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
vii
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 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 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>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 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 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
Bibliography
[1] A. Anastassopoulos and K. Chugg, ?Iterative equalization/decoding for TCM for
frequency-selective channels,? Proc. Asilomar, vol. 1, pp. 177-181, Nov. 1997.
[2] A.H. Sayed, Fundamentals of Adaptive Filtering, New York, NJ: Wiley, 2003.
[3] A. Lozano and C. Papadias, ?Layered space-time receivers for frequency-selective wire-
less channels,? IEEE Trans. Commun., vol. 50, pp. 65-73, Jan. 2002.
[4] C. Berrou, A. Glavieux, and P. Thitimajshima, ?Near Shannon limit error-correcting
coding and decoding: Turbo codes,? in Proc. IEEE Int. Conf. on Commun., Geneva,
Switzerland, May 1993.
[5] C. Douillard, M. J?z?quel, C. Berrou, A. Picart, P. Didier and A. Glavieux, ?Iterative
correction of intersymbol interference: Turbo-equalization,? Eur. Trans. Telecommun.,
vol. 6, pp. 507-511, Sept.-Oct. 1995.
[6] C. Komninakis, C. Fragouli, A.H. Sayed and R.D. Wesel, ?Multi-input multi-output
fading channel tracking and equalization using Kalman estimation,? IEEE Trans. Signal
Process., vol. 50, no. 5, pp. 1065-1076, May 2002.
[7] C. Paleologu, J. Benesty, and S. Ciochina, ?A Robust Variable Forgetting Factor Re-
cursive Least-Squares Algorithm for System Identification,? IEEE Signal Processing
Letters, vol. 15, pp. 597-600, 2008.
[8] D.K. Borah and B.D. Hart, ?Frequency-selective fading channel estimation with a poly-
nomial time-varying channel model,? IEEE Tran. Comm., vol. 47, no. 6, pp. 862-873,
Jun. 1999.
[9] E. Karami and M. Shiva, ?Blind multi-input?multi-output channel tracking using
decision-directed maximum-likelihood estimation,? IEEE Trans. Veh. Technol., vol. 56,
no. 3, pp. 1447-1454, May 2007.
[10] F. Vogelbruch and S. Haar, ?Low complexity turbo equalization based on soft feedback
interference cancelation,? IEEE Commun. Letters, vol. 9, no. 7, pp. 586-588, July 2005.
155
[11] G.B. Giannakis and C. Tepedelenlio?glu, ?Basis expansion models and diversity tech-
niques for blind identification and equalization of time-varying channels,? Proc. IEEE,
vol. 86, pp. 1969-1986, Nov. 1998.
[12] G. Bauch and N. Al-Dhahir, ?Reduced complexity space-time turbo equalization for
frequency-selective MIMO channels,? IEEE Trans. Wireless Commun., vol. 1, pp. 819-
828, Oct. 2002.
[13] G. Forney, ?Maximum-likelihood sequence estimation of digital sequences in the presence
of intersymbol interference,? IEEE Trans. Inform. Theory, vol. IT-18, pp. 363?378, May
1972.
[14] G. Leus, ?On the estimation of rapidly time-varying channels,? in Proc. European Signal
Proc. Conf., pp. 2227-2230, Vienna, Austria, Sept. 6-10, 2004.
[15] G.L. St?ber, Principles of Mobile Communication, 2nd ed., Boston, MA: Kluwer, 2002.
[16] H.S. Wang and P.-C. Chang, ?On verifying the first-order Markovian assumption for a
Rayleigh fading channel model,"IEEE Trans. Veh. Technol., vol. 45, no. 2, pp. 353-357,
May 1996.
[17] I. Barhumi, G. Leus, and M. Moonen, ?Time-varying FIR decision feedback equalization
of doubly-selective channels,? in Proc. IEEE Global Commun. Conf, pp. 2263-2268, San
Francisco, CA, Dec. 2003.
[18] I. Barhumi, G. Leus, and M. Moonen, ?Estimation and direct equalization of doubly
selective channels,? in EURASIP Journal on Applied Signal Process., article ID: 62831,
pp. 1-15, vol. 2006.
[19] J.G. Proakis, Digital Communications, 4th ed., NY: McGraw-Hill, 2001.
[20] J.G. Proakis and D.G. Manolakis, Digital Signal Processing, 3rd Ed., Upper Saddle
River, NJ: Prentice-Hall, 1996.
[21] J. Gao and H. Liu, ?Decision-directed estimation of MIMO time-varying Rayleigh fading
channels,? IEEE Trans. Signal Process., vol. 4, no. 4, pp. 1412-1417, Jul. 2005.
[22] J.K. Cavers, ?An analysis of pilot symbol assisted modulation for Rayleigh fading chan-
nels,? IEEE Trans. Veh. Technol., vol. 40, pp. 686-693, Nov. 1991.
156
[23] J.K. Tugnait and S. He, ?Recursive least-squares doubly-selective channel estimation
using exponential basis models and subblock-wise tracking,? in Proc. IEEE ICASSP
2008, Las Vegas, NV, pp. 2861-2864, Mar. 31 - Apr. 4, 2008.
[24] J. Lim, J. Song and K. Sung, ?Forward-backward time varying forgetting factor Kalman
Filter based DOA estimation algorithm for UAV (unmanned aerial vehicle) autoland-
ing,? in Proc. IEEE International Conference on Acoustics, Speech, and Signal Process-
ing 2002 (ICASSP ?02), vol. 4, pp. IV-3964 ? IV-3967, 2002.
[25] Junfeng Wang, ?A variable forgetting factor RLS adaptive filtering algorithm,? 3rd IEEE
International Symposium on Microwave, Antenna, Propagation and EMC Technologies
for Wireless Commun., pp. 1127-1130, Oct. 27-29, 2009.
[26] K.A.D. Teo, S. Ohno and T. Hinamoto, ?Kalman channel estimation based on over-
sampled polynomial model for OFDM over doubly-selective channel,? in Proc. IEEE
SPAWC, New York, NY, June 2005, pp. 116-120.
[27] K.E. Baddour and N.C. Beaulieu, ?Autoregressive modeling for fading channel simula-
tion,? IEEE Trans. Wireless Communications, vol. 4, pp. 1650-1662, July 2005.
[28] K. Huber and S. Haykin, ?Improved Bayesian MIMO channel tracking for wireless com-
munications: incorporating a dynamical model," IEEE Trans. Wireless Commun., vol.
5, pp. 2468-2476, Sept. 2006.
[29] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, ?Optimal Decoding of Linear Codes for
minimizing symbol error rate,? IEEE Trans. Inform. Theory, vol. IT-20, pp. 284?287,
March 1974.
[30] L. Yang, X. Ma, and G.B. Giannakis, ?Optimal training for MIMO fading channels
with time- and frequency- selectivity,? in Proc. IEEE Int. Conf. Acoust, Speech, Signal
Process.(ICASSP) 2004, vol. 3, pp. 17-21, May 2004.
[31] M. Dong, L. Tong and B.M. Sadler, ?Optimal insertion of pilot symbols for transmissions
over time-varying flat fading channels,? IEEE Trans. Signal Process., vol. 52, no. 5, pp.
1403-1418, May 2004.
[32] M.D. Srinath, P.K. Rajasekaran and R. Viswanathan, Introduction to Statistical Signal
Processing with Applications, Upper Saddle River, NJ: Prentice-Hall, 1996.
[33] M.F. Pop and N.C. Beaulieu, ?Limitations of sum-of-sinusoids fading channel simula-
tors,? IEEE Trans. Commun., vol. 49, no. 4, pp. 699-708, Apr. 2001.
157
[34] M.G. Pappas and J.E. Doss, ?Design of a Parallel Kalman Filter with Variable Forgetting
Factors,? American Control Conference, 1988, pp. 2368-2372, June 15-17, 1988.
[35] M. Martone, ?Wavelet-based separating kernels for array processing of cellular
DS/CDMA signals in fast fading,? IEEE Trans. Commun., vol. 48, no. 6, pp. 979-995,
Jun. 2000.
[36] M. Nissil? and S. Pasupathy, ?Adaptive Bayesian and EM-based detectors for frequency-
selective fading channels,? IEEE Trans. Commun., vol. 51, nov. 8, pp. 1325-1336, June
2003.
[37] M. T?chler and J. Hagenauer, ?EXIT charts of irregular codes,? in Proc. Conference on
Information Science and Systems, CISS 2002, March 20-22, 2002.
[38] M. T?chler, A. Singer and R. Koetter, ?Minimum mean squared error equalization using
a priori information,? IEEE Trans. Signal Process., vol. 50, no. 3, pp. 673-683, March
2002.
[39] M. T?chler, R. Koetter and A. Singer, ?Turbo equalization: Principles and new results,?
IEEE Trans. Commun., vol. 50, no. 5, pp. 754-767, May 2002.
[40] N. Al-Dhahir and J.M. Cioffi, ?MMSE decision-feedbck equalizers: finite-length results,?
IEEE Trans. Inf. Theory, vol. 41, no. 4, pp. 961-975, Jul. 1995.
[41] P. Banelli, R.C. Cannizzaro and L. Rugini, ?Data-aided Kalman tracking for channel
estimation in Doppler-affected OFDM systems,? in Proc. IEEE ICASSP 2007, vol. 3,
Honolulu, HI, April 2007, pp. 133-136.
[42] P. Stoica and R. Moses, Spectral Analysis of Signals, Upper Saddle River, NJ: Pearson
Prentice Hall, 2005.
[43] R. Bosisio, M. Nicoli and U. Spagnolini, ?Kalman filter of channel modes in time-varying
wireless systems,? in Proc. IEEE ICASSP ?05, vol. 3, pp. 785-788, Philadelphia, PA,
Mar. 18-23, 2005.
[44] R.C. Cannizzaro and P. Banelli and G. Leus, ?Adaptive channel estimation for OFDM
systems with Doppler spread,? in Proc. IEEE Workshop Signal Process. Advances in
Wireless Commun., Cannes, France, Jul. 2-5, 2006.
[45] R.H. Clarke, ?A statistical theory of mobile-radio reception,? Bell Syst. Tech. J., vol.
47, no. 6, pp. 957-1000, July-Aug. 1968.
158
[46] R. Koetter, A.C. Singer and M. T?chler, ?Turbo Equalization,? IEEE Signal Proc. Mag.,
vol. 21, pp. 67-80, 2004.
[47] R. Otnes and M. T?chler, ?EXIT chart analysis applied to adaptive turbo equalization,?
in Proc. IEEE Nordic Signal Proc. Symp., Hurtigruta, Troms?-Trondheim, Norway, Oct.
4-7, 2002.
[48] R. Otnes and M. T?chler, ?Iterative channel estimation for turbo equalization for time-
varying frequency-selective channels,? IEEE Trans. Wireless Commun., vol. 3, pp. 1918-
1923, Dec. 2004.
[49] R.R. Lopes and J.R. Berry, ?The soft-feedback equalizer for turbo equalization of highly
dispersive channels,? IEEE Trans. Commun., vol. 54, no. 5, pp. 783-788, May 2006.
[50] S. Haykin, Adaptive Filter Theory, 4rd Ed., Upper Saddle River, NJ: Prentice Hall,
2001.
[51] S. He, ?Doubly Selective Channel Estimation and Equalization using Superimposed
Training and Basis Expansion Models,? Ph.D. Dissertation, Auburn University, Auburn,
AL, 2007.
[52] S. He and J.K. Tugnait, ?Doubly-selective channel estimation using exponential basis
models and subblock tracking,? in Proc. IEEE GLOBECOM ?07, Washington, DC, Nov.
26-30, 2007.
[53] S. Kalyani and K. Giridhar, ?Mitigation of error propagation in decision directed OFDM
channel tracking using generalized M estimators,? IEEE Trans. Signal Process., vol. 55,
no. 5, pp. 1659-1672, May 2007.
[54] S. Lin and J.J. Costello, Error Control Coding., Englewood Cliffs, NJ: Prentice-Hall,
1983.
[55] S.M. Kay, Modern Spectral Analysis: Theory and Application, Englewoods Cliffs, NJ:
Prentice Hall, 1988.
[56] S. Roy and T.M. Duman, ?Soft input soft output Kalman equalizer for MIMO frequency
selective fading channels,? IEEE Trans. Wireless Commun., vol. 6, no. 2, pp. 506-514,
Feb. 2007.
[57] S. Song, A. Singer and K. Sung, ?Soft input channel estimation for turbo equalization,?
IEEE Trans. Signal Process., vol. 52, pp. 2885-2894, Oct. 2004.
159
[58] S. Song, J. Lim, S. Baek and K. Sung, ?Variable forgetting factor linear least squares
algorithm for frequency selective fading channel estimation,? IEEE Trans. on Veh. Tech-
nol., vol. 51, no. 3, pp. 613-616, May 2002.
[59] S. ten Brink, ?Convergence behavior of iterative decoded parallel concatenated codes,?
IEEE Trans. on Commun., vol. 49, no. 10, pp. 1727-1737, Oct. 2001.
[60] T.S. Rappaport, Wireless Communications: Principles and Practice, 2nd Ed., Upper
Saddle River, NJ: Prentice-Hall, 2002.
[61] T. S?derstr?m, P. Stoica, System Identification, Prentice-Hall Intern.: New York, NY,
1989.
[62] T. Zemen and C.F. Mecklenbr?uker, ?Time-variant channel equalization via discrete
prolate spheroidal Sequences,? in Proc. 37th Asilomar Conf. Signals, Syst. Comput.,
Pacific Grove, CA, invited paper, pp. 1288-1292, Nov. 2003.
[63] T. Zemen and C.F. Mecklenbr?uker, ?Time-variant channel estimation using discrete
prolate spheroidal sequences,? IEEE Trans. Signal Process., vol. 53, no. 9, pp. 3597-
3607, Sept. 2005.
[64] W. Chen and R. Zhang, ?Estimation of time and frequency selective channels in OFDM
systems: a Kalman filter structure,? in Proc. IEEE GLOBECOM ?04, vol. 2, pp. 800-
803. Dallas, TX, Nov. 29 - Dec. 3, 2004.
[65] W.C. Jakes, Microwave Mobile Communications, New York, NY: Wiley, 1974.
[66] W. Koch and A. Baier, ?Optimum and sub-optimum detection of coded data disturbed
by time varying intersymbol interference,? in Proc. of the IEEE Global Telecomm. Conf.,
Dec. 1990, pp. 1679?1684.
[67] X. Li and J.A. Ritcey, ?Bit-interleaved coded modulation with iterative decoding,? Proc.
IEEE Intern. Conf. Commun., pp. 858-864, June 1999.
[68] X. Li and T.F. Wong, ?Turbo equalization with nonlinear Kalman filtering for time-
varying frequency-selective fading channels,"IEEE Trans. Wireless Commun., vol. 6,
pp. 691-700, Feb. 2007.
[69] X. Ma and G.B. Giannakis, ?Maximum-diveristy transmissions over doubly selective
wireless channels,? IEEE Trans. Inf. Theory, vol. 49, no. 7, pp. 1832-1840, July 2003.
160
[70] X. Ma, G.B. Giannakis and S. Ohno, ?Optimal training for block transmissions over
doubly selective channels,? IEEE Trans. Signal Process., vol. 51, no. 5, pp. 1351-1366,
May 2003.
[71] X. Wang and H. Poor, ?Iterative (turbo) soft interference cancellation and decoding for
coded CDMA,? IEEE Trans. Commun., vol. 47, no. 7, pp. 1046-1061, July 1999.
[72] X. Wang and H. Poor, Wireless Communication Systems: Advanced Techiques for Signal
Reception, Upper Saddle River, NJ: Prentice Hall, 2004.
[73] X. Zhu and R. Murch, ?Layered space-time equalization for wireless MIMO systems,?
IEEE Trans. Wireless Commun., vol. 10, pp. 57-60, no. 3, Mar. 2003.
[74] Y. Li, B. Vucetic, and Y. Sato, ?Optimum soft-ouput detection for channel with inter-
symbol interference,? IEEE Trans. Inform. Theory, vol. 41, pp. 704?713, May 1995.
[75] Y.R. Zheng and C. Xiao, ?Simulation models with correct statistical properties for
Rayleigh fading channels,? IEEE Trans. Commun., vol?51, no. 6, pp. 920-928, June
2003.
[76] Z. Liu, X. Ma and G.B. Giannakis, ?Space-time coding and Kalman filtering for time-
selective fading channels,? IEEE Trans. Commun., vol. 50, pp. 183-186, Feb. 2002.
[77] Z. Tang, R.C. Cannizzaro, G. Leus and P. Banelli, ?Pilot-assisted time-varying channel
estimation for OFDM systems,? IEEE Trans. Signal Process., vol. 55, no. 5, pp. 2226-
2238, May 2007.
161