45D + 3) +u1D440u1D460(2u1D43F2u1D45D + 5u1D43Fu1D45D + 1) +u1D43F3u1D45D/3 +u1D43F2u1D45D
u1D453u1D450(DD-KF) =u1D4403 +u1D4402(3u1D441u1D45Au1D460 + 2) +u1D440[2(u1D441u1D45Au1D460)2 + 3u1D441u1D45Au1D460 + 1] + (u1D441u1D45Au1D460)3/3 + (u1D441u1D45Au1D460)2
u1D453u1D450(DFE) =2(u1D441u1D459u1D453)3 + (u1D441u1D459u1D453)2(?u1D459u1D453 + ?u1D459u1D44F +u1D43E + 1) + (u1D441u1D459u1D453)(?u1D459u1D453?u1D459u1D44F + ?u1D4592u1D44F +u1D43E?u1D459u1D44F +u1D43E)
+ 2(?u1D4593u1D44F +u1D43E?u1D4592u1D44F +u1D43E2?u1D459u1D44F) +u1D43E3/3 +u1D43E2u1D459u1D44F
whereu1D440u1D460 = u1D444(u1D43F+1) for SB-KF(BEMu1D447),u1D440 = u1D441u1D43Eu1D443(u1D43F+1) for DD-KF(ARu1D443) oru1D441u1D43Eu1D444(u1D43F+
1) for DD-KF(BEMu1D447), and u1D43Fu1D45D = u1D43F + 1,?u1D459u1D453 = u1D43E(u1D459u1D453 + u1D43F),?u1D459u1D44F = u1D43E(u1D459u1D44F + 1). The overall flop
counts over one block of u1D447u1D435 symbols for the subblock-wise and decision-directed Kalman
tracking schemes are then given by
u1D453u1D450(SB-KF(BEMu1D447)) = u1D441(u1D447u1D435/u1D45Au1D44F)u1D453u1D450(SB-KF) + [u1D447u1D435 ?u1D45Au1D461(u1D447u1D435/u1D45Au1D44F)]u1D453u1D450(DFE)
u1D453u1D450(DD-KF(ARu1D443)) = u1D447u1D435u1D453u1D450(DD-KF) + 2[u1D447u1D435 ?u1D45Au1D461(u1D447u1D435/u1D45Au1D44F)]u1D453u1D450(DFE),u1D45Au1D460 = 1
u1D453u1D450(DD-KF(BEMu1D447)) = (u1D447u1D435/u1D45Au1D460)u1D453u1D450(DD-KF) + 2[u1D447u1D435 ?u1D45Au1D461(u1D447u1D435/u1D45Au1D44F)]u1D453u1D450(DFE).
The comparative flop counts for each scheme with three receivers and two users(u1D441 =
3,u1D43E = 2) are compared in Table 6.3 over one block (u1D447u1D435 = 200).
Table 6.3: Subblock-wise and decision-directed MIMO channel tracking with DFE: compar-
ative flop count over one block of u1D447u1D435 symbols.
tracking schemes comparative flops
SB-KF(BEM400) 1
DD-KF(BEM400), u1D45Au1D460 = 4 30.18
DD-KF(BEM400), u1D45Au1D460 = 2 52.99
DD-KF(BEM400), u1D45Au1D460 = 1 98.81
DD-KF(AR9) 98.81
138
6.3.2 Simulation Examples
A random time- and frequency-selective Rayleigh fading MIMO channel is considered.
We assume h(u1D45B;u1D459) are zero-mean, complex Gaussian, and spatially white with autocorre-
lation u1D438
uni007B.alt02
h(u1D45B;u1D459)hu1D43B (u1D45B;u1D459)
uni007D.alt02
= u1D70E2?Iu1D441 for each u1D459. We take u1D43F = 2 (3 taps) in (6.21), and
u1D70E2? = 1/(u1D43F+ 1). For different u1D459?s, h(u1D45B;u1D459)?s are mutually independent and satisfy Jakes?
model. To this end, we simulate each single tap following [75] (with a correction in the
appendix of [63]).
We consider a communication system with carrier frequency of 2GHz, data rate of
40kBd (kilo-Bauds), therefore u1D447u1D460 = 25u1D707s, and a varying Doppler spread u1D453u1D451 = 400Hz, or
the normalized Doppler spread u1D453u1D451u1D447u1D460 = 0.01 (corresponding to a maximum mobile velocity
216km/h). The additive noise is zero-mean complex white Gaussian. The (receiver) SNR
refers to the average energy per symbol over one-sided noise spectral density.
We evaluate the performances of various schemes by considering their normalized chan-
nel mean square error (NCMSE) and their bit error rates (BER). The NCMSE is defined
as
NCMSE :=
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0
vextenddoublevextenddouble
vextenddouble?h(u1D456) (u1D45B;u1D459)?h(u1D456) (u1D45B;u1D459)
vextenddoublevextenddouble
vextenddouble2
uni2211.alt01u1D440u1D45F
u1D456=1
uni2211.alt01u1D447u1D441?1
u1D45B=0
uni2211.alt01u1D43F
u1D459=0?h(u1D456) (u1D45B;u1D459)?
2
where h(u1D456) (u1D45B;u1D459) is the true channel and ?h(u1D456) (u1D45B;u1D459) is the estimated channel at the u1D456-th Monte
Carlo run, among total u1D440u1D45F runs. In each run, a training mode of 200 symbols modulated
by binary phase-shift keying (BPSK) is followed by a decision-directed mode of quadrature
phase-shift keying (QPSK) 4000 symbols (u1D447u1D441 = 4000). All the simulation results are based
on 500 runs, and we consider the performances during the decision-directed mode only.
In the decision-directed mode, training sessions are also periodically sent to facilitate
the EW-RLS tracking. The TM training scheme of [70] is adopted, where each subblock of
equal length u1D45Au1D44F symbols consists of an information session of u1D45Au1D451 symbols and a succeeding
training session of u1D45Au1D461 symbols (u1D45Au1D44F = u1D45Au1D451 +u1D45Au1D461). The training session for each user contains
an impulse guarded by zeros (silent periods), which for the u1D458?th user has the structure
(u1D6FE > 0)
cu1D458,u1D45D :=
uni005B.alt02
01?((u1D458?1)(u1D43F+1)+u1D43F) u1D6FE 01?((u1D43E?u1D458)(u1D43F+1)+u1D43F)
uni005D.alt02
(6.38)
where u1D458 = 1,??? ,u1D43E. Therefore, u1D45Au1D461 = u1D43E(u1D43F+ 1) +u1D43F, which have to be devoted for training
and the remaining are available for information symbols. Note that cu1D458,u1D45D is the u1D458?th row
of the matrix cu1D45D. We assume that each information symbol has unit power, while at every
training session given by (6.38), we set u1D6FE =
uni221A.alt02
u1D43E(u1D43F+ 1) +u1D43F so that the average power per
symbol at training sessions is equal to that of information sessions.
We compared the following schemes:
139
0 5 10 15 20 25 30?25
?20
?15
?10
?5
0
5
10
SNR(dB)
NCMSE(dB)
QPSK,N=3,K=2,L=2,mb=100,T=400,Q=9,ms=2,fdTs=0.01,500Runs
 
 
SB?RLS?BEM
DD?KF?AR9
DD?RLS?BEM
DD?KF?BEM
PD?RLS?BEM
(a) NCMSE vs SNR
0 5 10 15 20 25 30
10?4
10?3
10?2
10?1
100
SNR(dB)
Bit Error Rate
QPSK,N=3,K=2,L=2,mb=100,T=400,Q=9,ms=2,fdTs=0.01,500Runs
 
 
SB?RLS?BEM
DD?KF?AR9
DD?RLS?BEM
DD?KF?BEM
PD?RLS?BEM
(b) BER vs SNR, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and equal-
ization delay u1D451 = 5
Figure 6.10: Decision-directed MIMO tracking: performance comparison for SNR?s, under
u1D441 = 3,u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 100.
1. The subblock-wise EW-RLS algorithm with u1D6FD = 1, which is one of the best subblock-
wise approaches in Chapter 4. [One can pick a different subblock-wise scheme using
SW-RLS algorithm or Kalman filter that has similar (or slightly worse) performances.]
In the simulations, we take the forgetting factor u1D706 = 0.5 for u1D45Au1D44F = 100 in EW-RLS
algorithm. This scheme is denoted by ?SB-RLS-BEM?.
2. The decision-directed channel tracking scheme in [6] using the u1D443-th order AR model,
i.e., the time-varying channel follows (assuming each independent channel tap has same
Doppler spread)
h(u1D45B;u1D459) =
u1D443uni2211.alt02
u1D456=1
Au1D456h(u1D45B?u1D456;u1D459) +w(u1D45B), (6.39)
where Au1D456 = u1D6FCu1D456I is the AR coefficients matrix and driving noise w(u1D45B) is zero-mean
complex Gaussian with autocorrelation,u1D438
uni007B.alt02
w(u1D45B)wu1D43B (u1D45B)
uni007D.alt02
= u1D70E2u1D464Iu1D441. Using Yule-Walker
equations, we take the AR coefficients and noise variance for u1D453u1D451u1D447u1D460 = 0.01. The channel
prediction stage is conducted by using (6.39) omitting the driving noise, i.e.,
?h(u1D45B;u1D459) = u1D443uni2211.alt02
u1D456=1
Au1D456?h(u1D45B?u1D456;u1D459). (6.40)
We take u1D443 = 9 to compare CE-BEM with u1D447 = 400,u1D444 = 9. This scheme is denoted by
?DD-KF-AR9?.
140
0 0.002 0.004 0.006 0.008 0.01?30
?25
?20
?15
?10
?5
0
fdTs
NCMSE(dB)
QPSK,N=3,K=2,L=2,mb=100,T=400,Q=9,ms=2,SNR=20dB,500Runs
 
 
SB?RLS?BEM
DD?KF?AR9
DD?RLS?BEM
DD?KF?BEM
(a) NCMSE vs u1D453u1D451u1D447u1D460
0 0.002 0.004 0.006 0.008 0.01
10?4
10?3
10?2
10?1
100
fdTs
Bit Error Rate
QPSK,N=3,K=2,L=2,mb=100,T=400,Q=9,ms=2,SNR=20dB,500Runs
 
 
SB?RLS?BEM
DD?KF?AR9
DD?RLS?BEM
DD?KF?BEM
(b) BER vs u1D453u1D451u1D447u1D460, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and equal-
ization delay u1D451 = 5
Figure 6.11: Decision-directed MIMO tracking: performance comparison for u1D453u1D451u1D447u1D460?s, under
u1D441 = 3,u1D43E = 2,SNR = 20dB,u1D45Au1D44F = 100.
3. The proposed decision-directed tracking scheme with different step sizes in the Kalman
tracking, denoting by ?DD-KF-BEM?. For the CE-BEM, we take the first-order AR co-
efficient for the BEM coefficients empirically via simulations,u1D6FC = 0.990,0.994 and 0.997
for u1D45Au1D460 = 4,2 and 1 respectively.
4. The proposed decision-directed tracking scheme with different step sizes in the extended
EW-RLS tracking with u1D6FD = 1, denoting by ?DD-RLS-BEM?. We take the same u1D6FC value
for each step size as the decision-directed Kalman tracking. We also take the forgetting
factor u1D706 = 0.93,0.96 and 0.98 for u1D45Au1D460 = 4,2 and 1 respectively.
5. Perfect symbol decisions are used as training for the extended EW-RLS channel track-
ing with step size u1D45Au1D460 = 2 and the same other setups as decision-directed RLS tracking.
First, the channel is estimated using exact knowledge of the information symbols. Then
this estimted channel is used to detect the ?unknown? information symbols. There-
fore the ?performance-gap? between ?decision-directed tracking? and ?perfect decision-
directed? in figures shows the performance deterioration by decision-directed approach.
[One can also use Kalman tracking with the perfect symbol decisions which has almost
the same performance.] This scheme provides the baseline for decision-directed track-
ing, denoted by ?PD-RLS-BEM?.
In the simulations, we consider a three-receiver and two-user scenario, i.e.,u1D441 = 3,u1D43E = 2
with the same transmitted power. We have u1D45Au1D461 = 8 with u1D6FE = ?8 for every user following the
141
0 5 10 15 20 25 30?25
?20
?15
?10
?5
0
5
10
SNR (dB)
NCMSE(dB)
DD?RLS?BEM,QPSK,N=3,K=2,L=2,mb=100,T=400,Q=9,fdTs=0.01,500Runs
 
 
ms=1
ms=2
ms=4
(a) NCMSE vs SNR
0 5 10 15 20 25 3010
?7
10?6
10?5
10?4
10?3
10?2
10?1
100
SNR (dB)
Bit Error Rate
DD?RLS?BEM,QPSK,N=3,K=2,L=2,mb=100,T=400,Q=9,fdTs=0.01,500Runs
 
 
ms=1
ms=2
ms=4
(b) BER vs SNR, with DFE of u1D459u1D453 = 8,u1D459u1D44F = 2 and equal-
ization delay u1D451 = 5
Figure 6.12: Decision-directed EW-RLS MIMO tracking: performances with different step
size u1D45Au1D460?s for SNR?s, under u1D441 = 3,u1D43E = 2,u1D453u1D451u1D447u1D460 = 0.01,u1D45Au1D44F = 100.
TM training scheme. For each of the above schemes, an MMSE-DFE described in Section
2.4.2 is employed at the receiver, using the obtained channel estimates, with u1D459u1D453 = 8, u1D459u1D44F = 2,
and the equalization delay u1D451 = 5 symbols.
In Fig. 6.10, the performances of the schemes are compared for different SNR?s, under
normalized Doppler spread u1D453u1D451u1D447u1D460 = 0.01. In Fig. 6.11, the performances are compared
over different normalized Doppler spread u1D453u1D451u1D447u1D460?s for fixed SNR = 20dB; we keep the AR(9)
coefficients and u1D444 = 9 for CE-BEM with u1D447 = 400 for the Doppler spread u1D453u1D451 = 400Hz.
It is clear the subblock-wise RLS tracking scheme does not work with a large subblock
size u1D45Au1D44F = 100 at all since it depends only on the training to estimate channels. Note
that the decision-directed schemes take the symbol-decisions as virtual training and hence
perform well with less training overhead. It is shown the CE-BEM-based decision-directed
approaches yield better performance than the symbol-wise AR-model-based decision-directed
approach [6]. Especially, the extended EW-RLS decision-directed tracking, a finite memory
algorithm with a forgetting factor u1D706, performs closely to the perfect decision-directed scheme
with high SNR?s. In Fig. 6.11, the performances for DD-RLS-BEM and DD-KF-BEM are
not so sensitive to the Doppler spread of the channel as AR-model-based scheme, which
means an upper bound, for instance u1D453u1D451 = 400Hz, is sufficient in practice with unknown exact
Doppler spreads.
142
The proposed BEM-based decision-directed tracking schemes update the BEM coeffi-
cients every step size (u1D45Au1D460 symbols) while the AR-model-based one updates the channel esti-
mates every symbol. Note that we takeu1D45Au1D460 = 2 for the BEM-based schemes (u1D447 = 400,u1D444 = 9)
in Fig. 6.10 and 6.11, where they have better performances with less computational complex-
ity than the AR(9)-based one. [See the flop counts in Table 6.3.] As shown in Fig. 6.12, a
?finer? channel tracking can be obtained by reducing step sizeu1D45Au1D460 although the computational
complexity increases.
6.4 Performance Analysis of Decision-Directed EW-RLS Tracking using CE-
BEM
The cost function for the EW-RLS algorithm can be rewritten for ?large? u1D45D(u1D45D? ?) as
u1D49Eu1D438u1D44A =
?uni2211.alt02
u1D456=0
u1D706u1D456?yu1D45Au1D460(u1D45D?u1D456)?C(u1D45D?u1D456)h?2. (6.41)
Let u1D441u1D460 denote the multiples of the step size u1D45Au1D460 on which we would like to base the estimate
of BEM coefficient h. It is clear that u1D438
uni007B.alt02
Cu1D43B(u1D456)C(u1D456)
uni007D.alt02
is periodic in u1D456 with period u1D447/u1D45Au1D460 as
CE-BEM is periodic with periodu1D447. In practice, we would like to have the memory length (in
symbols) to be less than the model period (recall that the channel is by no means periodic)
so that there are no deleterious effects due to the CE-BEM model for all time, i.e., u1D45Au1D460u1D441u1D460 ?u1D447
in order to avoid this periodicity. Let us pick u1D441u1D460 = u1D447/u1D45Au1D460. What other restriction should
we impose on u1D441u1D460? In the following we assume that the detected symbols resulting from
MMSE-DFE and used for EW-RLS tracking, are correct (no detection error).
A least-square solution for h to minimize u1D49Eu1D438u1D44A is given by [50, p. 796]
?h = A?1B (6.42)
where
A := 1u1D441
u1D460
?uni2211.alt02
u1D456=0
u1D706u1D456Cu1D43B(u1D45D?u1D456)C(u1D45D?u1D456), (6.43)
B := 1u1D441
u1D460
?uni2211.alt02
u1D456=0
u1D706u1D456Cu1D43B(u1D45D?u1D456)yu1D45Au1D460(u1D45D?u1D456). (6.44)
In order to analyze the behavior of h we need a model for yu1D45Au1D460(u1D456) for every step, u1D456 = u1D45D,u1D45D?
1,???. To this end, for the following analysis presented in this section, we assume the
143
following ?simplified? model:
yu1D45Au1D460(u1D45D?u1D456) = C(u1D45D?u1D456)h(u1D461u1D45F)(u1D45D?u1D456) +vu1D45Au1D460(u1D45D?u1D456). (6.45)
where h(u1D461u1D45F)(u1D45D?u1D456) is the ?true? BEM coefficient vector satisfying (u1D45A = 1,2,???)
h(u1D461u1D45F)(u1D45D?u1D456) := h(u1D461u1D45F)u1D45A for (u1D45A?1)u1D441u1D460 ?u1D45D?u1D456<u1D45Au1D441u1D460. (6.46)
That is, there exist some true BEM parameters that are ?fixed? over one BEM period of
u1D447 symbols (therefore, u1D441u1D460 = u1D447/u1D45Au1D460 steps), and are allowed to change over non-overlapping
periods. In this set-up, we estimate the most recent true BEM coefficient vector h(u1D461u1D45F)1 via
?h without it being unduly influenced by h(u1D461u1D45F)u1D45A ,u1D45A ? 2. Assuming the estimate ?h asymptot-
ically Gaussian distributed, we are trying to evaluate mean-square-error (MSE) of channel
estimation when the true channel follows CE-BEM.
The MSE of the channel estimate using CE-BEM is defined as
MSE
uni0028.alt02?
h
uni0029.alt02
:= u1D438
uni007B.alt03
u1D438
uni007B.alt03vextenddoublevextenddouble
vextenddouble?h?h(u1D461u1D45F)1
vextenddoublevextenddouble
vextenddouble2 ? h(u1D461u1D45F)
uni007D.alt03uni007D.alt03
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=:MSE1(?h)
+ 1u1D447
u1D447uni2211.alt02
u1D45B=1
u1D43Funi2211.alt02
u1D459=0
u1D438
uni007B.alt02
?eBEM(u1D45B;u1D459)?2
uni007D.alt02
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=:MSE2(?h)
, (6.47)
where MSE1
uni0028.alt02?
h
uni0029.alt02
comes from the channel estimation and MSE2
uni0028.alt02?
h
uni0029.alt02
is an mean square mod-
eling error that is intrinsic to the BEM representation. Note that MSE2
uni0028.alt02?
h
uni0029.alt02
has nothing to
do with channel estimation.
Following [51, Section 4.3], consider the BEM representation (6.2) as an approximation
where only u1D444 (out of total u1D447) basis functions are used to describe the channel. Then
MSE2
uni0028.alt02?
h
uni0029.alt02
is given by
MSE2
uni0028.alt02?
h
uni0029.alt02
= 1u1D447
u1D447uni2211.alt02
u1D45E=u1D444+1
u1D43Funi2211.alt02
u1D459=0
u1D438
uni007B.alt02
?hu1D45E(u1D459)?2
uni007D.alt02
= 1u1D447
u1D447uni2211.alt02
u1D45E=1
u1D43Funi2211.alt02
u1D459=0
u1D447?1uni2211.alt02
u1D45B1=0
u1D447?1uni2211.alt02
u1D45B2=0
u1D445?(u1D45B1,u1D45B2;u1D459)u1D452?u1D457 2u1D70B[u1D45E?(u1D447+1)/2]u1D447 (u1D45B1?u1D45B2) (6.48)
where u1D445?(u1D45B1,u1D45B2;u1D459) := u1D438
uni007B.alt02
hu1D43B(u1D45B1;u1D459)h(u1D45B2;u1D459)
uni007D.alt02
. For the modified Jakes? model,
u1D445?(u1D45B1,u1D45B2;u1D459) = u1D441u1D70E2?u1D43D0 (2u1D70Bu1D453u1D451u1D447u1D460(u1D45B1 ?u1D45B2))
144
where u1D43D0(?) denotes the zero-th order Bessel function of the first kind. After some manipu-
lations, we obtain
MSE2
uni0028.alt02?
h
uni0029.alt02
= u1D43F+ 1u1D4472
u1D447?1uni2211.alt02
u1D45B1=0
u1D447?1uni2211.alt02
u1D45B2=0
u1D441u1D70E2?u1D43D0 (2u1D70Bu1D453u1D451u1D447u1D460(u1D45B1 ?u1D45B2))
?
??
u1D447?u1D444+12uni2211.alt02
u1D45E=u1D444+12
u1D452u1D457 2u1D70Bu1D45E(u1D45B1?u1D45B2)u1D447
?
??
= u1D43F+ 1u1D447 u1D441u1D70E2?
uni005B.alt04
(u1D447 ?u1D444)?2
u1D447?1uni2211.alt02
u1D70F=1
uni007B.alt04
1? u1D70Fu1D447u1D43D0 (2u1D70Bu1D453u1D451u1D447u1D460u1D70F) sin
u1D70Bu1D70Fu1D444
u1D447
sin u1D70Bu1D70Fu1D447
uni007D.alt04uni005D.alt04
. (6.49)
For the mean square error of channel estimate MSE1, consider nonzero bias and the
channel estimation error can be written as
?h?h(u1D461u1D45F)1 = ?h??h+ ?h?h(u1D461u1D45F)1
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=bias
,
where ?h := u1D438
uni007B.alt02?
h
uni007D.alt02
?= h(u1D461u1D45F). Then MSE1 is given by
MSE1
uni0028.alt02?
h
uni0029.alt02
= u1D438
uni007B.alt03vextenddoublevextenddouble
vextenddouble?h??h
vextenddoublevextenddouble
vextenddouble2
uni007D.alt03
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=u1D461u1D45F{cov{?h}}
+u1D438
uni007B.alt03
u1D438
uni007B.alt03vextenddoublevextenddouble
vextenddouble?h?h(u1D461u1D45F)1
vextenddoublevextenddouble
vextenddouble2 ? h(u1D461u1D45F)
uni007D.alt03uni007D.alt03
(6.50)
The difference between the estimate ?h and its expected ?h can be determined by [61, Chapter
8, p.261]
?h??h = A?1?B (6.51)
where
?B := 1
u1D441u1D460
?uni2211.alt02
u1D456=0
u1D706u1D456Cu1D43B(u1D45D?u1D456)vu1D45Au1D460(u1D45D?u1D456). (6.52)
Then the covariance of channel estimate, cov
uni007B.alt02?
h
uni007D.alt02
in (6.50) is given by
cov
uni007B.alt02?
h
uni007D.alt02
= u1D438
uni007B.alt03uni0028.alt02
?h??huni0029.alt02uni0028.alt02?h??huni0029.alt02u1D43B
uni007D.alt03
= u1D438
uni007B.alt03
A?1?B?Bu1D43B(A?1)u1D43B
uni007D.alt03
. (6.53)
We can rewrite (6.43) as
A =
?uni2211.alt02
u1D45A=1
u1D706(u1D45A?1)u1D441u1D460Au1D45A, (6.54)
where
Au1D45A = 1u1D441
u1D460
u1D441u1D460?1uni2211.alt02
u1D456=0
u1D706u1D456Cu1D43B(u1D45D?u1D456?(u1D45A?1)u1D441u1D460)C(u1D45D?u1D456?(u1D45A?1)u1D441u1D460).
145
Now we establish that
limu1D441
u1D460??
u1D438
uni007B.alt02
?Au1D45A ?u1D438{Au1D45A}?2
uni007D.alt02
= 0. (6.55)
We have
u1D438
uni007B.alt02
?Au1D45A ?u1D438{Au1D45A}?2
uni007D.alt02
? 1u1D4412
u1D460
u1D441u1D460?1uni2211.alt02
u1D456=0
u1D441u1D460?1uni2211.alt02
u1D459=0
u1D706u1D456+u1D459
vextenddoublevextenddouble
vextenddoubleu1D438
uni007B.alt02
Yu1D43Bu1D456 Yu1D459
uni007D.alt02vextenddoublevextenddouble
vextenddouble, (6.56)
where
Yu1D456 := Xu1D456 ?u1D438{Xu1D456},
Xu1D456 := Cu1D43B(u1D45D?u1D456?(u1D45A?1)u1D441u1D460)C(u1D45D?u1D456?(u1D45A?1)u1D441u1D460).
Since the information sequence {u1D460(u1D45B)} is zero-mean white, it follows that u1D438
uni007B.alt02
Yu1D43Bu1D456 Yu1D459
uni007D.alt02
= 0 for
?u1D456?u1D459? > u1D4590 := 1 +
uni2308.alt02u1D43F?1
u1D45Au1D460
uni2309.alt02
. Setting u1D456?u1D459 = u1D457 and using the facts that
vextenddoublevextenddouble
vextenddoubleu1D438
uni007B.alt02
Yu1D43Bu1D456 Yu1D459
uni007D.alt02vextenddoublevextenddouble
vextenddouble ? u1D44F0 for
?u1D456?u1D459? ?u1D4590 and some 0 <u1D44F0 <?, and u1D706< 1, we can simplify (6.56) as
u1D438
uni007B.alt02
?Au1D45A ?u1D438{Au1D45A}?2
uni007D.alt02
? 1u1D4412
u1D460
u1D441u1D460?1uni2211.alt02
u1D457=?u1D441u1D460+1
u1D441u1D460?1uni2211.alt02
u1D459=0
u1D706u1D457+2u1D459
vextenddoublevextenddouble
vextenddoubleu1D438
uni007B.alt02
Yu1D43Bu1D457+u1D459Yu1D459
uni007D.alt02vextenddoublevextenddouble
vextenddouble
? 2u1D441u1D460 ?1u1D4412
u1D460
(2u1D4590 + 1)u1D44F0 ? 0 as u1D441u1D460 ? ?. (6.57)
Since u1D706< 1, in a similar fashion we can show that
limu1D441
u1D460??
u1D438
uni007B.alt02
?A?u1D438{A}?2
uni007D.alt02
= 0. (6.58)
Defining z(u1D45D) for the signal part and ?(u1D45D) for the BEM basis function part in C(u1D45D) as
z(u1D45D) :=
?
??
??
??
?
su1D447 (u1D45Du1D45Au1D460)
su1D447 (u1D45Du1D45Au1D460 + 1)
...
su1D447 ((u1D45D+ 1)u1D45Au1D460 ?1)
?
??
??
??
?
, (6.59)
?(u1D45D) := Iu1D43F+1 ?
uni005B.alt02
uD4D4 (u1D45Du1D45Au1D460) uD4D4 (u1D45Du1D45Au1D460 + 1) ??? uD4D4 ((u1D45D+ 1)u1D45Au1D460 ?1)
uni005D.alt02u1D43B
, (6.60)
we have C(u1D45D) = z(u1D45D)?(u1D45D). Note that we assumed the information sequence {u1D460(u1D45B)} is
mutually independent and identically distributed (i.i.d.) with zero mean and variance
u1D438{u1D460(u1D45B)u1D460? (u1D45B)} = u1D70E2u1D460. An important observation is that since u1D438
uni007B.alt02
Cu1D43B(u1D45D)C(u1D45D)
uni007D.alt02
is periodic
146
with period u1D441u1D460 = u1D447/u1D45Au1D460, u1D438{Au1D45A} is not a function of u1D45A. Therefore, we have
u1D438{A} =
?uni2211.alt02
u1D45A=0
u1D706(u1D45A?1)u1D441u1D460u1D438{Au1D45A} = 11?u1D706u1D441
u1D460
u1D438{Au1D45A}
= 11?u1D706u1D441
u1D460
u1D438
uni007B.alt04 1
u1D441u1D460
u1D441u1D460?1uni2211.alt02
u1D456=0
u1D706u1D456?u1D43B(u1D45D?u1D456)zu1D43B(u1D45D?u1D456)z(u1D45D?u1D456)?(u1D45D?u1D456)
uni007D.alt04
= 11?u1D706u1D441
u1D460
1
u1D441u1D460
u1D441u1D460?1uni2211.alt02
u1D456=0
u1D706u1D456?u1D43B(u1D45D?u1D456)u1D438
uni007B.alt02
zu1D43B(u1D45D?u1D456)z(u1D45D?u1D456)
uni007D.alt02
?(u1D45D?u1D456)
= u1D70E
2
u1D460
1?u1D706u1D441u1D460
1
u1D441u1D460
u1D441u1D460?1uni2211.alt02
u1D456=0
u1D706u1D456?u1D43B(u1D45D?u1D456)?(u1D45D?u1D456)
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=:A0
. (6.61)
and (i.p. stands for in probability)
limu1D441
u1D460??
A u1D456.u1D45D.= u1D438{A} = u1D70E
2
u1D460
1?u1D706u1D441u1D460 A0. (6.62)
Based on (6.62), we can rewrite (6.53) as
cov
uni007B.alt02?
h
uni007D.alt02
= u1D438{A}?1 cov
uni007B.alt02?
B
uni007D.alt02
u1D438
uni007B.alt02
Au1D43B
uni007D.alt02?1
. (6.63)
Assuming u1D463(u1D45B) is zero-mean, white noise, uncorrelated with u1D460(u1D45B), with u1D438{u1D463(u1D45B+u1D70F)u1D463?(u1D45B)} =
u1D70E2u1D463u1D6FF(u1D70F), the covariance of ?B is given as (using (6.52) and C(u1D45D?u1D456) = z(u1D45D?u1D456)?(u1D45D?u1D456))
cov
uni007B.alt02?
B
uni007D.alt02
= u1D438
uni007B.alt03
?B?Bu1D43B
uni007D.alt03
= 1u1D4412
u1D460
u1D438
??
?
??
?
?
?uni2211.alt02
u1D4561=0
u1D706u1D4561Cu1D43B(u1D45D?u1D4561)vu1D45Au1D460(u1D45D?u1D4561)
?
?
?
?
?uni2211.alt02
u1D4562=0
u1D706u1D4562Cu1D43B(u1D45D?u1D4562)vu1D45Au1D460(u1D45D?u1D4562)
?
?
u1D43B???
??
= 1u1D4412
u1D460
?uni2211.alt02
u1D4561=0
?uni2211.alt02
u1D4562=0
u1D706u1D4561+u1D4562u1D438
uni007B.alt02
Cu1D43B(u1D45D?u1D4561)
uni005B.alt02
vu1D45Au1D460(u1D45D?u1D4561)vu1D43Bu1D45Au1D460(u1D45D?u1D4562)
uni005D.alt02
C(u1D45D?u1D4562)
uni007D.alt02
= u1D70E
2
u1D463
u1D4412u1D460
?uni2211.alt02
u1D4561=0
?uni2211.alt02
u1D4562=0
u1D706u1D4561+u1D4562?u1D43B(u1D45D?u1D4561)u1D438
uni007B.alt02
zu1D43B(u1D45D?u1D4561)z(u1D45D?u1D4562)
uni007D.alt02
?(u1D45D?u1D4562)u1D6FF(u1D4561 ?u1D4562)
= u1D70E
2
u1D463
u1D4412u1D460
?uni2211.alt02
u1D456=0
u1D7062u1D456?u1D43B(u1D45D?u1D456)u1D438
uni007B.alt02
zu1D43B(u1D45D?u1D456)z(u1D45D?u1D456)
uni007D.alt02
?(u1D45D?u1D456)
= u1D70E
2
u1D463u1D70E
2
u1D460
u1D4412u1D460
?uni2211.alt02
u1D456=0
u1D7062u1D456?u1D43B(u1D45D?u1D456)?(u1D45D?u1D456) (6.64)
147
Using the periodicity, ?(u1D45D?u1D456?(u1D45A?1)u1D441u1D460) = ?(u1D45D?u1D456) for integer u1D45A?s, we can rewrite (6.64)
as
cov
uni007B.alt02?
B
uni007D.alt02
= u1D70E
2
u1D463u1D70E
2
u1D460
u1D4412u1D460
?uni2211.alt02
u1D45A=1
u1D706(u1D45A?1)2u1D441u1D460
uni005B.alt04u1D441
u1D460?1uni2211.alt02
u1D456=0
u1D7062u1D456?u1D43B(u1D45D?u1D456)?(u1D45D?u1D456)
uni005D.alt04
= u1D70E
2
u1D463u1D70E
2
u1D460
1?u1D7062u1D441u1D460
1
u1D4412u1D460
u1D441u1D460?1uni2211.alt02
u1D456=0
u1D7062u1D456?u1D43B(u1D45D?u1D456)?(u1D45D?u1D456)
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=:B0
(6.65)
Hence, using (6.62), (6.63) and (6.65), we have
cov
uni007B.alt02?
h
uni007D.alt02
= u1D70E
2
u1D463
u1D70E2u1D460
uni0028.alt041?u1D706u1D441
u1D460
1 +u1D706u1D441u1D460
uni0029.alt04
A?10 B0A?10 = 1SNR
uni0028.alt041?u1D706u1D441
u1D460
1 +u1D706u1D441u1D460
uni0029.alt04
A?10 B0A?10 . (6.66)
Considering the symbol decision errors in the decision-directed channel estimation, define
symbol error as
?u1D460u1D452(u1D45B)? := ?u1D460(u1D45B)? ?u1D460(u1D45B)? =
??
?
0 with probability 1?u1D443u1D460u1D452
u1D451 with probability u1D443u1D460u1D452 (6.67)
where u1D451 = 1 is the distance between two nearby symbols in the QPSK constellation diagram
and u1D443u1D460u1D452 is the symbol error rate. Then the variance of the symbol error is given by
u1D438{u1D460u1D452(u1D45B)u1D460?u1D452(u1D45B)} = u1D438
uni007B.alt02
?u1D460u1D452(u1D45B)?2
uni007D.alt02
= u1D4512u1D70E2u1D460u1D443u1D460u1D452 = u1D70E2u1D460u1D443u1D460u1D452. (6.68)
Using the symbol decision, one may rewrite the received signal as
y(u1D45B) =
u1D43Funi2211.alt02
u1D459=0
h(u1D45B;u1D459)u1D460(u1D45B?u1D459) +v(u1D45B)
=
u1D43Funi2211.alt02
u1D459=0
h(u1D45B;u1D459)[?u1D460(u1D45B?u1D459) +{u1D460(u1D45B?u1D459)? ?u1D460(u1D45B?u1D459)}] +v(u1D45B)
=
u1D43Funi2211.alt02
u1D459=0
h(u1D45B;u1D459)?u1D460(u1D45B?u1D459) +
u1D43Funi2211.alt02
u1D459=0
h(u1D45B;u1D459)u1D460u1D452 (u1D45B?u1D459) +v(u1D45B)
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=:vu1D452(u1D45B)
(6.69)
148
where vu1D452(u1D45B) is the ?effective? noise including the symbol decision errors. In order to com-
pensate the symbol decision errors, we take the effective noise variance u1D70E2u1D463u1D452 as
u1D438
uni007B.alt02
vu1D452(u1D45B)vu1D43Bu1D452 (u1D45B)
uni007D.alt02
= u1D70E2u1D463Iu1D441 +
u1D43Funi2211.alt02
u1D4591=0
u1D43Funi2211.alt02
u1D4592=0
u1D438
uni007B.alt02
h(u1D45B;u1D4591)hu1D43B(u1D45B;u1D4592)
uni007D.alt02
u1D438{u1D460u1D452(u1D45B?u1D4591)u1D460?u1D452(u1D45B?u1D4592)}
= u1D70E2u1D463Iu1D441 +u1D70E2u1D460u1D443u1D460u1D452
u1D43Funi2211.alt02
u1D459=0
u1D438
uni007B.alt02
h(u1D45B;u1D459)hu1D43B(u1D45B;u1D459)
uni007D.alt02
(6.70)
For zero-mean, complex Gaussian and spatially white channel with variance u1D70E2? = 1/(u1D43F+1),
it can be simplified as
u1D70E2u1D463u1D452 = u1D70E2u1D463 +u1D70E2u1D460u1D443u1D460u1D452. (6.71)
The probability of symbol error may be approximated as u1D443u1D460u1D452 ? 2u1D443u1D44Fu1D452 for high SNR?s (as is
necessary for practical QPSK systems), where u1D443u1D44Fu1D452 is a specific bit error rate. Using this
approximation, the effective noise variance is given by
u1D70E2u1D463u1D452 ?u1D70E2u1D463 +u1D70E2u1D460 ?2u1D443u1D44Fu1D452 (6.72)
and we can replace u1D70E2u1D463 in (6.66) with u1D70E2u1D463u1D452 in (6.72). For example, the effective variance of
noise is 3u1D70E2u1D463 for u1D443u1D44Fu1D452 = 0.01, SNR = 20dB.
Meanwhile, the other of MSE1 in (6.50) is a squared bias of channel estimate. Given
the true system model in (6.45), we can take the expected of ?h in (6.42) as
u1D438
uni007B.alt02
h ? h(u1D461u1D45F)
uni007D.alt02
= [u1D438{A}]?1u1D438
uni007B.alt02
B ? h(u1D461u1D45F)
uni007D.alt02
. (6.73)
Similarly using (6.44), (6.45) and a matrix A0 defined in (6.61), we have
u1D438
uni007B.alt02
B ? h(u1D461u1D45F)
uni007D.alt02
= u1D438
uni007B.alt04 1
u1D441u1D460
?uni2211.alt02
u1D456=0
u1D706u1D456Cu1D43B(u1D45D?u1D456)C(u1D45D?u1D456)h(u1D461u1D45F)(u1D45D?u1D456)
uni007D.alt04
+u1D438
uni007B.alt04 1
u1D441u1D460
?uni2211.alt02
u1D456=0
u1D706u1D456Cu1D43B(u1D45D?u1D456)vu1D45Au1D460(u1D45D?u1D456)
uni007D.alt04
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=0
= u1D70E
2
u1D460
u1D441u1D460
?uni2211.alt02
u1D456=0
u1D706u1D456?u1D43B(u1D45D?u1D456)?(u1D45D?u1D456)h(u1D461u1D45F)(u1D45D?u1D456)
= u1D70E2u1D460
?uni2211.alt02
u1D45A=1
u1D706(u1D45A?1)u1D441u1D460h(u1D461u1D45F)u1D45A
uni005B.alt04 1
u1D441u1D460
u1D441u1D460?1uni2211.alt02
u1D456=0
u1D706u1D456?u1D43B(u1D45D?u1D456)?(u1D45D?u1D456)
uni005D.alt04
= u1D70E2u1D460
?uni2211.alt02
u1D45A=1
u1D706(u1D45A?1)u1D441u1D460h(u1D461u1D45F)u1D45A A0. (6.74)
149
Using (6.61) and (6.74), we can rewrite (6.73) as
u1D438
uni007B.alt02
h ? h(u1D461u1D45F)
uni007D.alt02
= (1?u1D706u1D441u1D460)
uni005B.alt02
h(u1D461u1D45F)1 +u1D706u1D441u1D460h(u1D461u1D45F)2 +???
uni005D.alt02
, (6.75)
and it follows that for largeu1D441u1D460 (recall we are estimating the most recent true BEM coefficient
vector h(u1D461u1D45F)1 via ?h),
u1D438
uni007B.alt02?
h?h(u1D461u1D45F)1 ? h(u1D461u1D45F)
uni007D.alt02?
= u1D706u1D441u1D460
uni005B.alt02
h(u1D461u1D45F)2 ?h(u1D461u1D45F)1
uni005D.alt02
. (6.76)
For WSSUS channel, where h(u1D461u1D45F)1 and h(u1D461u1D45F)2 have the same statistics and assuming that
u1D438
uni007B.alt03vextenddoublevextenddouble
vextenddoubleh(u1D461u1D45F)1
vextenddoublevextenddouble
vextenddouble2
uni007D.alt03
= u1D438
uni007B.alt03vextenddoublevextenddouble
vextenddoubleh(u1D461u1D45F)2
vextenddoublevextenddouble
vextenddouble2
uni007D.alt03
= 1, the squared bias of channel estimate in (6.50) can be
simplified as
u1D438
uni007B.alt03
u1D438
uni007B.alt03vextenddoublevextenddouble
vextenddouble?h?h(u1D461u1D45F)1
vextenddoublevextenddouble
vextenddouble2 ? h(u1D461u1D45F)
uni007D.alt03uni007D.alt03
= u1D7062u1D441u1D460u1D438
uni007B.alt03uni005B.alt02
h(u1D461u1D45F)2 ?h(u1D461u1D45F)1
uni005D.alt02u1D43B uni005B.alt02
h(u1D461u1D45F)2 ?h(u1D461u1D45F)1
uni005D.alt02uni007D.alt03
= u1D7062u1D441u1D460
uni0028.alt03
u1D438
uni007B.alt03vextenddoublevextenddouble
vextenddoubleh(u1D461u1D45F)1
vextenddoublevextenddouble
vextenddouble2
uni007D.alt03
+u1D438
uni007B.alt03vextenddoublevextenddouble
vextenddoubleh(u1D461u1D45F)2
vextenddoublevextenddouble
vextenddouble2
uni007D.alt03uni0029.alt03
= 2u1D7062u1D441u1D460 (6.77)
So far, we obtain the MSE1 of the channel estimate using (6.50), (6.66) and (6.77),
MSE1
uni0028.alt02?
h
uni0029.alt02
= u1D461u1D45F
uni007B.alt02
cov
uni007B.alt02?
h
uni007D.alt02uni007D.alt02
+ 2u1D7062u1D441u1D460, (6.78)
and the MSE of channel estimate based on CE-BEM by (6.47), (6.49) and (6.78).
0.9 0.92 0.94 0.96 0.98 1?15
?10
?5
0
5
10
DD/EW?RLS,N=2,K=1,L=2,T=400,Q=9,fdTs=0.01,SNR=14dB,Pbe=0.001
?
MSE of Channel Estimation (dB)
 
 
ms=1
ms=2
ms=4
(a) u1D443u1D44Fu1D452 = 0.02
0.9 0.92 0.94 0.96 0.98 1?15
?10
?5
0
5
10
DD/EW?RLS,N=2,K=1,L=2,T=400,Q=9,fdTs=0.01,SNR=14dB,Pbe=BER
?
MSE of Channel Estimation (dB)
 
 
ms=1
ms=2
ms=4
(b) u1D443u1D44Fu1D452 = BER in Fig. 6.9
Figure 6.13: Decision-directed EW-RLS tracking: MSE with different step size u1D45Au1D460?s for
forgetting factor u1D706?s, under u1D441 = 2,u1D453u1D451u1D447u1D460 = 0.01,SNR = 14u1D451u1D435.
150
In Fig. 6.13a, a theoretical MSE of EW-RLS decision-directed channel tracking is
shown for u1D441 = 2,u1D43E = 1,SNR = 14dB over different u1D706 values in the fast-fading channel
(u1D453u1D451 = 400u1D43Bu1D467). We set u1D447 = 400,u1D444 = 9 for CE-BEM and u1D45Au1D460 = 1,2,and 4 for the step sizes.
We prespecified u1D443u1D44Fu1D452 = 0.002 in (6.72) for the effective noise variance including the symbol
decision errors. One can pick a suitable u1D706 to get the minimum-mean-square-error (MMSE)
for one?s step size. The MMSE?s of different u1D45Au1D460?s are similar although the respective suitable
range of u1D706?s are different. It is clear the smaller step size (i.e., u1D45Au1D460 = 1) is more sensitive to u1D706
than the larger ones (i.e., u1D45Au1D460 = 4) although the decision-directed channel tracking with the
smaller one has more complexity.
In Fig. 6.13b, the actual BER values based on 500 Monte Carlo runs in Fig. 6.9 are use
for u1D443u1D44Fu1D452 in (6.72) to obtain the more realistic analysis. The MSE curves shrink inside since
larger u1D443u1D44Fu1D452?s increase the covariance of channel estimates.
6.5 Conclusions
A decision-directed channel tracking approach is investigated for doubly-selective chan-
nel in this paper. We exploited a CE-BEM to capture the overall time-variations of the
channel, and an AR model to update the BEM coefficients. Since the time-varying nature
of the channel is typically well captured in the CE-BEM by the known exponential basis
functions, the time-variations of the unknown BEM coefficients are likely much slower than
those of channels. Our decision-directed scheme tracks the slow-varying BEM coefficients via
Kalman filtering and extended exponentially-weighted RLS, using the detected information
symbols of MMSE-DFE as pseudo-trainings. A time gap, arising from the decision-directed
tracking, occurred between the symbol decisions and the required channel estimates of the
DFE, which could be successfully bridged by channel prediction using the CE-BEM and
the estimated BEM coefficients. Simulation examples illustrated that our decision-directed
approach has superior performance over the AR channel model-based tracking scheme of [6];
also, it achieves comparable performances to the subblock tracking scheme proposed in Chap-
ter 3 and 4, with much less training overhead.
151
Chapter 7
Summary and Future Work
7.1 Summary of Original Work
An adaptive channel estimation approach, exploiting TM training and subblock-wise
tracking, was presented for frequency-selective time-varying channels. One block of u1D447u1D435 sym-
bols comprises several subblocks of u1D45Au1D44F symbols. We employ the CE-BEM to describe the
channel. Rather than track each channel tap, we estimate the BEM coefficients subblock-by-
subblock and then regenerate the channel via CE-BEM with the estimated BEM coefficients.
In this way, the modeling mismatch introduced by the conventionally used symbol-wise AR
channel model can be greatly reduced, and hence better performance can be achieved in fast-
fading environments. The performance of symbol-wise channel tracking in [68,76] (without
turbo equalization procedure) deteriorates as Doppler spread increases. Since the time-
varying nature of the channel is well captured in the CE-BEM by the known exponential
basis functions, the time variations of the unknown BEM coefficients are likely much slower
than those of the real channel and thus more convenient to track. The existing BEM-
based block-wise channel tracking in [70] works well only with the oversampled CE-BEM,
which makes it hard to satisfy the parameter identifiability requirement. The BEM-based
subblock-wise tracking is insensitive to the parameter identifiability and maintains a satis-
factory performance with a flexible subblock size (i.e., a flexible training overhead). Hence
the subblock-wise channel tracking using CE-BEM outperforms symbol-wise or block-wise
channel estimation schemes.
In Chapter 3, the Kalman filter-based subblock-wise tracking was proposed, where we
assume the BEM coefficients follow a first-order AR model. We analyzed the choice of the
AR(1) coefficient for the subblock-wise Kalman channel estimation. However, this first-order
AR assumption is an arbitrary a priori model that is not necessarily true for a real-world
channel and possibly incurs modeling error in channel estimation. With no a priori models
for the BEM coefficients, we considered adaptive filtering algorithms with finite memory:
EW-RLS and SW-RLS algorithms. It was shown that they are superior to Kalman filtering
for the subblock-wise channel tracking. In particular, subblock-wise EW-RLS tracking is a
good alternative since it has the same computational complexity as Kalman tracking. We
also analyze the theoretical choice of the forgetting factoru1D706in EW-RLS for the subblock-wise
tracking using CE-BEM.
152
In Chapter 5, we investigated the BEM-based approach to coded modulation commu-
nication systems using turbo equalization receiver. Based on the subblock-wise BEM-based
approach, we extended the turbo equalization approach of [68] using AR modeling of chan-
nels to channels using CE-BEM where an adaptive turbo equalizer with non-linear Kalman
filtering is coupled with a soft-in soft-out (SfiSfo) decoder to iteratively perform equalization
and decoding using soft information feedback. Assuming the BEM coefficients follow an
AR(1) model, the adaptive equalizer jointly optimizes the estimates of the BEM coefficients
and data symbols, thereby automatically accounting for the correlation between the chan-
nel estimates and data symbols. The proposed CE-BEM-based turbo equalizer has better
performances than the AR-model-based approach in [68], even with the same computational
complexity. We need only a few iterations to have a performance close to the optimal maxi-
mum a posteriori (MAP) approach or the turbo equalizer with perfect knowledge of the true
channels. Since the time-variations of the BEM coefficients are likely much slower than those
of the real channels, the BEM-based approach is not so sensitive to the Doppler spread as
the AR-model-based one. We also provided EXIT chart analyses that can be used to infer
performance comparisons without running the actual simulations.
The decision-directed tracking of doubly-selective channels using CE-BEM was also
investigated. In [6], the decision-directed scheme was proposed that relies on the AR model
only. Acting as virtual training symbols, the information symbol decisions, provided by a
DFE (with delayu1D451? 0) utilizing the estimated channel, were used to enhance the estimation
of the BEM coefficients, so that much of the spectrum resource allocated to training can
be saved. Although a time gap still exists between the available symbol decisions and the
channel estimates required by the DFE, it can be successfully bridged by the CE-BEM-based
channel prediction, without incurring much estimation variance. Similar to the other BEM-
based approaches, we exploited CE-BEM to capture the overall time-variations of the channel
and the first-order AR model to update the BEM coefficients. Our decision-directed scheme
updates the BEM coefficients for the step size u1D45Au1D460, which is much smaller than the BEM
period (i.e., u1D45Au1D460 ? u1D447), via the Kalman filtering or the extended EW-RLS algorithm. With
a larger subblock size (i.e., a small training overhead), the CE-BEM-based decision-directed
tracking outperforms the subblock-wise tracking of Chapter 3 and Chapter 4. The AR-
model-based decision-directed scheme in [6] with the comparable complexity (lettingu1D443 = u1D444)
performs better only for the small range of slow fading. Since the BEM-based decision-
directed tracking updates the BEM coefficients for the step size u1D45Au1D460 ? 1 symbol, we can
reduce the computational complexity by increasing the step size (e.g., u1D45Au1D460 = 2 or 4 symbols)
with little loss in performance. It was shown that the approach of [6] based on an AR model
153
only has serious performance loss with increasing Doppler spread. However, the CE-BEM-
based decision-directed approach maintains satisfactory performance and thereby is more
reliable in practice, where we do not know the actual Doppler spread. We also presented a
performance analysis to provide guidelines for the theoretical choice of the forgetting factor
u1D706 for the decision-directed tracking using CE-BEM.
All the channel estimation and equalization schemes were extended to MIMO systems,
and simulation examples were provided for performance comparisons based on Monte Carlo
simulations.
7.2 Suggested Future Work
So far we have discussed doubly-selective channel estimation and equalization using
exponential basis models, including subblock-wise channel tracking, turbo equalization and
decision-directed tracking. Future work may include the following areas.
First, one should investigate other training schemes for the subblock-wise tracking in
Chapter 3 and 4 that is more efficient than the time-multiplexed (TM) training. The TM
training may waste resources for long channel length or multiple users. Since the subblock-
wise channel estimation depends only on the training session, it is important to find a training
scheme that can save system resources and overcome the user interference in a MIMO system.
For the turbo equalization using the extended Kalman filter in Chapter 5, one may
reduce the complexity by increasing the number of symbols for one recursion of the channel
estimation and detection algorithm, namely, use step size u1D45Au1D460 > 1. Since the neighboring
symbols have very similar BEM coefficients in CE-BEM with large block size (u1D447u1D435 ? u1D45Au1D460),
we do not need to update the BEM coefficient every symbol. Although the algorithm can
be more ?complicated?, the overall computational complexity decreases with little loss of
performances; we already have similar examples in decision-directed tracking with u1D45Au1D460 =
1,2 and 4 in Chapter 6.
In Section 6.4, the optimal value for the forgetting factor in EW-RLS decision-directed
tracking varies with SNR; as SNR increases, the optimalu1D706becomes smaller (farther from one)
due to the lower covariance of the channel estimate in (6.66). One can expect improvement
in performance by using a variable forgetting factor dependent on the measured SNR. For
instance, [7,25,58] investigated the variable forgetting factor RLS algorithm. The Kalman
filter with a variable forgetting factor has also been proposed in some papers [24,34].
Finally, one may use other basis expansion models instead of the Fourier basis function
based CE-BEM, i.e., discrete prolate spheroidal basis expansion model (DPS-BEM) in [62,63]
or the orthogonal polynomial basis expansion model (OP-BEM) in [8,26].
154
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