CHANNEL ESTIMATION AND EQUALIZATION FOR DOUBLY-SELECTIVE CHANNELS USING
BASIS EXPANSION MODELS
Except where reference is made to the work of others, the work described in this dissertation is my
own or was done in collaboration with my advisory committee. This dissertation does not include
proprietary or classified information.
Liying Song
Certificate of Approval:
Stanley J. Reeves
Professor
Electrical and Computer Engineering
Jitendra K. Tugnait, Chair
James B. Davis Professor
Electrical and Computer Engineering
Shiwen Mao
Assistant Professor
Electrical and Computer Engineering
Joe F. Pittman
Interim Dean
Graduate School
CHANNEL ESTIMATION AND EQUALIZATION FOR DOUBLY-SELECTIVE CHANNELS USING
BASIS EXPANSION MODELS
Liying Song
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
May 10, 2008
CHANNEL ESTIMATION AND EQUALIZATION FOR DOUBLY-SELECTIVE CHANNELS USING
BASIS EXPANSION MODELS
Liying Song
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
VITA
Liying Song, daughter of Zhoutian Song and Shuqin Qian, was born in Daqing, Heilongjiang
Province, China, on May 25, 1980. She received her B. E. and M. S. degree in Electrical Engineer-
ing from University of Electronics Science and Technology of China in 2001 and 2004, respectively.
Since August 2004 she has been a Ph. D student in Electrical and Computer Engineering Depart-
ment of Auburn University.
Her research interests are in wireless communications and signal processing, including channel
estimation and equalization for time- and frequency-selective channels, OFDM for wireless com-
munications, and MIMO technique.
iv
DISSERTATION ABSTRACT
CHANNEL ESTIMATION AND EQUALIZATION FOR DOUBLY-SELECTIVE CHANNELS USING
BASIS EXPANSION MODELS
Liying Song
Doctor of Philosophy, May 10, 2008
(M. S., Univ. of Electronics Sci. and Tech. of China, 2001)
(B. E., Univ. of Electronics Sci. and Tech. of China, 2004)
166 Typed Pages
Directed by Jitendra K. Tugnait
The nature of the wireless channels places fundamental limitations on the performance of wire-
less communication systems. In addition to the frequency-selectivity characteristics caused by mul-
tipath propagation, the high-rate wireless and mobile links often exhibit time-selectivity charac-
teristics caused by the user?s mobility, so-called doubly-selective wireless channels. The quality
of channel acquisition has a major impact on the overall system performance. Therefore, reliable
estimation of doubly-selective channels is well motivated. Equalization is used at the receiver to
compensate for intersymbol interference created by multipath propagation and improve received
signal quality. Equalizers should be adaptive since the channel is time-varying.
In this dissertation, channel estimation and equalization for doubly-selective channels are con-
sidered in Chapter 2 (under single input single output models) and Chapter 3 (under multiple input
multiple output models), where the time-varying channel is assumed to be well described by basis
expansion models (BEM). Our focus is on time-multiplexed training for channel estimation where
the training symbols are periodically inserted and use all transmitted power during their transmis-
sion.
v
The linear equalization and decision feedback equalization (DFE) of doubly-selective channels
modeled via BEMs are introduced in Chapter 4. There has been much interest in designing time-
variant serial finite impulse response (FIR) linear and DFE equalizers using complex exponential
(CE-) BEMs for equalizers in addition to using CE-BEM for modeling the channel itself. In this
dissertation we show that the Kalman filter formulation of the linear equalizer and an alternative
formulation of the FIR DFE based on a CE-BEM channel model yields the same or an improved
BER at a lower computational cost, without incurring the approximation error inherent in CE-BEM
modeling of equalizers.
In Chapter 5, an adaptive channel estimation scheme, exploiting the oversampled complex
exponential basis expansion model (CE-BEM), is presented for doubly-selective channels where
we track the BEM coefficients via a multiple model approach in this dissertation. We propose to
use a multiple model framework where several candidate Doppler spread values are used to cover
the range from zero to an upper bound, which leads to multiple CE-BEM channel models, each
corresponding to an assumed value of the Doppler spread. Subsequently, the well known interacting
multiple model (IMM) algorithm is used for symbol detection based on multiple state-space models
corresponding to the multiple estimated channels.
Orthogonal Frequency-Division Multiplexing (OFDM), a digital multi-carrier modulation scheme,
has developed into a popular scheme for wideband wireless communication due to its high spec-
tral efficiency and simple equalization. We extend the optimum time-multiplexed training based
channel estimation introduced in Chapter 2 to OFDM systems under doubly-selective channels in
Chapter 6. Compared to the traditional frequency-domain training design, the main advantages of
time-domain training for OFDM system is that the information symbols are not contaminated by
the training symbols as in the frequency-domain training case.
vi
ACKNOWLEDGMENTS
It has been a great pleasure working with the faculty, staff, and students at the Electrical and
Computer Engineering Department, Auburn University, during my tenure as a doctoral student.
Completing this work is definitely a high point in my academic career. I could not have come this
far without the assistance of many individuals and I want to express my deepest appreciation to
them.
My first and most earnest thanks go to my advisor, Dr. Jitendra K. Tugnait for his support,
guidance, and kindness. In every sense, none of this work would have been possible without him.
I would like to sincerely thank my committee members Dr. Stanley J. Reeves and Dr. Shiwen
Mao who have given their time to read this manuscript and offered valuable advice during my
graduate studies. Many thanks to Dr. Douglas A. Leonard who served as my outside reader for
providing valuable comments that improved the contents of this dissertation.
I gratefully acknowledge Dr. Xiaoli Ma who gave me the chance to start my PhD journey at
Auburn University. I also wish to thank my colleagues and friends whose encouragement over the
years has been invaluable.
My final, and most heartfelt, acknowledgment must go to my parents. They have raised me
and have been giving their love and support all the time.
I would end this list of acknowledgments not with a ?thank you? but with a warm memory.
vii
Style manual or journal used Journal of Approximation Theory (together with the style known
as ?aums?). Bibliography follows van Leunen?s A Handbook for Scholars.
Computer software used The document preparation package TEX (specifically LATEX) together
with the departmental style-file aums.sty.
viii
TABLE OF CONTENTS
LIST OF FIGURES xii
LIST OF TABLES xv
1 INTRODUCTION 1
1.1 Characteristics and Representations of Wireless Communication Channels . . . . . 1
1.1.1 Jakes? Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Complex Exponential Basis Expansion Model (CE-BEM) . . . . . . . . . 5
1.1.3 Discrete Prolate Spheroidal Basis Expansion Model (DPS-BEM) . . . . . . 7
1.1.4 Modeling Error Comparison between CE-BEM and DPS-BEM . . . . . . 9
1.2 Channel Estimation and Equalization Approaches . . . . . . . . . . . . . . . . . . 11
1.2.1 Channel Estimation Approaches . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Equalization Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Optimal Maximum Likelihood Detector (Viterbi Decoder) . . . . . . . . . . . . . 14
1.4 Orthogonal Frequency Division Multiplexing (OFDM) . . . . . . . . . . . . . . . 17
1.5 Outline and Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 SISO DOUBLY-SELECTIVE CHANNEL ESTIMATION USING DISCRETE PROLATE
SPHEROIDAL BASIS EXPANSION MODELS AND TIME-MULTIPLEXED TRAINING 22
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 Least-squares Channel Estimation . . . . . . . . . . . . . . . . . . . . . . 29
2.3.2 Linear Minimum Mean-Square-Error Channel Estimation . . . . . . . . . 30
2.3.3 Channel Estimation Error . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Optimum Training Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5 Training Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.6.1 Example 1: Approximation Errors . . . . . . . . . . . . . . . . . . . . . 41
2.6.2 Example 2: DPS-BEM Channel Estimation Performance . . . . . . . . . . 44
2.6.3 Example 3: CE-BEM versus DPS-BEM . . . . . . . . . . . . . . . . . . . 46
2.6.4 Example 4. Training Power Allocation . . . . . . . . . . . . . . . . . . . 47
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
ix
3 MIMO DOUBLY-SELECTIVE CHANNEL ESTIMATION USING DISCRETE PROLATE
SPHEROIDAL BASIS EXPANSION MODELS AND TIME-MULTIPLEXED TRAINING 50
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Multiuser MIMO Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.1 Linear Least-Squares Channel Estimator . . . . . . . . . . . . . . . . . . . 55
3.3.2 Linear MMSE Channel Estimator . . . . . . . . . . . . . . . . . . . . . . 56
3.3.3 Channel Estimation Error . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4 Optimum Training Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 Training Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.6.1 Example 1: Approximation Errors . . . . . . . . . . . . . . . . . . . . . . 70
3.6.2 Example 2: Channel Estimation Performance . . . . . . . . . . . . . . . . 71
3.6.3 Example 3: CE-BEM versus DPS-BEM . . . . . . . . . . . . . . . . . . . 74
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4 TIME-VARYING EQUALIZATION FOR DOUBLY-SELECTIVE CHANNELS 76
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Linear Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.2 Existing Linear Time-Varying MMSE Equalizers . . . . . . . . . . . . . . 80
4.2.3 Kalman Filter with Equalization Delay d . . . . . . . . . . . . . . . . . . 83
4.2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3 Decision Feedback Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.3.2 Time-Varying FIR MMSE DFEs . . . . . . . . . . . . . . . . . . . . . . . 94
4.3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5 A MULTIPLE MODEL APPROACH TO DOUBLY-SELECTIVE CHANNEL ESTIMATION US-
ING EXPONENTIAL BASIS MODELS 104
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2 System Model and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.2 Block-Adaptive Channel Estimation [48] . . . . . . . . . . . . . . . . . . 107
5.2.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Multiple Model Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.1 Multiple Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.2 Interacting Multiple Model Data Detection . . . . . . . . . . . . . . . . . 110
5.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4.1 Example 1: IMM with Three Models . . . . . . . . . . . . . . . . . . . . 113
5.4.2 Example 2: IMM with Two Models . . . . . . . . . . . . . . . . . . . . . 115
x
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6 DOUBLY-SELECTIVE CHANNEL ESTIMATION FOR OFDM SYSTEMS USING DPS-BEM
AND TIME-MULTIPLEXED TRAINING 123
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.3 Doubly-Selective Channel Estimation for OFDM Systems Using Frequency-
Domain Training [62] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4.1 Example 1: Channel Estimation Performance . . . . . . . . . . . . . . . . 129
6.4.2 Example 2: CE-BEM versus DPS-BEM . . . . . . . . . . . . . . . . . . . 130
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7 SUMMARY AND FUTURE WORK 133
7.1 Summary of Original Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2 Possible Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
BIBLIOGRAPHY 138
A ASYMPTOTIC DPS-BEM/ SLEPIAN SEQUENCES 145
B MATHEMATICAL NOTATIONS 147
C ABBREVIATIONS 149
xi
LIST OF FIGURES
1.1 Basis expansion modeling error comparison between CE-BEM and DPS-BEM models 9
1.2 Subchannels are 312 kHz wide in 802.11a and HyperLAN II . . . . . . . . . . . . 18
1.3 An OFDM transceiver diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Channel estimation errors (with and without modeling errors):Numerical results de-
rived from QF = 2d?DmaxNe+1 and QS = d2?DmaxNe+1, SNR=20dB . . . . 41
2.2 MSE lower bound comparison between LS (2:44) and MMSE (2:45) estimators . . 42
2.3 Comparison between the LS channel estimation MSE lower bound in (2:44) and
simulation results in (2:65), QS = d2?DmaxNe+1 . . . . . . . . . . . . . . . . . 42
2.4 Comparison between the MMSE channel estimation MSE lower bound in (2:45)
and simulation results in (2:65), QS = d2?DmaxNe+1 . . . . . . . . . . . . . . . 43
2.5 LS channel estimation MSE with varying maximum normalized Doppler band-
width, QS = d2?DmaxNe+1,QF = 2d?DmaxNe+1 . . . . . . . . . . . . . . . . 44
2.6 BER with varying maximum normalized Doppler bandwidth using BPSK modula-
tion, QS = d2?DmaxNe+1,QF = 2d?DmaxNe+1 . . . . . . . . . . . . . . . . 45
2.7 BER with varying maximum normalized Doppler bandwidth using QPSK modula-
tion, QS = d2?DmaxNe+1,QF = 2d?DmaxNe+1 . . . . . . . . . . . . . . . . 46
2.8 Simulations-based BER versus fl for SNR=15dB . . . . . . . . . . . . . . . . . . 47
2.9 Theoretical flopt (2:64) versus received signal SNR . . . . . . . . . . . . . . . . . 48
3.1 Channel estimation errors (with and without modeling errors): numerical results
derived from QF = 2d?DmaxNe+1 and QS = d2?DmaxNe+1, SNR=25dB . . 67
3.2 MSE lower bound comparison between LS (3.37) and MMSE (3.38) estimators . . 68
3.3 Comparison between channel estimation MSEs lower bound in (3.37) and simula-
tion results (3.58) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
xii
3.4 Comparison between channel estimation MSEs lower bound in (3.38) and simula-
tion results (3.58) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 Channel estimation MSE with varying maximum normalized Doppler bandwidth,
QF = 2d?DmaxNe and QS = d2?DmaxNe . . . . . . . . . . . . . . . . . . . . . 70
3.6 Channel estimation MSE with varying maximum normalized Doppler bandwidth,
QF = 2d?DmaxNe+1 and QS = d2?DmaxNe+1 . . . . . . . . . . . . . . . . . 71
3.7 BER with varying maximum normalized Doppler bandwidth for BPSK modulation,
QF = 2d?DmaxNe and QS = d2?DmaxNe . . . . . . . . . . . . . . . . . . . . . 72
3.8 BER with varying maximum normalized Doppler bandwidth for QPSK modulation,
QF = 2d?DmaxNe and QS = d2?DmaxNe . . . . . . . . . . . . . . . . . . . . . 73
3.9 BER with varying maximum normalized Doppler bandwidth for BPSK modulation,
QF = 2d?DmaxNe+1 and QS = d2?DmaxNe+1 . . . . . . . . . . . . . . . . . 73
3.10 BER with varying maximum normalized Doppler bandwidth for QPSK modulation,
QF = 2d?DmaxNe+1 and QS = d2?DmaxNe+1 . . . . . . . . . . . . . . . . . 74
4.1 BER performance versus SNR using BPSK modulation, averaged over 10,000 runs.
BLM denotes the method of [5]; FRESH-OPT and FRESH-SUBOPT are methods
of [68]. N = 50, P = 100, Q = 4, L = 3, d = 5, Q0 = 18, L0 = 6 . . . . . . . . . 88
4.2 BER performance versus SNR using QPSK modulation, averaged over 10,000 runs 89
4.3 BER performance versus SNR using BPSK modulation, averaged over 1,000 runs.
BLM denotes the method of [5]; FRESH-OPT and FRESH-SUBOPT are methods
of [68]. Kalman filter is based on P = 800 when Q = 4 and on P = 1600
when Q = 8. P = N corresponds to critically sampled CE-BEM and P = 2N
corresponds to oversampled CE-BEM. N = 800, P = 800 or 1600, Q = 4 or
Q = 8, L = 3, d = 10, Q0 = 12, L0 = 12 . . . . . . . . . . . . . . . . . . . . . . 90
4.4 BER performance versus SNR using QPSK modulation, averaged over 1,000 runs . 91
4.5 BER performance versus SNR For DPS-BEM and CE-BEM channel modeling us-
ing Kalman filtering, BPSK modulation . . . . . . . . . . . . . . . . . . . . . . . 91
4.6 BER performance versus SNR For DPS-BEM and CE-BEM channel modeling us-
ing Kalman filtering, QPSK modulation . . . . . . . . . . . . . . . . . . . . . . . 92
xiii
4.7 BER performance versus SNR, averaged over 1,000 runs, BPSK signal. N = 800,
P = 1600, Q = 8, L = 3, d=5, Q0=12, lf=12, Q00 = 4, lb = 3. R denotes the
number of receive antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.8 BER performance versus SNR, averaged over 1,000 runs, BPSK signal. N = 800,
P = 1600, Q = 8, L = 3, d=5, Q0=12, lf=12, Q00 = 4, lb = 3. R denotes the
number of receive antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.1 IMM with three models, BER vs SNR with BPSK information symbols. . . . . . . 113
5.2 IMM with three models, BER vs SNR with QPSK information symbols. . . . . . . 114
5.3 IMM with two models. BER vs SNR with BPSK information symbols. Step shape
time-varying Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.4 IMM with two models. BER vs SNR with QPSK information symbols. Step shape
time-varying Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.5 Average mode probability of IMM. Step shape time-varying channel. SNR=20dB . 119
5.6 IMM with two models. BER vs SNR with BPSK information symbols. Linear
Shape Time-Varying Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.7 IMM with two models. BER vs SNR with QPSK information symbols. Linear
Shape Time-Varying Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.8 Average mode probability of IMM. Linear shape time-varying channel. SNR=20dB 120
5.9 Average mode probability of IMM. Time-invariant channel. With training insertion.
SNR=20dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.10 Average mode probability of IMM. Time-invariant channel. Without training inser-
tion. SNR=20dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1 OFDM transceiver block diagram in [62] . . . . . . . . . . . . . . . . . . . . . . . 123
6.2 Proposed OFDM transceiver block diagram . . . . . . . . . . . . . . . . . . . . . 124
6.3 LS Channel estimation MSE comparison between TDE and FDE with varying ?Dmax129
6.4 LS Channel estimation MSE comparison between DPS-BEM and CE-BEM with
varying ?Dmax, TDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.5 BER comparison between DPS-BEM and CE-BEM with varying ?Dmax, TDE . . 131
xiv
LIST OF TABLES
4.1 Operation summary for Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2 Computation complexity; N = 50, P = 2N = 100 . . . . . . . . . . . . . . . . . 87
4.3 Computation complexity; N = 800, P = 2N = 1600 . . . . . . . . . . . . . . . . 88
4.4 Computation complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.1 Summary of the IMM algorithm (one cycle) . . . . . . . . . . . . . . . . . . . . . 111
xv
CHAPTER 1
INTRODUCTION
The doubly-selective channel estimation and equalization algorithms described in this disser-
tation involve some background material in the wireless communication area. We will first briefly
introduce these topics in this chapter.
Starting with the characteristics and representations of wireless channels, two basis expansion
models (BEM) used in this dissertation are introduced and compared. The pros and cons of different
channel estimation and equalization approaches are then discussed. The fundamentals of the Viterbi
decoder, a commonly used information detector in this dissertation, and the principles of Orthogonal
Frequency-Division Multiplexing (OFDM) technology are also introduced.
1.1 Characteristics and Representations of Wireless Communication Channels
In wireless telecommunications, multipath is the propagation phenomenon that results in radio
signals? reaching the receiving antenna by two or more paths. Causes of multipath include atmo-
spheric ducting, ionospheric reflection and refraction, and reflection from terrestrial objects, such
as mountains and buildings.
The effects of multipath include constructive and destructive interference, and phase shifting of
the signal. This causes Rayleigh fading, named after Lord Rayleigh. Rayleigh fading is a statistical
model for the effect of a propagation environment on a radio signal, such as that used by wireless
devices. It assumes that the power of a signal that has passed through such a transmission medium
(also called a communication channel) will vary randomly, or fade, according to a Rayleigh distri-
bution - the radial component of the sum of two uncorrelated Gaussian random variables. Rayleigh
1
fading is used to refer to the rapid fluctuations of the received signal in both dimensions ? time and
frequency. If we assume that fading is caused by the superposition of a large number of independent
scattered components, then the in-phase and quadrature components of the received signal can be
assumed to be independent zero-mean Gaussian processes. The envelope A of the received signal
has a Rayleigh probability density function (pdf) given by
fA (a) :=
8>
<
>:
a
2 exp
?
? a22 2
?
a ? 0;
0 a < 0
(1.1)
with 2 being the time-average power of the received signal before envelope detection. The phase
of the received signal is uniformly distributed with pdf
f? ( ) := 12?; 2 [0;2?): (1.2)
The autocorrelation function of the received signal for two-dimensional isotropic scattering and an
omnidirectional receiving antenna is given by [17]
RA(?) = 2 cos(!c?)J0(!m?) (1.3)
where !c is the carrier radian frequency, J0(?) is the zero-order Bessel function of the first kind and
!m is the maximum Doppler radian frequency spread. The autocorrelation function of the Rayleigh
fading channels is periodic in lag and its envelope decays slowly after the initial zero-crossing.
In a multipath propagation environment, several delayed and scaled versions of the transmitted
signal arrive at the receiver. The span of path delays is called delay spread. Delay spread causes
frequency-selective fading as the channel acts like a tapped delay line filter. Time-selective fading
2
due to scatter or transmitter/receiver motion results in a Doppler spread, i.e., a pure tone spreads
over a finite spectral bandwidth. The frequency- and time- selective Rayleigh fading channel is the
channel model we consider in this proposal.
1.1.1 Jakes? Model
As described above, a Rayleigh fading channel itself can be modeled by generating the real
and imaginary parts of a complex number according to independent normal Gaussian variables.
Any model simulating the Rayleigh fading channel has to exhibit the statistical behaviors given in
(1.1) - (1.3).
In [34], Jakes popularized a time-varying model for Rayleigh fading based on summing sinu-
soids. The model supposes the received signal g(t) at time t is
g(t) = E0
LX
l=1
Cl cos(!ct+!mtcosAl +'l) (1.4)
where E0 is the amplitude of the transmitted cosine wave, Cl is a random variable representing
the attenuation of the l-th path, Al is a random variable representing the angle of arrival of the l-th
ray with respect to the direction of motion of the receiver, 'l is a random variable representing the
phase shift undergone by the l-th ray. Note that the stochastic signal g(t) representing the flat fading
signal can be characterized by L sets of triples (Cl;Al;'l). The random variables Cl, Al, and 'l
are assumed statistically independent.
3
To reduce the complexity, the simplified Jakes? model selects:
Cl = 1pL; (1:5a)
Al = 2?lL ; (1:5b)
'l = 0; (1:5c)
where l = 1;2;:::;L. Furthermore, L is of the form L = 4M +2 where M is a positive integer.
However, the simplifying relationships forced in (1.5) make this simulation model determinis-
tic and wide-sense nonstationary [53]. Various modifications of Jakes? model have been proposed,
which we call the family of Jakes? simulators. Among the Jakes? simulator family, [81] is worthy
of mention since it generates a wide-sense stationary process and its second-order correlation statis-
tics match desired reference model (1.4) exactly. Following [81], the normalized low-pass fading
process of the statistical sum-of-sinusoids simulation model is defined by
X (t) = Xc (t)+jXs (t); (1:6a)
Xc (t) = 2pM
MX
l=1
cos(#l)cos(!mtcosfil +`); (1:6b)
Xs (t) = 2pM
MX
l=1
sin(#l)cos(!mtcosfil +`) (1:6c)
with
fil = 2?l?? + 4M ; l = 1;2;:::;M
where fil, `l, and #l are statistically independent and uniformly distributed over [??;?) for all l.
As M !1, the envelope jXj is Rayleigh distributed and the phase ?X (t) is uniformly distributed
4
over [??;?), for which the pdf?s are given by
fjXj(x) = xexp
?x
2
2
?
; x ? 0;
f?X ( ) = 12?; 2 [??;?):
A minor defect, however, occurs in model (1:6a) when !m = 0 or the Doppler spread is small: A
Rayleigh distribution cannot be guaranteed [77]. This problem can be easily resolved by replacing
a common phase ` by `l, which is also uniformly distributed over [??;?) for all l. The simulation
model is revised as [77]:
X (t) = Xc (t)+jXs (t); (1:7a)
Xc (t) = 2pM
MX
l=1
cos(#l)cos(!mtcosfil +`l); (1:7b)
Xs (t) = 2pM
MX
l=1
sin(#l)cos(!mtcosfil +`l): (1:7c)
1.1.2 Complex Exponential Basis Expansion Model (CE-BEM)
Statistical modeling of the channel is well motivated when time-varying path delays arise due
to a large number of scatterers. Deterministic basis expansion models have gained popularity for
wireless applications, especially when the multipath is caused by a few strong reflectors and path
delays exhibit variations due to the mobiles [26]. The time-varying taps are expressed as a super-
position of time-varying bases (complex exponentials when modeling Doppler effects) with time
invariant coefficients. By assigning time variations to the bases, rapidly fading channels with coher-
ence time as small as a few tens of symbols can be captured.
5
Consider a time-varying channel with impulse response h(t;?) (response at time t to an unit
impulse at time t??) which includes transmit-receive filters as well as doubly-selective propagation
effects. Let s(t) denote the complex baseband, continuous-time input signal (with symbol duration
Ts), and x(t) denote the complex baseband, continuous-time received signal. The noise-free re-
ceived signal x(t) is the convolution of s(t) and h(t;?) [41]:
x(t) =
Z 1
0
h(t;?)s(t??)d?: (1:8)
Let H(f;?) = R1?1h(t;?)e?j2?ftdt be the Fourier transform of h(t;?); H(f;?) is the delay-
Doppler spreading function of the channel. If jH(f;?)j ? 0 for j?j > ?d, then ?d is called the
(multipath) delay-spread of the channel; if jH(f;?)j ? 0 for jfj > fd, the fd is called the Doppler
spread of the channel [13]. If s(t); x(t) and h(t;?) in (1:8) are sampled at symbol rate, then by
[41], for t = nTs 2 [t0;t0 +NTs), the sampled signal x(n) := x(t)jt=nTs has the representation
x(n) =
LX
l=0
h(n;l)s(n?l) (1:9)
where Ts is the symbol duration. Over the block interval of [t0;t0 + NTs), the channel impulse
response h(n;l) can be represented using QF + 1 coefficients fwq(l)gQFq=0, which remain invariant
during this block but are allowed to change for the next block, and QF + 1 Fourier basis functions
that are used to describe the temporal variation of the channel and are common for each block.
Then for the block of [t0;t0+NTs), the discrete-time baseband equivalent channel model based on
complex exponential basis expansion can be described as [13, 41]:
h(n;l) =
QFX
q=0
wq(l)ej!qn; (1:10)
6
where
!q := 2?T (q ? QF2 ); q = 0;1;:::;QF; (1:11a)
L := b?d=Tsc; (1:11b)
QF ? 2dfdNTse: (1:11c)
There are two slightly different CE-BEMs involved in this dissertation. One is referred to as the
critically-sampled CE-BEM because the BEM period equals the length of the observed window.
The other uses a longer BEM period and is thus referred to as the oversampled CE-BEM. Section
4.2.1 provides the reader more details about oversampled CE-BEM. Note that the basis functions in
critically-sampled CE-BEM are orthogonal, while in over-sampled CE-BEM they are not.
1.1.3 Discrete Prolate Spheroidal Basis Expansion Model (DPS-BEM)
It has been known that the Fourier basis function based CE-BEM model has the major draw-
back that the rectangular window associated with the discrete Fourier transform (DFT) introduces
spectral leakage [44], which results in the floor in the bit error rate (BER) [3]. In [76, 77], the BEM
coefficients are expanded by the orthogonal discrete prolate spheroidal (DPS) sequences resulting
in a basis expansion model (DPS-BEM). The DPS sequences have a double orthogonality prop-
erty: They are orthogonal over the finite set f0;:::;N ?1g and the infinite set f?1;:::;1g = Z
simultaneously. This remarkable property enables parameter estimation without the drawbacks of
windowing in the case of the Fourier basis expansion.
Let Ts denote the symbol interval. For a channel with a multipath delay spread of ?d sec
and a Doppler spread of fd Hz, ?Dmax := fdTs is the maximum normalized Doppler bandwidth.
An ideal basis function should have at least two properties: It is band-limited to the normalized
7
frequency range [?fdTs;fdTs]; its energy is time-concentrated in a certain time interval [0;N ?1].
Thus, given the maximum normalized Doppler bandwidth fdTs and the window size N, we seek a
sequence ?(n) to maximize
? =
PN?1
n=0 j?(n)j
2
P1
m=?1j?(m)j2
(1:12)
with the band-limited constraint
?(n) =
Z fdTs
?fdTs
?(f)ej2?fndf (1:13)
where
?(f) =
1X
m=?1
?(m)e?j2?fm: (1:14)
The DPS sequences f?q(n)g give us the solution to the constrained maximization problem [58]. In
DPS-BEM, the DPS vectors ?q := [?q(0);:::;?q(N ? 1)]T 2 RN (called Slepian sequences in
[77], which are time-windowed DPS sequences) with elements ?q(n) for n 2 f0;:::;N ?1g, are
eigenvectors of the matrix C 2RN?N, fulfilling [77]
C?q = ?q?q; (1:15)
where ?1 ? ?1 ????? ?N are eigenvalues of matrix C. The (y;z) entries in matrix C are defined
as:
[C]y;z = sin[2?(y ?z)?Dmax]?(y ?z) ; (1:16)
8
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10?3
10?4
10?3
10?2
10?1
?Dmax
Channel Estiamtion MSE
CE?BEM
DPS?BEM
Figure 1.1: Basis expansion modeling error comparison between CE-BEM and DPS-BEM models
where y;z 2f0;:::;N ?1g. In this case one takes QS ?d2?DmaxNe [77, 58].
1.1.4 Modeling Error Comparison between CE-BEM and DPS-BEM
In the following, we use generic notations Q and ?q(n) to indicate that the expressions are
applicable to both CE-BEM and DPS-BEM basis functions. In a basis expansion representation
over a time-block n = f0;1;:::;N ?1g, it is assumed that [26]
h(n;l) =
QX
q=0
wq(l)?q(n): (1:17)
9
However, the true channel may not be exactly equal to this basis expansion since modeling error
always occurs, so that we have to revise (1:17) as
h(n;l) =
QX
q=0
wq(l)?q(n)+eBEM(n;l); (1:18)
where eBEM(n;l) is the basis expansion modeling error. By the orthogonality principle, eBEM(n;l)
should be orthogonal to the given basis set f?q(n)gQq=0 when the square error PN?1n=0 je(n;l)j2 is
minimized. Then
N?1X
n=0
h(n;l)??q0(n) = wq0(l)
N?1X
n=0
j?q0(n)j2: (1:19)
Since the orthogonal basis functions of CE-BEM and DPS-BEM satisfy PN?1n=0 j?q0(n)j2 = 1, the
BEM coefficient is derived as
wq0(l) =
N?1X
n=0
h(n;l)??q0(n): (1:20)
Based on (1:19) and (1:20), the least squares approximation of the channel is given by
^h(n;l) =
QX
q0=0
wq0(l)?q0(n): (1:21)
Fig. 1.1 shows the modeling error comparison between CE-BEM and DPS-BEM models. In
this example, an SISO doubly-selective Rayleigh fading channel defined in (1.7) that is based on the
modified Jakes model is considered. Since the Jakes model is more realistic as a practical wireless
channel, we are trying to approximate a 3-tap Jakes channel by two BEMs: CE-BEM and DPS-
BEM. We choose a data record length of 400 symbols, Ts = 25?s, average over Mfi = 1000
realizations of randomly generated channels, and plot the channel estimation mean square error
10
(MSE), which is defined as:
MSE := 1M
fiN
MfiX
fi=1
N?1X
n=0
LX
l=0
khfi(n;l)?^hfi(n;l)k2: (1:22)
In Fig 1.1, the basis function dimensions QF and QS change with the maximum Doppler bandwidth
?Dmax := fdTs, and SNR=20dB. From the results, we notice that the channel estimation MSE of
DPS-BEM is usually several orders of magnitude lower than that of the CE-BEM. Clearly DPS-
BEM is better than CE-BEM when approximating a band-limited time-varying channel.
1.2 Channel Estimation and Equalization Approaches
1.2.1 Channel Estimation Approaches
For channel state information (CSI) acquisition, three classes of approaches are available: the
training-based approach, the blind approach and the semi-blind approach. In the following, we
briefly describe these three approaches.
In conventional training-based approaches, training sequences (known to the receiver) are time-
multiplexed with the information sequence [21, 32, 63, 79]. Training symbols can be placed either
at the beginning of each burst (as a preamble) or regularly throughout the burst. In rapidly fading
or quasi-static fading channels, preamble-based training may not work well. This motivates em-
bedding training symbols in each transmitted block, instead of concentrating them at the preamble.
More recently, a superimposed training approach has been explored where the training sequences
are added (superimposed) at a low power to the information sequence at the transmitter before
modulation and transmission [65]. There is no loss in information rate, but the channel estima-
tion using superimposed training will be interfered with by information symbols. The conventional
11
time-multiplexed training approach is attractive especially when it decouples symbol detection from
channel estimation and thus simplifies the receiver implementation and relaxes the required channel
identifiability conditions [52].
Blind approaches have been proposed to mitigate the multipath effects in wireless commu-
nications. Blind equalization algorithms are usually based on optimization procedures trying to
minimize some nonlinear functional(s) of the received samples [66]. Compared with a training-
based approach, a blind approach avoids training and thus makes an efficient use of the available
bandwidth [66, 25]. But on the other hand, blind algorithms typically require longer data records
and entail higher complexity [66]. Other major drawbacks of blind approaches are the slow conver-
gence time of the equalizer and possible misconvergence which takes place when the convergence
process reaches a local minimum of the functional to be minimized. For certain applications these
disadvantages can be unacceptable.
Semi-blind approaches use a combination of training and blind cost functions. A training
sequence is used in a semi-blind approach. At the receiver, the channel estimation depends on both
the known training sequence and the unknown information sequence. This way, in addition to the
information carried by the training symbols, the unknown information sequence is also exploited
to enhance the channel estimation performance. Using a semi-blind approach allows the length of
the training sequence to be shortened compared to the traditional training-based approaches [27,
78]. Compared with training-based approaches and blind approaches, semi-blind approaches have a
relative better bandwidth efficientcy than training-based approaches and converge faster than blind
approaches.
12
1.2.2 Equalization Approaches
With the emergence of next-generation wireless mobile communications, multimedia services
have increasing demands for high data rates and high mobility. The high data rates give rise to
frequency selectivity, while the mobility and carrier offset introduce time selectivity. To confront
the doubly-selective effects of wireless channels, equalizers are usually employed at the receiver
end. The existing equalizers can be divided into two types: block equalizers and serial equalizers.
The interest in block equalization can be motivated as follows. When transmission channels are
affected by both frequency and time selectivity, reliable communication can be achieved by dividing
the information data stream into short blocks and by adding a header of known data to each block.
The known symbols allow us to obtain the reliable channel identification and to prevent interference
between two adjacent blocks. Receiver processing can be carried out on a block-by-block basis so
that if the transmission channel does not change appreciably during the transmission of each block,
the receiver has to cope with the frequency selectivity only. Block equalization strategies are used
to compensate for this channel impairment [36, 18]. However, block equalizers are usually complex
to design since the inversion of a large matrix is required. Especially, since a doubly-selective
channel can not be diagonalized by a channel-independent transformation, the implementation of
block time-varying (TV) equalizers, which collect and process in blocks all the available data in the
received frame, leads to a very high computational complexity [5].
Serial equalizers process few data at a time and provide a flexible trade-off between complexity
and performance [66].
13
1.3 Optimal Maximum Likelihood Detector (Viterbi Decoder)
In a communication system, the role of channel estimation is to aid in extracting the desired
information data from the distorted receive symbols. Next we will briefly review the commonly
used symbol detection technique - Viterbi detector.
The Viterbi algorithm, originally introduced as a method for decoding convolutional codes,
has become one of the most commonly used detectors in digital communications. Forney [23] has
shown in 1972 that the algorithm solves maximum likelihood sequence detection (MLSD) of a
pulse amplitude modulated (PAM) sequence of symbols with finite intersymbol interference (ISI)
and memoryless noise. The algorithm has earned its place in almost every modern digital commu-
nications textbook where it is recognized as the optimal sequence detector for memoryless noise.
Consider a SIMO (single-input multiple-output) FIR (finite impulse response) linear channel
with R outputs and discrete-time impulse response h(n;l). Let fs(n)g denote the input sequence
to the SIMO channel. The noisy channel output is given by
y(n) =
LX
l=0
h(n;l)s(n?l)+?(n) (1:23)
where L+1 is the multipath channel length and ?(n) is the white complex Gaussian noise. Assume
that the white Gaussian noise ?(n) is uncorrelated with fs(n)g, with mean Ef?(n)g = 0 and
Ef[?(n+?)][?(n)]Hg = 2?IR?(?).
Given s(n), y(n) is a R-dim Gaussian random vector with mean PLl=0h(l)s(n?l) and vari-
ance 2?IR. The joint probability density function (pdf) ofy(n) givenfs(n);s(n?1);:::;s(n?L)g
14
is
p(y(n)js(n);:::;s(n?L)) = 1(?
?)R
exp
8<
:?
1
2?
y(n)?
LX
l=0
h(n;l)s(n?l)
29=
; (1:24)
where s(n) = 0 for n < 0. The joint pdf of the random vectors fy(0); y(1); :::; y(N ? 1)g
given the transmitted sequence fs(0); s(1); :::; s(N ?1)g is
p(y(0);:::;y(N ?1)js(0);:::;s(N ?1))
=
N?1Y
n=0
1
(? ?)R exp
8<
:?
1
2?
y(n)?
LX
l=0
h(l)s(n?l)
29=
;;
= 1(?
?)NR
exp
8<
:?
1
2?
N?1X
n=0
y(n)?
LX
l=0
h(l)s(n?l)
29=
;: (1:25)
Taking the logarithm on both sides of the equation above, we have
ln p( y(0);:::;y(N ?1)js(0);:::;s(N ?1))
= ?NRln(? ?)? 1 2
?
N?1X
n=0
y(n)?
LX
l=0
h(n;l)s(n?l)
2
: (1:26)
The ML (maximum likelihood) estimate of the input sequence fs(0); :::; s(N ? 1)g is the one
that maximizes
p(y(0);:::;y(N ?1) j s(0);:::;s(N ?1));
or equivalently maximizes
ln p(y(0);:::;y(N ?1) j s(0);:::;s(N ?1));
15
or minimizes the Euclidean distance
N?1X
n=0
y(n)?
LX
l=0
h(n;l)s(n?l)
2
:
This MLSE (maximum likelihood sequence estimation) criterion is equivalent to the problem of
estimating the state of a discrete-time ?finite-state machine?. In this case, the finite-state machine
is the discrete-time channel with coefficients fh(n;l)g and its state at any time n is represented by
the L most recent input symbols
Staten = (s(n); s(n?1);:::; s(n?L+1))
where s(n) = 0 for n < 0. If the input symbols are M-ary, the finite-state machine has ML states.
Consequently, the channel is described by an ML-state trellis and the Viterbi algorithm may be used
to determine the most probable path through the trellis. In brief, we describe the Viterbi algorithm
in the following 3 steps:
Step 1. We begin with y(L), from which we compute the ML+1 metrics
LX
n=0
y(n)?
LX
l=0
h(n;l)s(n?l)
2
:
The ML+1 possible sequences are divided into ML groups according to the ML states. From each
group, we pick the one with the minimum metric, i.e., the most probable sequence, and assign to
the surviving sequence the metric
PM0(s(L); :::; s(1)) = min
s(0)
8<
:
LX
n=0
y(n)?
LX
l=0
h(n;l)s(n?l)
2 9=
;:
16
The M ?1 remaining sequences from each of the ML groups are discarded.
Step 2. Upon reception of y(L+n), n ? 1, compute the ML+1 metrics
y(L+n)?
LX
l=0
h(n;l)s(L+n?l)
2
+PMn?1(s(L+n?1); :::; s(n)):
Again, the ML+1 sequences are divided into ML groups corresponding to the ML possible state
(s(L + n);:::;s(n + 1)) and the most probable sequence from each group is selected while the
other M ?1 sequences are discarded. The surviving metrics are
PMn(s(L+n); :::; s(n+1))
= min
s(n)
8
<
:
y(L+n)?
LX
l=0
h(n;l)s(L+n?l)
2
+PMn?1(s(L+n?1); :::; s(n))
9
=
;:
(1:27)
Step 3. If y(L+n) is the last received sample, from the ML survivor sequences, pick the one
as the ML (maximum likelihood) sequence estimator which has the minimum metric; otherwise, set
n = n+1 and then go to step 2.
1.4 Orthogonal Frequency Division Multiplexing (OFDM)
Frequency division multiplexing is a technology that transmits multiple signals simultaneously
over a single transmission path, such as a cable or wireless system. Each signal travels within its
own unique frequency range (carrier), which is modulated by the data (text, voice, video, etc.).
Orthogonal frequency division multiplexing (OFDM) distributes the data over a large number
of subchannels that are spaced apart at precise frequencies (see Fig. 1.2). This spacing provides the
?orthogonality? in this technique which prevents the demodulators from seeing frequencies other
17
subchannelmag
nitu
de carrier
channel
Figure 1.2: Subchannels are 312 kHz wide in 802.11a and HyperLAN II
than their own. The benefits of OFDM are high spectral efficiency, resiliency to radio frequency
(RF) interference, and lower multipath distortion. This is useful because in a typical terrestrial
broadcasting scenario there are multipath-channels (i.e. the transmitted signal arrives at the receiver
using various paths of different length). Since multiple versions of the signal interfere with each
other (intersymbol interference) it becomes very hard to extract the original information.
OFDM has already been included in digital audio/video broadcasting (DAB/DVB) standards
in Europe, and has been successfully applied to high-speed digital subscriber line (DSL) modems in
the United States. Recently, it has also been proposed for digital cable television systems and local
area mobile wireless networks, such as those specified in the IEEE802.11a, and the HIPERLAN/2
standards [15]. By implementing an inverse fast Fourier transform (IFFT) at the transmitter and
a fast Fourier transform (FFT) at the receiver, OFDM converts an intersymbol interference (ISI)
channel into parallel ISI-free subchannels with gains equal to the channel?s frequency response val-
ues on the FFT grid. To eliminate interblock interference (IBI) between successive IFFT-processed
blocks, a cyclic prefix (CP) of length no less than the channel order is inserted per transmitted block.
Discarding the CP at the receiver not only suppresses IBI, but also converts the linear channel con-
volution into circular convolution, which facilitates diagonalization of the associated channel matrix
(see, e.g., [71]). An OFDM transceiver diagram is shown in Fig. 1.3.
18
P/S QAM
decoder
invert
channel
=
frequencydomain
equalizer
S/P
quadrature
amplitude
modulation
(QAM)
encoder
N-IFFT
add
cyclic
prefix
P/S D/A +transmit
filter
N-FFT S/P
remove
cyclic
prefix
TRANSMITTER
RECEIVER
N subchannels N complex samples
N complex samplesN subchannels
receive
filter
+
A/D
multipath channel
Bits
00110
Figure 1.3: An OFDM transceiver diagram
1.5 Outline and Contribution
The dissertation is organized in the following chapters and the author?s contributions are as
follows:
Chapter 2: The channel estimation for doubly-selective channels is considered using time-
multiplexed training. The time-varying channel is assumed to be well-described by a basis expan-
sion model using discrete prolate spheroidal sequences as the bases (DPS-BEM). The popular linear
least squares and minimum mean-square-error approaches are exploited to estimate the basis expan-
sion coefficients. Computer simulations based on Monte Carlo runs are provided. With the channel
estimation MSE and bit error rate as the performance measures, we find that the channel estima-
tion based on DPS-BEM significantly outperforms the more widely used complex exponential basis
expansion model-based channel estimation.
19
Certain aspects of pilot symbol aided modulation (PSAM) parameter design for DPS-BEM-
based doubly-selective channels is investigated, following the CE-BEM results in [41, 75]. The
optimum time-multiplexed training structure design based on an asymptotic DPS expression is pre-
sented by minimizing the DPS-BEM-based channel estimation MSE. The training power allocation
problem is also addressed in this chapter.
Chapter 3: The channel estimation approaches proposed in Chapter 2 are then extended to
multiuser MIMO systems in this chapter. We suppose there are multiple users at the transmit side,
and each user has one antenna.
Chapter 4: The conventional Kalman filter is used as a time-varying (TV) minimum mean-
square-error (MMSE) equalizer for doubly-selective channels. A formulation of FIR decision feed-
back equalizer is proposed for doubly-selective channels. The proposed equalizers have the main
advantages that they do not incur the approximation error inherent in BEM modeling of equalizers.
The BER performance and the computational complexity of the proposed design are investigated
by means of Monte Carlo computer simulations, and compared with the existing BEM-based TV
equalizers.
Chapter 5: An adaptive channel estimation scheme, exploiting the oversampled CE-BEM, is
presented for doubly-selective channels where we track the BEM coefficients via a multiple model
approach. In the past work the number of BEM coefficients used to model the doubly-selective
channels for channel estimation has been based on an upper bound on the channel Doppler spread.
The higher the Doppler spread, the more the number of BEM coefficients leading to a higher channel
estimation variance. In this chapter we propose to use a multiple model framework where several
candidate Doppler spread values are used to cover the range from zero to an upper bound, leading to
multiple CE-BEM channel models, each corresponding to an assumed value of the Doppler spread.
20
Subsequently the well known interacting multiple model (IMM) algorithm is used for symbol de-
tection based on multiple state-space models corresponding to the multiple estimated channels.
Chapter 6: The pilot-aided doubly-selective channel estimation for OFDM systems is con-
sidered in this chapter. The time-varying channel is described by CE-BEM or DPS-BEM. The
?optimum? training strategies proposed in Chapter 2 are applied to OFDM systems under doubly-
selective channels. Compared to the traditional frequency-domain training design, the main ad-
vantages of time-domain training for OFDM system is that the information symbols are not con-
taminated by the training symbols as in the frequency-domain training case. The performance of
frequency-domain training-based channel estimation and time-domain training-based channel esti-
mation is presented and compared.
Chapter 7: The dissertation concludes in Chapter 7 with future research topics suggested.
21
CHAPTER 2
SISO DOUBLY-SELECTIVE CHANNEL ESTIMATION USING DISCRETE PROLATE SPHEROIDAL
BASIS EXPANSION MODELS AND TIME-MULTIPLEXED TRAINING
2.1 Introduction
Doubly-selective channel estimation using complex exponential basis expansion model (CE-
BEM) and time-multiplexed training is considered in [41, 75], 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 [44], the bit error rate (BER) suffers an error floor [3,
76]. In [76, 77], 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
significantly outperform CE-BEM-based approaches for the doubly-selective channel estimation
and data detection.
To acquire the channel state information at the receiver, training symbols are usually peri-
odically inserted during transmission, which is known as pilot symbol aided modulation (PSAM)
[16]. Optimization of the PSAM for CE-BEM based doubly-selective channel models has been
considered in [41, 75] where the time-multiplexed training sequence is designed to minimize the
channel estimation mean-square-error (MSE). In the case of CE-BEM with independent basis ex-
pansion coefficients, minimizing the channel estimation MSE is also shown (in [41, 75]) to be
equivalent to maximizing a lower bound on the estimated channel-based average capacity. No such
considerations are to be found in [76, 77] where the doubly-selective is represented by DPS-BEM.
The linear MMSE channel estimator for DPS-BEM-based MIMO-OFDM (multiple input multiple
22
output-orthogonal frequency division multiplexing) doubly-selective channels is introduced in [49],
but the training design problem is not considered.
In this chapter, we consider the channel estimation for doubly-selective single input single
output (SISO) channels described by DPS-BEM. Both linear least squares (LS) and minimum mean-
square-error (MMSE) estimators are presented and compared.
Our system model is exactly as in [41] except that instead of CE-BEM as in [41] we use DPS-
BEM. In [41, 75], the linear MMSE channel estimator is used, which requires knowledge of the
noise variance and of the covariance matrix of the channel basis expansion coefficients. While the
former may be known at the receiver, the latter is seldom known. In [41], the latter is assumed to
be known and diagonal. For the Jakes? model, the basis expansion coefficients for a given tap are
not mutually uncorrelated, hence the diagonal assumption does not always hold true. The linear LS
channel estimator does not need to know the covariance matrix or make any assumption regarding
its nature. On the other hand, the performance of the MMSE channel estimator is better than that of
the LS estimator; however, the difference is negligible at high SNRs. [41] does not consider the LS
channel estimation whereas [77] has used the LS channel estimator but not the MMSE estimator.
Certain aspects of PSAM parameter design for DPS-BEM-based doubly-selective channels
is also considered, following the CE-BEM results in [41, 75]. Since the Slepian sequences as a
solution to (1:15) are hard to work with analytically, we give asymptotic DPS expressions based on
some heuristic considerations. Then the optimum time-multiplexed training structure design based
on asymptotic DPS expressions is presented by minimizing the DPS-BEM-based channel estimation
MSE. The training power allocation problem is finally addressed.
23
2.2 System Model
Consider a doubly-selective single input single output (SISO) finite impulse response (FIR)
linear channel. Let h(n;l) denote the symbol-rate impulse response (the channel response at time
n to a unit input at time n ? l), where n 2 [0;1;:::;N ? 1] and l 2 [0;L] capture the time-
and frequency- selectivity of the channel, respectively. Over a time-block of size N, given N
orthonormal functions of ?q(n), the following representation is always true
h(n;l) =
N?1X
q=0
wq(l)?q(n); (2:1)
where ?q(n) is the q-th basis function and the basis expansion coefficient wq(l) is fixed over the data
block. As the above representation is not parsimonious, the following BEM is used to approximate
model (2:1):
hBEM(n;l) =
QX
q=0
wq(l)?q(n); (2:2)
where only Q + 1 << N basis functions are involved. Hence, the channel modeling error can be
expressed as:
eBEM(n;l) = h(n;l)?hBEM(n;l) =
N?1X
q=Q+1
wq(l)?q(n): (2:3)
Let Ts denote the symbol interval. For a channel with a multipath delay spread of ?d sec and a
Doppler spread of fd Hz, in the complex exponential basis expansion model (CE-BEM) [56, 38],
24
one takes
?(F)q (n) := ej!qn;
!q := 2?(q ? QF2 )=N;
L := b?d=Tsc and QF := 2d?DmaxNe
where ?Dmax := fdTs is the maximum normalized Doppler bandwidth. After normalization to unit
norm, we have
?(F)q (n) = 1pNe(i2?(q?QF=2)n=N): (2:4)
In the discrete prolate spheroidal sequence-based BEM (DPS-BEM), the DPS vectors ?(S)q 2
RN (called Slepian sequences in [77], which are time-windowed DPS sequences) with elements
?(S)q (n) for n 2f0;:::;N ?1g, are eigenvectors of the matrix C 2RN?N, fulfilling [77]
C?(S)q = ?q?(S)q ; (2:5)
where ?q are eigenvalues of matrix C. The (y;z) entries in matrix C are defined as:
[C]y;z = sin[2?(y ?z)?Dmax]?(y ?z) ; (2:6)
where y;z 2f0;:::;N ?1g. In this case one takes QS ?d2?DmaxNe [77, 58].
In the following, as in [77], we will use a general notation for the basis expansion quantities
?q(n), Q, wq, and h to indicate that all expressions are applicable to any set of orthonormal basis
functions ?q(n).
25
Let fs(n)g denotes transmitter?s information sequence. Using the discrete-time baseband
equivalent channel model, the received sequence at the receive antenna can be written as
x(n) =
LX
l=0
hBEM(n;l)s(n?l)+?(n); (2:7)
where ?(n) is the additive complex Gaussian noise at the receive antenna, with zero-mean and
variance 2?. Plugging (2:2) into (2:7), we can rewrite x(n) as:
x(n) =
QX
q=0
?q(n)
" LX
l=0
wq(l)s(n?l)
#
+?(n): (2:8)
We consider block transmission as in [41], where transmitted symbols are collected into N?1
blocks with s = [s(0);s(1);:::;s(N ? 1)]T as the 0th block and received x(n)?s are also collect
into blocks with x = [x(0);x(1);:::;x(N ? 1)]T as 0th block. To avoid inter-block interference
(IBI), as in [41], L guard zeros are inserted in each block at the transmitter. Then the matrix-vector
input-output relationship of (2:8) is given by
x =
QX
q=0
D?qWqs+?; (2:9)
where ? is defined similarly to x, D?q = diag[?q] with ?q := [?q(0);?q(1);:::;?q(N?1)]T , and
Wq?s are N ?N lower triangular Toeplitz matrices with 1st column [wq(0);wq(1);:::;wq(L);
0;:::;0]T .
26
2.3 Channel Estimation
Since ?q(n) are known by the receiver, the objective of the channel estimator is to find basis
expansion coefficients in (2:2) from the received samples corresponding to the training symbols.
The proposed channel estimation relies on time-multiplexed training symbols at known positions.
As in [41], each transmitted block s consists of J segments (sub-blocks) of training and in-
formation symbols b(n) and c(n), respectively, and each segment has the same length. Then the
general structure of s is
s = [bT1 ;cT1 ;:::;bTJ;cTJ]T; (2:10)
where bj with length Nb and cj with length Nc, for all j 2 [1;J], denote training and information
symbol sub-blocks, respectively. Therefore, N = J(Nb + Nc) with Nb > L. Let M = Nb + Nc
denote the sub-block size. Obviously, the first L symbols in the ?training part? of the jth subblock
of the received signal are contaminated by information symbols in the preceding (j?1)th subblock.
In the similar way, the first L symbols in the ?information part? of the jth subblock of the received
signal are also contaminated by the last L training symbols in the current jth subblock. In order to
avoid the inter-subblock interference (ISBI) so that the channel estimation is decoupled from data
detection, we will choose the first and the last L symbols in each training subblock to be zeros, as
in [41].
Definebj := [b((j?1)M);b((j?1)M+1);:::;b((j?1)M+Nb?1)]T . Further define ?D?q;j =
diag[ ??q;j] where ??q;j = [?q((j?1)M +L);?q((j?1)M +L+1);:::;?q((j?1)M +Nb?1)]T .
Then the ISBI free received subblock can be written as
?xb;j =
QX
q=0
?D?
q;j
?Wqbj + ??b;j; (2:11)
27
where ?x(r)b;j := [xb((j ?1)M + L);xb((j ?1)M + L + 1);:::;xb((j ?1)M + Nb ?1)]T , ??b;j is
defined similarly, and (Nb ?L)?Nb matrix ?Wq is given by
?Wq =
2
66
66
64
wq(L) ::: wq(0)
... ... ...
wq(L) ::: wq(0)
3
77
77
75: (2:12)
Gathering training symbols per block, we obtain
?xb =
QX
q=0
2
66
66
64
?D?
q;1
?Wqb1
...
?D?
q;J
?WqbJ
3
77
77
75+ ??b: (2:13)
According to the commutativity property of convolution, we have ?Wqbj = Bjwq with wq :=
[wq(0);:::;wq(L)]T and Bj a (Nb ?L)?(L+1) Toeplitz matrix given by
Bj :=
2
66
66
64
bj(L) ::: bj(0)
... ... ...
bj(Nb ?1) ::: bj(Nb ?L?1)
3
77
77
75; (2:14)
where bj(l) := b((j ?1)M +l). Therefore, (2:13) can be rewritten as
?xb = 'w + ??b; (2:15)
28
with simple substitutions, where the [J(Nb ?L)]?[(Q+1)(L+1)] matrix
' :=
2
66
66
64
?D?
0;1B1 :::
?D?
Q;1B1
... ... ...
?D?
0;JBJ :::
?D?
Q;JBJ
3
77
77
75; (2:16)
and
w := [wT0 ;wT1 ;:::;wTQ]T: (2:17)
2.3.1 Least-squares Channel Estimation
The linear least-squares (LS) channel estimator based on (2:15) is
^wLS = ?LS?xb; (2:18)
where ?LS = ('H')?1'H. Define the estimation error of BEM parameters as ~wLS := w? ^wLS.
Then the covariance matrix of ~wLS is
R~wLS := E[ ~wLS ~wHLS] = 2??'H'??1 : (2:19)
As a result, the MSE of ~wLS is
2~wLS := tr(R~wLS) = 2?tr?('H')?1?: (2:20)
29
Using [52, Lemma 1], 2~wLS is lower bounded by (S := K(Q+1)(L+1))
2~wLS ? 2?
SX
k=1
1
[('H')]k;k =
2
?
SX
k=1
1
['H']k;k (2:21)
where the equality holds if and only if 'H' is a diagonal matrix. By the arithmetic-geometric
mean inequality [31, p. 535],
2?
SX
k=1
1
['H']k;k ?
2
?S
? SY
k=1
1
['H']k;k
!1=S
(2:22)
where the equality holds iff ?'H'?k;k are all equal. Equivalently, we need 'H' to be a diagonal
matrix with all its diagonal entries equal.
2.3.2 Linear Minimum Mean-Square-Error Channel Estimation
The linear minimum mean-square-error (MMSE) channel estimator based on (2:15) is
^wMMSE = ?MMSE?xb; (2:23)
where ?MMSE = 2?R?1w +('H')?1'H. This estimator requires Rw := E?wwH? to be known
at the receiver. Define the estimation error of BEM parameters as ~wMMSE := w? ^wMMSE. Then
the covariance matrix of ~wMMSE is
R~wMMSE := E[ ~wMMSE ~wHMMSE] =
?
R?1w + 1 2
?
('H')
??1
: (2:24)
30
As a result, the MSE of ~wMMSE is
2~wMMSE := tr(R~wMMSE) = tr
"
R?1w + 1 2
?
('H')
??1#
: (2:25)
Since our analysis allows for correlated channels, the BEM coefficients wq(l) are not necessar-
ily independent. Eigen-decomposition of Rw yields Rw = Uw?wU?1w where U?1w = UHw . Since
tr(AB) = tr(BA), (2:25) can be rewritten as
2~wMMSE = tr
"
??1w + 1 2
?
U?1w ('H')Uw
??1#
: (2:26)
Similar to (2:21), by [52, Lemma 1], 2~wMMSE is lower bounded by
2~wMMSE ?
X
i
1h
??1w + 1 2
?
U?1w ('H')Uw
i
i;i
(2:27)
where the equality holds if and only if U?1w ('H')Uw is a diagonal matrix. This is true if 'H' is
a diagonal matrix with all its diagonal elements equal. Similar as in (2:21)-(2:22), 2~wMMSE is lower
bounded by
2~wMMSE ? S ?
2
64Y
i
1h
??1w + 1 2
?
U?1w ('H')Uw
i
i;i
3
75
1
S
(2:28)
where the equality holds if and only if ??1w + 1 2
?
U?1w ?IR ?('H')?Uw is a diagonal matrix
with all its diagonal entries equal. Equivalently, we need 'H' to be a diagonal matrix with all its
diagonal entries equal provided that diagonal ?w has all its diagonal entries equal; unfortunately,
the latter is not necessarily true (at least for a channel tap with Jakes? spectrum).
31
2.3.3 Channel Estimation Error
Given estimated BEM parameters ^wq(l) via LS or MMSE estimators, the channel impulse
response is then given by:
^hBEM(n;l) =
QX
q=0
^wq(l)?q(n): (2:29)
There are two sources of channel estimation error: one is from the difference between hBEM(n;l)
and ^hBEM(n;l), the other is from the channel modeling error eBEM(n;l) in (2:3). Therefore, the
mean square value of channel estimation error is expressed as:
2~h = N?1
N?1X
n=0
LX
l=0
E
?
hBEM(n;l)?^hBEM(n;l)+eBEM(n;l)
2 : (2:30)
Since the following orthogonality is true for both CE-BEM and DPS-BEM models
E
?h
hBEM(n;l)?^hBEM(n;l)
iH
eBEM(n;l)
= 0; (2:31)
we have
2~h = 2~w +N?1
N?1X
n=0
LX
l=0
E
n
[eBEM(n;l)]H eBEM(n;l)
o
| {z }
2BEM
: (2:32)
The channel modeling error 2BEM has an analytic expression from Niedzwiecki?s results in [51]:
2BEM ? 1?
Z ?
0
?(n;!)trfShh(!)gd!; (2:33)
where ?(n;!) is the instantaneous parameter matching characteristics of a basis function estimator:
?(n;!) = j1?H(n;!)j2; (2:34)
32
H(n;!) is the instantaneous frequency response of the basis expansion estimator:
H(n;!) = fT(n)
N?1X
n0=0
f(n)e?i!(n?n0);
f(n) := [?0(n);?1(n);:::;?Q(n)]T; (2:35)
the power spectral density Shh(!) is derived from the autocorrelation of h(n):
Rhh(?) := E[h?(n+?)h(n)];
Rhh(?) = 12?
Z ?
??
Shh(!)ei!?d!: (2:36)
The modeling error for CE-BEM (referred to as 2CE) and DPS-BEM (referred to as 2DPS) models
are analyzed in [77]. We have also shown some simulation results earlier in Chapter 1 (Fig. 1.1).
For a 3-tap Jakes? channel, 2DPS is several orders of magnitude smaller than 2CE.
2.4 Optimum Training Design
Observe from (2:21) that in order to achieve the lower bound of 2~wLS, The LS estimator re-
quires 'H' to be a diagonal matrix with all its diagonal entries equal. Observe from (2:27) and
(2:28) that in order to achieve the lower bound of 2~wMMSE, The MMSE estimator also requires
'H' to be a diagonal matrix to achieve (2:27) and for both it and ?w to be diagonal with all their
respective diagonal entries equal to achieve (2:28). Unfortunately, in general, the diagonal ?w does
not necessarily have all its diagonal entries equal. We will design the training schemes to make
'H' to be a diagonal matrix with all its diagonal entries equal which will achieve the lower bound
33
in (2:22) for the LS channel estimator and in (2:27) (but not in (2:28)) for the linear MMSE channel
estimator.
Suppose that we choose
'H' = fiI(Q+1)(L+1) for some fi > 0 (2:37)
Then by (2:16) we must have
JX
j=1
BHj ?DH?q1;j ?D?q2;jBj = fiI?(q1 ?q2): (2:38)
For CE-BEM, it turns out that [41]
JX
j=1
?DH?
q1;j
?D?
q2;j =
1
MINb?L?(q1 ?q2) (2:39)
Note that in [41], (2:4) is not normalized; here it is. By (2:16) and (2:38), for all k?s and q?s, the
training sequence should be designed to satisfy
JX
j=1
BHj ?DH?q1;j ?D?q2;jBj =
JX
j=1
?DH?
q1;j
?D?
q2;j: (2:40)
Under (2:39), following [41] and [75], we pick Nb = 2L + 1 and bTj = [0TL;bk;0TL]T where 0L is
a size L null column, in which case 'H' = b2MI. Therefore, with Pb := Jb2 denoting the total
training power, we can obtain fi = PbN (recall that N = JM).
34
Plugging (2:37) into (2:22), (2:28) and (2:32), the LS and MMSE lower bounds of channel
estimation error are derived as
2~h
LS
= N
2?
Pb (L+1)(QF +1)+
2
CE; (2:41)
2~h
MMSE
=
(L+1)(QF+1)X
t=1
?
???1t + Pb
2?N
??1
+ 2CE; (2:42)
where ??t is the t-th diagonal entry of matrix ?w, i.e. ??t is the t-th eigenvalue of Rw.
For DPS-BEM, we will use ?large? N approximation from the Appendix A for DPS-BEM
basis functions to obtain an expression for the modeled part of channel estimation error. Using
heuristic asymptotic (A:6), we can easily establish (note that asymptotic (A:6) corresponds to (2:4)
and (A:3))
JX
j=1
?DH?
q1;j
?D?
q2;j ?
1
MINb?L?(q1 ?q2): (2:43)
Therefore, for DPS-BEM mimicking (2:41) and (2:42), the lower bounds of channel estimation
error are
2~h
LS
? N
2?
Pb (L+1)(QS +1)+
2
DPS; (2:44)
2~h
MMSE
?
(L+1)(QS+1)X
t=1
?
???1t + Pb
2?N
??1
+ 2DPS: (2:45)
Remark 2.1 Note that (2:43) is critical for (2:44) and (2:45) to hold true and for the proposed
training designs to be valid. The asymptotic Slepian sequences in (A:6) are only ?heuristic.? But
we can ?verify? (2:43) by computing the results in (2:20) and (2:22) numerically and compare them
with the results in (2:44). This is done in Section 2.6.
35
2.5 Training Power Allocation
We assume that the time-varying channel h(n;l) is zero-mean, wide sense stationary (in n
with fixed l) complex Gaussian with the same variance 2h. We also assume that the channel taps
are mutually independent, i.e. h(n;l) is wide sense stationary uncorrelated scattering (WSSUS). To
simplify the expressions, in this section we assume that the channel modeling error eBEM in (2:3) is
zero.
The received information symbols at the received antenna can be expressed as
xc(n) =
LX
l=0
^h(n;l)c(n?l)
| {z }
:=xs(n)
+
LX
l=0
[h(n;l)?^h(n;l)]c(n?l)+?(n)
| {z }
=x?(n)
; (2:46)
where ^h(n;l) = PQq=0 ^wq(l)?q(n) is used for data detection. Therefore, the signal power is given
by
2xs(n) := E'jxs(n)j2?
= E
8<
:
2
4
flfl
flfl
fl
LX
l=0
^h(n;l)c(n?l)
flfl
flfl
fl
23
5
9=
;
= ?Pc
LX
l=0
?
Ew
?
E
?flfl
flh(n;l)?^h(n;l)
flfl
fl
2jw +E'jh(n;l)j2??
= ?Pc
h
2~h(n)+(L+1) 2h
i
; (2:47)
where the average power of information symbols is
?Pc := E'jc(n)j2?; (2:48)
36
and the effective noise power is:
2x?(n) = E
8<
:
2
4
flfl
flfl
fl
LX
l=0
[h(n;l)?^h(n;l)]c(n?l)
flfl
flfl
fl
23
5
9=
;+E
'j?(n)j2?
= ?Pc 2~h(n)+ 2?; (2:49)
where 2~h(n) := PLl=0 Ew
?
E
?flfl
flh(n;l)?^h(n;l)
flfl
fl
2jw . Define
Wl := [w0(l);w1(l);:::;wQ(l)]T;
W := [W0;W1;:::;WL]T; (2:50)
By (2:17) and (2:50), we can get the following relationship:
W = ?w; (2:51)
where
? =
2
66
66
66
66
66
66
66
66
66
66
66
64
1
Lz}|{
0:::0 ::: :::
0 0:::0 1
Lz}|{
0:::0 ::: :::
::: ::: :::
0 1
Lz}|{
0:::0 ::: ::: :::
0 0
Lz}|{
0:::0 1
Lz}|{
0:::0 :::
... ... ...
::: ::: ::: ::: ::: 1
3
77
77
77
77
77
77
77
77
77
77
77
75
(Q+1)(L+1)?(Q+1)(L+1)
: (2:52)
37
We also find that ?H? is always an identity matrix. Then 2~h(n) can be rewritten as
2~h(n) =
LX
l=0
EW
?
E
?flfl
flh(n;l)?^h(n;l)
flfl
fl
2jW
= tr
n
?(n)EW
n
covf ^W; ^WjWg
o
?(n)H
o
; (2:54)
where
?(n) := [?0(n);?1(n);:::;?Q(n)];
?(n) := I(L+1) ??(n):
Based on (2:50) and (2:51), we have
EW
n
covf ^W; ^WjWg
o
= ?EW fcovf ^w; ^wjWgg| {z }
R~w
?H: (2:55)
Based on the orthonormality of ?q(n), we have
NX
n=1
?(n)H?(n) = I(L+1)(Q+1): (2:56)
Therefore, the time-averaged of ? 2~h over information subblocks in the current block is
? 2~h := (N ?JNb)?1
X
n
2~h(n) ? N?1
N?1X
n=0
2~h(n) = 1Ntr'?R~w?H? = 1N 2~w (2:57)
38
Similarly, the time averaged signal and noise powers turn out to be
? 2xs := 1N
NX
n=1
2xs(n) = ?Pc[? 2~h +(L+1) 2h];
? 2x? := 1N
NX
n=1
2x?(n) = ?Pc? 2~h + 2?: (2:58)
Therefore, we obtain an effective average SNR for (2:46) as
SNRd = ?
2xs
? 2x?: (2:59)
Define the total information power and received signal power Pc := JNc ?Pc and P := Pb+Pc,
respectively. Define the training power overhead
fl := PbP
c +Pb
:
Our objective is to maximize SNR with respect to fl under the constraint of a fixed P. Thus,
incorporating those constraint-carrying variables into (2:59) and using the developed expression for
average signal and noise powers in (2:59), we obtain the unconstrained cost
SNRd(fl) =
(1?fl)P
JNc [?
2~
h +(L+1)
2
h]
(1?fl)P
JNc ?
2~
h +
2? : (2:60)
Using the lower bound of the LS estimator in (2:44) (due to the lower bound of the MMSE estimator,
the closed form of the optimal fl can not be obtained), we can explicitly write (2:59) as
SNRd(fl) = f1fl
2 +f2fl +f3
g1fl +g2 ; (2:61)
39
where
f1 = ?P(L+1)
2
h
NcJ ;
f2 = P(L+1)
2
h
NcJ ?
(L+1)(Q+1) 2?
NcJ ;
f3 = (L+1)(Q+1)
2?
NcJ ;
g1 = 2? ? (L+1)(Q+1)
2?
NcJ ;
g2 = (L+1)(Q+1)
2?
NcJ : (2:62)
Setting the first derivative of SNRd(fl) with respect to fl to zero, we obtain a quadratic equation in
fl
fl2 +2g2g
1
fl + f2g2 ?f3g1f
1g1
= 0 (2:63)
with two roots, one of which is negative (fl < 0), and hence is excluded. The other root is given by
flopt = g2g
1
"
?1+
s
1+ g1(f3g1 ?f2g2)g2
2f1
#
: (2:64)
2.6 Numerical Examples
In the following examples, we use binary phase shift keying (BPSK) and quadrature phase
shift keying (QPSK) modulation. Each transmitted block has J = 10 subblocks, and each subblock
has Nc = 30 information symbols and Nb = 2L + 1 training symbols with optimal structure
[0L;b;0L], b > 0. A doubly-selective Rayleigh fading channel h(n;l) is simulated according to
[77, 81] with the channel order L = 2, carrier frequency of 2GHz, data rate of 40 kbps, and thus,
symbol duration Ts = 25?s. Therefore, each tap of the generated time-variant channel has a Jakes?
40
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10?3
10?3
10?2
10?1
100
?Dmax
(2.20)
(2.22) right hand side
(2.41)
(2.44)
(2.20)+?DPS2
Figure 2.1: Channel estimation errors (with and without modeling errors):Numerical results derived
from QF = 2d?DmaxNe+1 and QS = d2?DmaxNe+1, SNR=20dB
spectrum; it is not generated using the assumed BEM modeling. Also, 3 taps of the channel are
mutually independent. Depending on different maximum Doppler spread fds, a varying maximum
normalized one-sided Doppler bandwidth ?Dmax = fdTs can be derived. Given ?Dmax, Jakes?
spectrum and other information, we can calculate Rw and therefore, ?w and ??ts, needed in (2:28),
(2:45) and elsewhere. The SNR refers to the ratio of total signal and training power to the total noise
power, each per block.
2.6.1 Example 1: Approximation Errors
As noted earlier in Remark 2.1, here we want to show the influence of approximation errors in
?D?
q;j when the true Slepian sequences instead of the approximations in (A:6) are used. We compute
(2:20) and (2:22) numerically with true Slepian sequences generated by (2:5), then compare them
41
0 5 10 15 20 25 3010
?4
10?3
10?2
10?1
100
101
SNR(dB)
Channel Estimation MSE
fd=40Hz, QS=2, LS Bound
fd=40Hz, QS=2, MMSE Bound
fd=100Hz, QS=3, LS Bound
fd=100Hz, QS=3, MMSE Bound
Figure 2.2: MSE lower bound comparison between LS (2:44) and MMSE (2:45) estimators
0 5 10 15 20 25 3010
?4
10?3
10?2
10?1
100
101
SNR(dB)
Channel Estimation MSE
40Hz, QS=2, simu, MSE
40Hz, QS=2, simu, MSE??
40Hz, QS=2, simu, MSE+?
100Hz, QS=3, simu, MSE
100Hz, QS=3, simu, MSE??
100Hz, QS=3, simu, MSE+?
40Hz, QS=2, LS Bound
100Hz, QS=3, LS Bound
Figure 2.3: Comparison between the LS channel estimation MSE lower bound in (2:44) and simu-
lation results in (2:65), QS = d2?DmaxNe+1
42
0 5 10 15 20 25 3010
?4
10?3
10?2
10?1
100
101
SNR(dB)
Channel Estimation MSE
40Hz, QS=2, simu, MSE
40Hz, QS=2, simu, MSE+?
40Hz, QS=2, simu, MSE??
100Hz, QS=3, simu, MSE
100Hz, QS=3, simu, MSE??
100Hz, QS=3, simu, MSE+?
40Hz, QS=2, MMSE Bound
100Hz, QS=3, MMSE Bound
Figure 2.4: Comparison between the MMSE channel estimation MSE lower bound in (2:45) and
simulation results in (2:65), QS = d2?DmaxNe+1
with (2:41) and (2:44). For the parameters stated earlier in this section (J=10, Nb=8, Nc=30,
M = Nb + Nc=38, N = JM =380) the results are shown in Fig. 2.1, where SNR=20dB and the
dimensions QF = 2d?DmaxNe+1 and QS = d2?DmaxNe+1 change with the maximum Doppler
bandwidth. [Here the minimum QF = 2d?DmaxNe and QS = d2?DmaxNe are not taken since a
slightly higher values of Q?s can significantly reduce the modeling error. This is suggested in [77]].
From Fig. 2.1 we see that the CE-BEM lower bound in (2:41) is high due to ?large? modeling error
2CE even though the error for the modeled part is minimum, whereas the results in (2:20), (2:22)
and (2:44) are close to each other due to ?small? modeling error 2DPS and ?close-to- minimum?
(compare curves for (2:20) (exact error) and (2:22) (lower bound)) error for the modeled part. [We
note that 2DPS and 2CE were obtained via Monte Carlo averaging, similar to [77].]
43
0 5 10 15 2010
?3
10?2
10?1
100
101
SNR(dB)
Channel Estimation MSE
?Dmax(S) =0.001
?Dmax(S) =0.003
?Dmax(S) =0.005
?Dmax(F) =0.001
?Dmax(F) =0.003
?Dmax(F) =0.005
Figure 2.5: LS channel estimation MSE with varying maximum normalized Doppler bandwidth,
QS = d2?DmaxNe+1,QF = 2d?DmaxNe+1
2.6.2 Example 2: DPS-BEM Channel Estimation Performance
In this case we pick b = 1 in training. The LS and MMSE estimators are used to estimate
w, and then the channel is estimated as in (2:29). Based on Mr Monte Carlo runs, the channel
estimation MSE is calculated as (hk corresponds to the kth run)
MSE = (MrN)?1
MrX
k=1
N?1X
n=0
LX
l=0
j^hk(n;l)?hk(n;l)j2: (2:65)
In Fig. 2.2, the lower bounds in (2:44) and (2:45) are plotted using QS = 2 for fd = 40Hz
and using QS = 3 for fd = 100Hz. [The MMSE bound needs ??ts which requires knowledge of
the Doppler spread. Note also that for both fd = 40Hz and fd = 100Hz, the minimum QS =
d2?DmaxNe are actually QS = 1 and QS = 2, respectively; however, a slightly higher value of
44
0 5 10 15 2010
?4
10?3
10?2
10?1
100
SNR(dB)
BER ?Dmax(S) =0.001
?Dmax(S) =0.003
?Dmax(S) =0.005
?Dmax(F) =0.001
?Dmax(F) =0.003
?Dmax(F) =0.005
Figure 2.6: BER with varying maximum normalized Doppler bandwidth using BPSK modulation,
QS = d2?DmaxNe+1,QF = 2d?DmaxNe+1
QS = d2?DmaxNe + 1 = 3 for fd = 100Hz yields the smaller modeling error, which becomes
significant at high SNRs]. Then they are compared with the simulation results (averaged over 200
Monte Carlo runs) in Figs. 2.3 and 2.4 for fd = 40Hz and 100Hz; also shown are ? bounds on
the simulation averages. It can be seen that the theoretical results are consistent with the simulation
results for fd =40Hz indicating that the optimal pilot design does minimize the channel MSE when
using DPS-BEM. For fd=100 Hz, there is a ?small gap? between theory and simulations at high
SNRs which is probably attributable to modeling error (?small? but nonzero). Furthermore, the
performance of the MMSE estimator outperforms that of the LS estimator at low SNR, and they
converge to each other at high SNR.
45
0 5 10 15 20
10?2
10?1
100
SNR(dB)
BER ?Dmax(S) =0.001
?Dmax(S) =0.003
?Dmax(S) =0.005
?Dmax(F) =0.001
?Dmax(F) =0.003
?Dmax(F) =0.005
Figure 2.7: BER with varying maximum normalized Doppler bandwidth using QPSK modulation,
QS = d2?DmaxNe+1,QF = 2d?DmaxNe+1
2.6.3 Example 3: CE-BEM versus DPS-BEM
In Fig. 2.5, the LS channel estimation MSE (2:65) versus SNR under different maximum
Doppler bandwidths (?(S)Dmax is for DPS-BEM, ?(F)Dmax is for CE-BEM) are plotted. It is clear that
the MSE of DPS-BEM is consistently smaller than that of CE-BEM. Fig. 2.6 takes BER (average
over 2000 Monte Carlo runs) as a performance measure to compare the performance between DPS-
BEM and CE-BEM. A Kalman filter formulation is used for information detection after the channel
estimation. Comparing with Fig. 2.5 makes it quite clear that the significantly reduced MSE for the
DPS-BEM channel estimation leads to a pronounced reduction in BER compared to the CE-BEM
case.
46
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910
?4
10?3
10?2
10?1
100
?
BER
vDmax=0.001
vDmax=0.003
vDmax=0.007
vDmax=0.01
Figure 2.8: Simulations-based BER versus fl for SNR=15dB
The plotting in Fig. 2.7 is exactly the same as Fig. 2.6 except that the information sequences
are QPSK signals. It is seen from Figs. 2.6 and 2.7 that the BER by using QPSK information
symbols are worse than that using BPSK symbols, as expected.
2.6.4 Example 4. Training Power Allocation
Here we vary training power (by varying b) with fixed total transmitted power. The BER versus
optimum fl based on simulation results (averaged over 1000 Monte Carlo runs) is shown in Fig. 2.8
for SNR=15dB where we used BPSK modulation and a Viterbi detector based on the estimated
channel for data detection. We also varied ?Dmax according to different maximum Doppler spread
fd. In Fig. 2.9, we plot the optimum theoretical values of fl (derived in (2:64)) versus the received
signal SNR. Comparing Figs. 2.8 and 2.9, we see that the two show mutually consistent results
47
5 10 15 20 25 300.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
SNR(dB)
? opt
vDmax=0.001
vDmax=0.003
vDmax=0.007
vDmax=0.01
Figure 2.9: Theoretical flopt (2:64) versus received signal SNR
supporting our theoretical results: the optimal (simulations based) fl inferred from Fig. 2.8 is in
good agreement with the theoretical flopt of Fig. 2.9.
2.7 Conclusion
The channel estimation for doubly-selective channels was considered using time-multiplexed
training. The time-varying channel was assumed to be well-described by a basis expansion model
using discrete prolate spheroidal sequences as bases. Training designs for time-multiplexed training
based on minimization of the channel estimation mean-square-error were investigated. Both least
squares and minimum mean-square-error approaches were exploited to estimate the basis expansion
coefficients. Computer simulation examples were presented where the channel was generated via
48
Jakes? models with different Doppler spreads. In these examples the DPS-BEM model significantly
outperforms the more widely used complex exponential basis expansion model.
49
CHAPTER 3
MIMO DOUBLY-SELECTIVE CHANNEL ESTIMATION USING DISCRETE PROLATE
SPHEROIDAL BASIS EXPANSION MODELS AND TIME-MULTIPLEXED TRAINING
3.1 Introduction
The prospect of extraordinary improvements in the capacity of wireless networks has drawn
considerable attention to multiple-input multiple-output (MIMO) communication techniques. MIMO
methods employ multiple transmitter and receiver antennas to increase the data rate and to achieve
spatial diversity. Traditionally, multiple antennas have been used on the receiver side to combat the
multipath fading. The receive antennas see independently faded versions of the same signal. The
receiver combines these signals so that the resultant signal exhibits considerably reduced amplitude
variability (fading) in comparison with the signal at any one antenna. This is called diversity gain.
Diversity is characterized by the number of independently fading branches, also known as the di-
versity order and is equal to the number of receive antennas in single-input-multiple output (SIMO)
channels. However, recent advances have shown that using multiple antennas at both the transmit-
ter and the receiver can significantly increase the data rate and improve the performance [22, 24].
By employing multiple transmitter antennas, multiple spatial channels are supported in the same
frequency band, thus data can be transmitted in parallel which results in an increased data rate.
In this chapter, the approaches proposed in Chapter 2 are extended to the MIMO system. We
consider the problem of channel estimation for the doubly-selective MIMO channels described by
DPS-BEM. Time-multiplexed training is used to estimate the channel. Illustrative simulation exam-
ples are provided to show the performance of channel estimation and data detection under a MIMO
system.
50
3.2 Multiuser MIMO Channels
Consider a multiuser channel with K users and R receive antennas leading to a MIMO system
with K inputs and R outputs. Let h(r;k)(n;l) denote the symbol-rate impulse response (the channel
response at time n to a unit input at time n?l) of the doubly-selective MIMO FIR linear channels
between the kth user?s transmit antenna and the rth receive antenna, where n 2 [0;1;:::;N ? 1]
and l 2 [0;L] capture the time- and frequency- selectivity of the channel, respectively. In a general
basis expansion representation over a time-block, the following is always true:
h(r;k)(n;l) =
N?1X
q=0
w(r;k)q (l)?q(n); (3.1)
where ?q(n) is the q-th basis function and the basis expansion coefficient w(r;k)q (l) is fixed over the
data block. As the above representation is not parsimonious, the following basis expansion model
is used to approximate model (3.1):
h(r;k)BEM(n;l) =
QX
q=0
w(r;k)q (l)?q(n); (3.2)
where only Q + 1 ? N basis functions are involved. Hence, the channel modeling error between
h(r;k)(n;l) and h(r;k)BEM(n;l) can be expressed as:
e(r;k)BEM(n;l) := h(r;k)(n;l)?h(r;k)BEM(n;l) =
N?1X
q=Q+1
w(r;k)q (l)?q(n): (3.3)
51
Let fsk(n)g denote the kth transmitter?s information sequence. The received sequence at the
rth receive antenna can be written as
x(r)(n) =
KX
k=1
LX
l=0
h(r;k)(n;l)sk(n?l)+?(r)(n); (3.4)
where ?(r)(n) is the additive complex Gaussian noise at the rth receive antenna, with zero-mean
and variance 2?. Plugging (3.2) into (3.4), we can rewrite x(r)(n) as:
x(r)(n) =
KX
k=1
QX
q=0
?q(n)
" LX
l=0
w(r;k)q (l)sk(n?l)
#
+?(r)(n): (3.5)
We consider block transmission as in [41], where transmitted symbols for the kth transmitter
are collected into N ? 1 blocks with sk = [sk(0);sk(1);:::;sk(N ? 1)]T as the 0th block and
received x(r)(n)?s are also collect into blocks with x(r) = [x(r)(0);x(r)(1);:::;x(r)(N ? 1)]T as
0th block. To avoid inter-block interference (IBI), as in [41], L guard zeros are inserted in each
block at the transmitter. Then the matrix-vector input-output relationship of (3.5) is given by
x(r) =
KX
k=1
QX
q=0
D?qW(r;k)q sk +?(r); (3.6)
where ?(r) is defined similarly to x(r), D?q = diag[?q] with ?q := [?q(0);?q(1);:::;?q(N ?1)],
and W(r;k)q s are N ?N lower triangular Toeplitz matrices with 1st column
h
w(r;k)q (0);w(r;k)q (1);
:::;w(r;k)q (L);0;:::;0
iT
.
52
3.3 Channel Estimation
As in [41], each transmitted block sk consists of J segments (sub-blocks) of training and
information symbols bk(n) and ck(n), respectively, and each segment has the same length. Then
the general structure of sk is
sk = [bTk;1;cTk;1;:::;bTk;J;cTk;J]T; (3.7)
where bk;j with length Nb and ck;j with length Nc, for all j 2 [1;J], denote training and informa-
tion symbol sub-blocks, respectively. Therefore, N = J(Nb+Nc) with Nb > L. Let M = Nb+Nc
denote the sub-block size. Obviously, the first L symbols in the ?training part? of the jth subblock
of the received signal are contaminated by information symbols in the preceding (j?1)th subblock.
In the similar way, the first L symbols in the ?information part? of the jth subblock of the received
signal are also contaminated by the last L training symbols in the current jth subblock. In order to
avoid the inter-subblock interference (ISBI) so that the channel estimation is decoupled from data
detection, we will choose the first and the last L symbols in each training subblock to be zeros, as
in [41].
Define bk;j := [bk((j?1)M);bk((j?1)M +1);:::;bk((j?1)M +Nb?1)]T . Further define
?D?
q;j = diag[
??q;j] where ??q;j = [?q((j?1)M +L);?q((j?1)M +L+1);:::;?q((j?1)M +
Nb ?1)]T . Then the ISBI free received subblock can be written as
?x(r)b;j =
KX
k=1
QX
q=0
?D?
q;j
?W(r;k)q bk;j + ??(r)b;j; (3.8)
53
where ?x(r)b;j := [xb((j ?1)M + L);xb((j ?1)M + L + 1);:::;xb((j ?1)M + Nb ?1)]T , ??b;j is
defined similarly, and (Nb ?L)?Nb matrix ?W(r;k)q is given by
?W(r;k)q =
2
66
66
64
w(r;k)q (L) ::: w(r;k)q (0)
... ... ...
w(r;k)q (L) ::: w(r;k)q (0)
3
77
77
75:
Gathering training symbols per block, we obtain
?x(r)b =
KX
k=1
QX
q=0
2
66
66
64
?D?
q;1
?W(r;k)q bk;1
...
?D?
q;J
?W(r;k)q bk;J
3
77
77
75+ ??
(r)
b : (3.9)
According to the commutativity property of convolution, we have ?W(r;k)q bk;j = Bk;jw(r;k)q
with w(r;k)q := [w(r;k)q (0);:::;w(r;k)q (L)]T and Bk;j a (Nb ?L)?(L+1) Toeplitz matrix given by
Bk;j :=
2
66
66
64
bk;j(L) ::: bk;j(0)
... ... ...
bk;j(Nb ?1) ::: bk;j(Nb ?L?1)
3
77
77
75; (3.10)
where bk;j(l) := bk((j ?1)M +l). Therefore, (4.9) can be rewritten as
?x(r)b =
KX
k=1
'kw(r;k) + ??(r)b = 'w(r) + ??(r)b ; (3.11)
54
with simple substitutions, where the [J(Nb ?L)]?[(Q+1)(L+1)] matrix
'k :=
2
66
66
64
?D?
0;1Bk;1 :::
?D?
Q;1Bk;1
... ... ...
?D?
0;JBk;J :::
?D?
Q;JBk;J
3
77
77
75;
' := ['1;:::;'K] (3.12)
and
w(r;k) :=
2
66
66
64
w(r;k)0
...
w(r;k)Q
3
77
77
75; w
(r) :=
2
66
66
64
w(r;1)
...
w(r;K)
3
77
77
75: (3.13)
Since the matrix ' is common for all receivers, after collecting all ?x(r)b and w(r) for different r, we
get
xb = (IR ?')w +?b; (3.14)
where w is defined as
w :=
?
w(1)T ??? w(R)T
?T
; (3.15)
xb and ?b are defined similarly.
3.3.1 Linear Least-Squares Channel Estimator
The linear least-squares (LS) channel estimator based on (3.14) is
^wLS = ?LSxb; (3.16)
55
where ?LS = (IR ? ('H'))?1(IR ?'H). Define the estimation error of BEM parameters as
~wLS := w? ^wLS, then the covariance matrix of ~wLS is
R~wLS := E[ ~wLS ~wHLS] = 2??IR ?('H')??1 : (3.17)
As a result, the MSE of ~wLS is
2~wLS := tr(R~wLS) = 2?tr
h?
IR ?('H')??1
i
: (3.18)
Using [52, Lemma 1], 2~wLS is lower bounded by (S := K(Q+1)(L+1))
2~wLS ? 2?
RSX
i=1
1
[IR ?('H')]i;i = R
2
?
SX
i=1
1
['H']i;i (3.19)
where the equality holds if and only if 'H' is a diagonal matrix. By the arithmetic-geometric
mean inequality [33, p. 535],
R 2?
SX
i=1
1
['H']i;i ? R
2
?S
? SY
i=1
1
['H']i;i
!1=S
(3.20)
where the equality holds iff ?'H'?i;i are all equal. Equivalently, we need 'H' to be a diagonal
matrix with all its diagonal entries equal.
3.3.2 Linear MMSE Channel Estimator
The linear minimum mean-square-error (MMSE) channel estimator based on (3.14) is
^wMMSE = ?MMSExb; (3.21)
56
where ?MMSE = ? 2?R?1w +(IR ?('H'))??1 (IR ? 'H). This estimator requires Rw :=
E?wwH?to be known at the receiver. Define the estimation error of BEM parameters as ~wMMSE :=
w? ^wMMSE. Then the covariance matrix of ~wMMSE is
R~wMMSE := E[ ~wMMSE ~wHMMSE]
=
?
R?1w + 1 2
?
?I
R ?('H')
???1 : (3.22)
As a result, the MSE of ~wMMSE is
2~wMMSE := tr(R~wMMSE)
= tr
"
R?1w + 1 2
?
?I
R ?('H')
???1
#
(3.23)
Since our analysis allows for correlated channels, the BEM coefficients w(r;k)q (l) are not nec-
essarily independent. Eigen-decomposition of Rw yields Rw = Uw?wU?1w where U?1w = UHw .
Since tr(AB) = tr(BA), (3.23) can be rewritten as
2~wMMSE = tr
"
??1w + 1 2
?
U?1w ?IR ?('H')?Uw
??1#
: (3.24)
Similar as in (3.19), by [52, Lemma 1], 2~wMMSE is lower bounded by
2~wMMSE ?
X
i
1h
??1w + 1 2
?
U?1w (IR ?('H'))Uw
i
i;i
(3.25)
where the equality holds if and only if U?1w ?IR ?('H')?Uw is a diagonal matrix. This is true if
'H' is a diagonal matrix with all its diagonal elements equal. Similar to (3.19)-(3.20), 2~wMMSE is
57
lower bounded by
2~wMMSE ? (RS)?
2
64Y
i
1h
??1w + 1 2
?
U?1w (IR ?('H'))Uw
i
i;i
3
75
1
RS
(3.26)
where the equality holds if and only if ??1w + 1 2
?
U?1w ?IR ?('H')?Uw is a diagonal matrix
with all its diagonal entries equal. Equivalently, we need 'H' to be a diagonal matrix with all its
diagonal entries equal provided that diagonal ?w has all its diagonal entries equal; unfortunately,
the latter is not necessarily true (at least for a channel tap with Jakes? spectrum).
3.3.3 Channel Estimation Error
After deriving the estimated BEM parameters ^w(r;k)q (l) by the LS or MMSE estimators, the
channel impulse response is then given by:
^h(r;k)BEM(n;l) =
QX
q=0
^w(r;k)q (l)?q(n): (3.27)
Here we consider that the channel estimation error is from two sources: one is from the difference
between h(r;k)BEM(n;l) and ^h(r;k)BEM(n;l), the other is from the channel modeling error e(r;k)BEM(n;l) in
(3.3). Therefore, the mean square error of channel estimation error is expressed as:
2~h = N?1
RX
r=1
KX
k=1
N?1X
n=0
LX
l=1
E
?
h(r;k)BEM(n;l)?^h(r;k)BEM(n;l)+e(r;k)BEM(n;l)
2 : (3.28)
Since the following orthogonality is true for both CE-BEM and DPS-BEM models
E
?h
h(r;k)BEM(n;l)?^h(r;k)BEM(n;l)
iH
e(r;k)BEM(n;l)
= 0;
58
we have
2~h = 2~w +N?1
RX
r=1
KX
k=1
N?1X
n=0
LX
l=0
E
?h
e(r;k)BEM(n;l)
iH
e(r;k)BEM(n;l)
| {z }
:= 2BEM
: (3.29)
For simulations presented in Section 3.6 we calculate 2BEM by averaging over Monte Carlo runs.
3.4 Optimum Training Design
Observe from (3.19) that in order to achieve the lower bound of 2~wLS, The LS estimator re-
quires 'H' to be a diagonal matrix with all its diagonal entries equal. Observe from (3.25) and
(3.26) that in order to achieve the lower bound of 2~wMMSE, The MMSE estimator also requires
'H' to be a diagonal matrix to achieve (3.25), and for both it and ?w to be diagonal with all their
respective diagonal entries equal to achieve (3.26). Unfortunately, in general, the diagonal ?w does
not necessarily have all its diagonal entries equal. We will design the training schemes to make
'H' to be a diagonal matrix with all its diagonal entries equal which will achieve the lower bound
in (3.19) for the LS channel estimator and in (3.25) (but not in (3.26)) for the linear MMSE channel
estimator.
Suppose that we choose
'Hk1'k2 =
8>
<
>:
fiI(Q+1)(L+1) ; for k1 = k2
0 ; for k1 6= k2;
(3.30)
59
or, equivalently
PJ
j=1B
H
k1;j ?D
H
?q1;j ?D?q2;jBk2;j = fiI ; for k1 = k2;q1 = q2
PJ
j=1B
H
k1;j ?D
H
?q1;j ?D?q2;jBk2;j = 0 ; for k1 = k2;q1 6= q2
PJ
j=1B
H
k1;j ?D
H
?q1;j ?D?q2;jBk2;j = 0 ; for k1 6= k2;for all q
(3.31)
for some fi > 0.
For CE-BEM, it is shown in [41] that
JX
j=1
?DH?
q1;j
?D?
q2;j =
8>
<
>:
1
MINb?L ; for q1 = q2
0 ; for q1 6= q2
(3.32)
By (3.12) and (3.31), for all k?s and q?s, the training sequence should be designed to satisfy
JX
j=1
BHk1;j ?DH?q1;j ?D?q2;jBk2;j =
JX
j=1
?DH?
q1;j
?D?
q2;j (3.33)
for some > 0. Under (3.32), following [41] and [75], we pick Nb = K(L + 1) + K and
bTk;j = [0k(L+1)?1;bk;0(K?k)(L+1)+L] where 0k(L+1)?1 is a size k(L+1)?1 null column, in
which case 'Hk 'k = b2kMI. Therefore, with Pbk := Jb2k denoting the total training power from the
kth transmitter, we can obtain fi = PbkN (recall that N = JM). Since fi should not be a function of
k, we take bk = b leading to Pbk := Pb for all k.
60
Plugging (3.30) into (3.20), (3.26) and (3.29), the LS and MMSE lower bounds of channel
estimation error are derived as
2~h
LS
= 2~wLS + 2CE = N
2?
Pb RK(L+1)(QF +1)+
2
CE; (3.34)
2~h
MMSE
= 2~wMMSE + 2CE =
RK(L+1)(QF+1)X
t=1
?
???1t + Pb
2?N
??1
+ 2CE; (3.35)
where ??t is the t-th diagonal entry of matrix ?w, i.e. ??t is the t-th eigenvalue of Rw.
For DPS-BEM, using the heuristic asymptotic (A:6), we can easily establish
JX
j=1
?DH?
q1;j
?D?
q2;j =
8>
<
>:
1
MI(Nb?L) ; for q1 = q2
0 ; for q1 6= q2:
(3.36)
Therefore, for DPS-BEM mimicking CE-BEM, the lower bounds of the channel estimation error
are
2~h
LS
= N
2?
Pb RK(L+1)(QS +1)+
2
DPS; (3.37)
2~h
MMSE
=
RK(L+1)(QS+1)X
t=1
?
???1t + Pb
2?N
??1
+ 2DPS: (3.38)
Remark 3.1. Note that (3.36) is critical for (3.37) and (3.38) to hold true and for the proposed
training designs to be valid. The asymptotic Slepian sequences in (A:6) are only ?heuristic.? But
we can ?verify? (3.36) by computing the results in (3.18) and (3.20) numerically and compare them
with the results in (3.37). This is done so in Sec. 3.6.
61
3.5 Training Power Allocation
We assume that the time-varying channel h(r;k)(n;l) is zero-mean, WSS complex Gaussian
with the same variance 2h for each tap. We also assume that the channel taps are mutually indepen-
dent, i.e. h(r;k)(n;l) is WSSUS. To further simplify the analysis, we assume the channel correlation
matrices are the same across r 2 [1;:::;R]. The received information symbols at the rth received
antenna can be expressed as
x(r)c (n) =
KX
k=1
LX
l=0
^h(r;k)(n;l)ck(n?l)
| {z }
:=xs(n)
+
KX
k=1
LX
l=0
[h(r;k)(n;l)?^h(r;k)(n;l)]ck(n?l)+?(r)(n)
| {z }
=x?(n)
;
(3.39)
where ^h(r;k)(n;l) = PQq=0 ^w(r;k)q (l)?q(n) is used for data detection. Therefore, similar to the single
user case in Section 2.5, the signal power of all receivers is given by
2xs(n) =
RX
r=1
E'jxs(n)j2?
=
RX
r=1
E
8<
:
KX
k=1
2
4
flfl
flfl
fl
LX
l=0
^h(r;k)(n;l)ck(n?l)
flfl
flfl
fl
23
5
9=
;
= ?Pc
RX
r=1
KX
k=1
LX
l=0
?
Ew
?
E
?flfl
flh(r;k)(n;l)?^h(r;k)(n;l)
flfl
fl
2jw +Enjh(r;k)(n;l)j2o?
= ?Pc
h
2~h(n)+RK(L+1) 2h
i
; (3.40)
where the average information power per user:
?Pc := E'jck(n)j2?: (3.41)
62
The effective noise power is:
2x?(n) =
RX
r=1
E
8<
:
KX
k=1
2
4
flfl
flfl
fl
LX
l=0
[h(r;k)(n;l)?^h(r;k)(n;l)]ck(n?l)
flfl
flfl
fl
23
5
9=
;+
RX
r=1
E
n
j?(r)(n)j2
o
= ?Pc
RX
r=1
" KX
k=1
LX
l=0
Ew
?
E
?flfl
flh(r;k)(n;l)?^h(r;k)(n;l)
flfl
fl
2jw
#
+R 2?
= ?Pc 2~h(n)+R 2? (3.42)
where 2~h(n) := PRr=1PKk=1PLl=0 Ew
?
E
?flfl
flh(r;k)(n;l)?^h(r;k)(n;l)
flfl
fl
2jw . Define
W(r;k)l := [w(r;k)0 (l);w(r;k)1 (l);:::;w(r;k)Q (l)]T;
W(r;k) := [W(r;k)0 ;W(r;k)1 ;:::;W(r;k)L ]T;
W(r) := [W(r;1);W(r;2);:::;W(r;K)]T;
W := [W(1);W(2);:::;W(R)]T: (3.43)
By (3.15) and (3.43), we can get the following relationship:
W = ?w; (3.44)
63
where
? =
2
66
66
66
66
66
66
66
66
66
66
66
64
1
Lz}|{
0:::0 ::: :::
0 0:::0 1
Lz}|{
0:::0 ::: :::
::: ::: :::
0 1
Lz}|{
0:::0 ::: ::: :::
0 0
Lz}|{
0:::0 1
Lz}|{
0:::0 :::
... ... ...
::: ::: ::: ::: ::: 1
3
77
77
77
77
77
77
77
77
77
77
77
75
(Q+1)(L+1)?(Q+1)(L+1)
(3.45)
and
? = IRK ??: (3.46)
We also find that ?H? is always an identity matrix. Then 2~h(n) can be rewritten as
2~h(n) =
RX
r=1
KX
k=1
LX
l=0
EW
?
E
?flfl
flh(r;k)(n;l)?^h(r;k)(n;l)
flfl
fl
2jW
= tr
n
?(n)EW
n
covf ^W; ^WjWg
o
?(n)H
o
; (3.47)
where
?(n) := [?0(n);?1(n);:::;?Q(n)];
?(n) := IRK(L+1) ??(n):
64
Based on (3.43) and (3.44), we have
EW
n
covf ^W; ^WjWg
o
= ?EW fcovf ^w; ^wjWgg| {z }
R~w
?H: (3.48)
Based on the orthonormality of ?q(n), we have
NX
n=1
?(n)H?(n) = IRK(L+1)(Q+1): (3.49)
Therefore, the time-averaged of ? 2~h over length N is
? 2~h := (N ?JNb)?1
X
n
2~h(n) ? N?1
N?1X
n=0
2~h(n) = 1Ntr'?R~w?H? = 1N 2~w: (3.50)
In a similar way, the time averaged signal and noise power can be expressed as
? 2xc = 1N
NX
n=1
2xc(n) = ?Pc[? 2~h +RK(L+1) 2h];
? 2xn = 1N
NX
n=1
2xn(n) = ?Pc? 2~h +R 2?: (3.51)
Therefore, we obtain an effective average SNR of (3.39) as
SNRd = ?
2xc
? 2xn: (3.52)
Define the total information power and received signal power Pc := JNc ?Pc and P := Pb+Pc,
respectively. Define the training power overhead fl := PbPc+Pb . Our objective is to maximize SNR
with respect to fl under the constraint of a fixed P. Thus, incorporating those constraint-carrying
65
variables into (3.52) and using the developed expression for average signal and noise powers in
(3.52), we obtain the unconstrained cost
SNRd(fl) =
(1?fl)P
JNc [?
2~
h +RK(L+1)
2
h]
(1?fl)P
JNc ?
2~
h +
2? : (3.53)
Using the lower bound of the LS estimator in (3.37) (due to the lower bound of the MMSE estimator,
the closed form of the optimal fl can not be obtained), we can explicitly write (3.52) as
SNRd(fl) = f1fl
2 +f2fl +f3
g1fl +g2 ; (3.54)
where
f1 = ?PRK(L+1)
2
h
NcJ ;
f2 = PRK(L+1)
2
h
NcJ ?
RK(L+1)(Q+1) 2?
NcJ ;
f3 = RK(L+1)(Q+1)
2?
NcJ ;
g1 = 2? ? RK(L+1)(Q+1)
2?
NcJ ;
g2 = RK(L+1)(Q+1)
2?
NcJ : (3.55)
Setting the first derivative of SNR(fl) with respect to fl to zero, we obtain a quadratic equation in fl
fl2 +2g2g
1
fl + f2g2 ?f3g1f
1g1
= 0 (3.56)
66
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10?3
10?3
10?2
10?1
100
?Dmax
(3.18)
(3.20) righ hand side
(3.34)
(3.37)
(3.18)+?DPS2
Figure 3.1: Channel estimation errors (with and without modeling errors): numerical results derived
from QF = 2d?DmaxNe+1 and QS = d2?DmaxNe+1, SNR=25dB
the two roots, one of which is negative (fl < 0), and hence is excluded. The other root is given by
flopt = g2g
1
"
?1+
s
1+ g1(f3g1 ?f2g2)g2
2f1
#
: (3.57)
3.6 Numerical Examples
In the following examples we consider a multiuser system with K = 2 users and R = 2
receivers. We use binary phase shift keying (BPSK) and quadrature phase shift keying (QPSK)
modulation. Each transmitted block has J = 10 subblocks, and each subblock has Nc = 30 infor-
mation symbols and Nb = 3L+2 training symbols with optimal structure ?0k(L+1)?1;b;
67
0 5 10 15 20 25 3010
?3
10?2
10?1
100
101
SNR(dB)
Channel Estimation MSE
fd=40Hz, QS=2, LS Bound
fd=40Hz, QS=3, MMSE Bound
fd=100Hz, QS=2, LS Bound
fd=100Hz, QS=3, MMSE Bound
Figure 3.2: MSE lower bound comparison between LS (3.37) and MMSE (3.38) estimators
0 5 10 15 20 25 3010
?3
10?2
10?1
100
101
SNR(dB)
Channel Estimation MSE
40Hz, QS=2, simu, MSE
40Hz, QS=2, sinu, MSE+?
40Hz, QS=2, sinu, MSE??
40Hz, QS=2, LS Bound
100Hz, QS=3, simu, MSE
100Hz, QS=3, simu, MSE+?
100Hz, QS=3, simu, MSE??
100Hz, QS=3, LS Bound
Figure 3.3: Comparison between channel estimation MSEs lower bound in (3.37) and simulation
results (3.58)
68
0 5 10 15 20 25 3010
?3
10?2
10?1
100
101
SNR(dB)
Channel Esitamation MSE
40Hz, QS=2, simu, MSE
40Hz, QS=2, simu, MSE+?
40Hz, QS=2, simu, MSE??
40Hz, QS=2, MMSE Bound
100Hz, QS=3, simu, MSE
100Hz, QS=3, simu, MSE+?
100Hz, QS=3, simu, MSE??
100Hz, QS=3, MMSE Bound
Figure 3.4: Comparison between channel estimation MSEs lower bound in (3.38) and simulation
results (3.58)
0(K?k)(L+1)+L?, b > 0. A doubly-selective Rayleigh fading channel h(r;k)(n;l) is simulated ac-
cording to [77, 81] with channel order L = 2, carrier frequency of 2GHz, data rate of 40 kbps,
and thus, symbol duration Ts = 25?s. Therefore, each tap of the generated time-variant channel
has a Jakes? spectrum; it is not generated using the assumed BEM modeling. Also, 3 taps of the
channel are mutually independent. Depending on different maximum Doppler spread fds, a varying
maximum normalized one-sided Doppler bandwidth ?Dmax = fdTs can be derived. Given ?Dmax,
Jakes? spectrum and other information, we can calculate Rw and therefore, ?w and ??ts, needed in
(3.25), (3.38) and elsewhere. A Kalman filter formulation is used for information detection after
the channel estimation. The SNR refers to 1= 2? where the information sequence power per user is
normalized to one and the channel power (per user) is also normalized to unity.
69
0 5 10 15 2010
?3
10?2
10?1
100
101
SNR(dB)
Channel Estimation MSE
?Dmax(S) =0
?Dmax(S) =0.001
?Dmax(S) =0.003
?Dmax(S) =0.005
?Dmax(F) =0
?Dmax(F) =0.001
?Dmax(F) =0.003
?Dmax(F) =0.005
Figure 3.5: Channel estimation MSE with varying maximum normalized Doppler bandwidth, QF =
2d?DmaxNe and QS = d2?DmaxNe
3.6.1 Example 1: Approximation Errors
As noted earlier in Remark 3.1, here we want to show the influence of approximation errors in
?D?
q;j when the true Slepian sequences instead of the approximations in (A:6) are used. We com-
pute (3.18) and (3.20) numerically with true Slepian sequences generated by (2:5), then compare
them with (3.34) and (3.37). For the parameters stated earlier in this section (J=10, Nb=8, Nc=30,
M = Nb + Nc=38, N = JM =380) the results are shown in Fig. 3.1, where the dimensions
QF = 2d?DmaxNe and QS = d2?DmaxNe change with the maximum Doppler bandwidth and
SNR=25dB. From Fig. 3.1 we see that the CE-BEM lower bound in (3.34) is high due to ?large?
modeling error 2rmCE even though the error for the modeled part is minimum, whereas the re-
sults in (3.18), (3.20) and (3.37) are close to each other due to ?small? modeling error 2DPS and
?close-to-minimum? (compare curves for (3.18) (exact error) and (3.20) (lower bound)) error for
70
0 5 10 15 2010
?2
10?1
100
101
SNR(dB)
Channel Estimation MSE
?Dmax(S) =0.001
?Dmax(S) =0.003
?Dmax(S) =0.005
?Dmax(F) =0.001
?Dmax(F) =0.003
?Dmax(F) =0.005
Figure 3.6: Channel estimation MSE with varying maximum normalized Doppler bandwidth, QF =
2d?DmaxNe+1 and QS = d2?DmaxNe+1
the modeled part. [We note that 2DPS and 2CE were obtained via Monte Carlo averaging, similar
as in [77].]
3.6.2 Example 2: Channel Estimation Performance
In this case we pick b = 1 in training. The LS and MMSE estimators are used to estimate
w, and then the channel is estimated as ^h(r;k)(n;l) = PQq=0 ^w(r;k)q (l)?q(n). Based on Mr Monte
Carlo runs, the channel estimation MSE is calculated as
MSE = (MrN)?1
MrX
fi=1
RX
r=1
KX
k=1
N?1X
n=0
LX
l=0
j^h(r;k)fi (n;l)?h(r;k)fi (n;l)j2: (3.58)
71
0 5 10 15 2010
?4
10?3
10?2
10?1
100
SNR(dB)
BER
?Dmax(S) =0
?Dmax(S) =0.001
?Dmax(S) =0.003
?Dmax(S) =0.005
?Dmax(F) =0
?Dmax(F) =0.001
?Dmax(F) =0.003
?Dmax(F) =0.005
Figure 3.7: BER with varying maximum normalized Doppler bandwidth for BPSK modulation,
QF = 2d?DmaxNe and QS = d2?DmaxNe
In Fig. 3.2, the lower bounds in (3.37) and (3.38) are plotted using QS = 2 for fd = 40Hz
and using QS = 3 for fd = 100Hz. [The MMSE bound needs ??ts which requires knowledge of
the Doppler spread. Note also that for both fd = 40Hz and fd = 100Hz, the minimum QS =
d2?DmaxNe are actually QS = 1 and QS = 2, respectively; however, a slightly higher value of
QS = d2?DmaxNe+ 1 = 3 for fd = 100Hz yields smaller modeling error which has been proved
in Example 1]. Then they are compared with the simulation results (averaged over 200 Monte Carlo
runs) in Figs. 3.3 and 3.4 for fd = 40Hz and 100Hz; also shown are ? bounds on the simulation
averages. It can be seen that the theoretical results are consistent with the simulation results for
fd =40Hz indicating that the optimal pilot design does minimize channel MSEs when using DPS-
BEM. For fd=100 Hz, there is a ?small gap? between theory and simulations at high SNRs which
is probably attributable to modeling error (?small? but nonzero). Furthermore, the performance of
72
0 5 10 15 2010
?2
10?1
100
SNR(dB)
BER ?Dmax(S) =0.001
?Dmax(S) =0.003
?Dmax(S) =0.005
?Dmax(F) =0.001
?Dmax(F) =0.003
?Dmax(F) =0.005
Figure 3.8: BER with varying maximum normalized Doppler bandwidth for QPSK modulation,
QF = 2d?DmaxNe and QS = d2?DmaxNe
0 5 10 15 2010
?4
10?3
10?2
10?1
100
SNR(dB)
BER ?Dmax
(S) =0.001
?Dmax(S) =0.003
?Dmax(S) =0.005
?Dmax(F) =0.001
?Dmax(F) =0.003
?Dmax(F) =0.005
Figure 3.9: BER with varying maximum normalized Doppler bandwidth for BPSK modulation,
QF = 2d?DmaxNe+1 and QS = d2?DmaxNe+1
73
0 5 10 15 2010
?3
10?2
10?1
100
SNR(dB)
BER ?Dmax(S) =0.001
?Dmax(S) =0.003
?Dmax(S) =0.005
?Dmax(F) =0.001
?Dmax(F) =0.003
?Dmax(F) =0.005
Figure 3.10: BER with varying maximum normalized Doppler bandwidth for QPSK modulation,
QF = 2d?DmaxNe+1 and QS = d2?DmaxNe+1
MMSE estimator outperforms that of LS estimator at low SNR, and they converge to each other at
high SNR.
3.6.3 Example 3: CE-BEM versus DPS-BEM
In Fig. 3.5, the LS channel estimation MSE (3.58) versus SNR under different maximum
Doppler bandwidths (?(S)Dmax is for DPS-BEM, ?(F)Dmax is for CE-BEM) are plotted. It is clear that the
MSE of DPS-BEM is consistently smaller than that of CE-BEM. However, we notice that the chan-
nel estimation error based on DPS-BEM suffers an error floor, too. This can be alleviated by taking
QS to be larger than d2?DmaxNe at the cost of more computations, as Fig. 3.6 shows. Fig. 3.7 and
Fig. 3.9 take BER (average over 2000 Monte Carlo runs) as a performance measure to compare the
performance between DPS-BEM and CE-BEM. Comparing with Fig. 3.5 and Fig. 3.6 make it quite
74
clear that the drastically reduced MSE for the DPS-BEM channel estimation leads to a pronounced
reduction in BER compared to the CE-BEM case. The BER performances under QPSK modulation
are plotted in Fig. 3.8 and Fig. 3.10, which are worse than those of BPSK modulation, as expected.
3.7 Conclusion
In this chapter, the channel estimation approaches and optimum training design proposed in
Chapter 2 were extended to multiuser MIMO system. Illustrative simulation examples were pro-
vided to show the performance of channel estimation and data detection under MIMO system. The
system model and performance analysis in Chapter 3 are consistent with those in Chapter 2. The
numerical results of MIMO system (Chapter 3) outperforms the SISO system (Chapter 2) due to the
diversity gain.
75
CHAPTER 4
TIME-VARYING EQUALIZATION FOR DOUBLY-SELECTIVE CHANNELS
4.1 Introduction
Well-known techniques of equalization include (a) linear equalization (LE), (b) decision feed-
back equalization (DFE), and (c) maximum likelihood sequence estimation (MLSE) [66]. Linear
equalizers are the simplest computationally whereas MLSE yields (near-)optimum performance but
is computationally demanding. DFE typically provides a good compromise between complexity
and performance. In this chapter we consider LE and DFE for doubly-selective channels modeled
via basis expansion models (BEM).
Design of equalizers for wireless communication systems over doubly-selective channels has
been studied in the literature [26, 67, 45, 46, 3, 5, 68]. These equalizers may be divided into two
types: block equalizers and serial equalizers. The block equalizers are usually complex to design
since inversion of a large matrix is required. Especially, since a doubly-selective channel can not be
diagonalized by a channel-independent transformation, the implementation of block time-varying
(TV) equalizers, which collect and process in blocks all the available data in the received frame,
leads to a very high computational complexity [5]. On the other hand, serial equalizers are more
favored since they process few data at a time and provide a flexible trade-off between complexity
and performance [66]. The TV serial equalizers in [45, 46, 3, 5, 68] rely on a particular basis
expansion model (BEM) of the TV channel impulse response, namely complex exponential (CE)
BEM, and the knowledge of the BEM coefficients at the receiver. Furthermore, they also use a
CE-BEM representation for the TV equalizer; therefore, when a low-order CE-BEM model is used
for the equalizer, it incurs an approximation error inherent in CE-BEM modeling of equalizers.
76
Thus, their performances are highly related to the choice of the number of BEM coefficients and the
equalizer order.
It is well known that the Kalman filter, a linear recursive minimum mean-square-error (MMSE)
filter, is the best linear MMSE detector for a given detection delay. It provides a symbol detection
method of obtaining the true MMSE performance within an implementable structure having a fi-
nite number of filter weights. The Kalman filter has been widely used in channel estimation and
equalization over wireless channels [74]. In this chapter we exploit the Kalman filter as a TV linear
MMSE equalizer for doubly-selective channels. An alternative formulation of the FIR DFE based
on a CE-BEM channel model is also proposed for doubly-selective channels.
The main advantages of the proposed LE and DFE are fourfold: (i) The design process does not
rely on a specific basis expansion model for the underlying channel; therefore, it can be applied to
any doubly-selective channel model. (ii) The proposed equalizers rely solely on the channel model
and therefore, do not incur any approximation error inherent in CE-BEM modeling of equalizers.
(iii) Only one parameter, the equalizer delay, will influence the performance of the Kalman filter;
only three parameters, the equalizer delay, and feedforward and feedback filter lengths, will influ-
ence the performance of proposed DFE. (iv) The computational complexity in terms of the number
of flops turns out to be much lower than existing BEM-based TV equalizers. The simulation results
will show that the proposed equalizers can achieve the same or an improved bit error rate (BER)
performance than the equalizers in [45, 46, 3, 5, 68], without incurring the approximation error
inherent in BEM modeling of equalizers.
77
4.2 Linear Equalization
4.2.1 System Model
Consider a doubly-selective finite impulse response (FIR) linear channel with single input and
multiple outputs (SIMO). The discrete time impulse response h(r)(n;l) denotes the time-varying
impulse response of channel that includes transmit-receive filters as well as doubly-selective propa-
gation effects between the transmit antenna and the rth receive antenna, where n 2 [0;1;:::;N ?1]
and l 2 [0;L] capture the time- and frequency- selectivity of the channel, respectively. Let s(n)
denote the transmitted symbols which is input to the SIMO channel, the received sequence at the
rth receive antenna can be written as
x(r)(n) =
LX
l=0
h(r)(n;l)s(n?l)+?(r)(n); (4.1)
where ?(r)(n) is the additive complex Gaussian noise at the rth receive antenna, with zero-mean
and variance 2?.
In [45, 46, 3, 5, 68], the designs of TV equalizers depend on a BEM of the time-varying
channel, given by [26]:
h(r)(n;l) =
QX
q=0
w(r)q (l)?q(n); (4.2)
where w(r)q (l) is the basis expansion model parameter (coefficient), ?q(n) is the basis expansion
function, and Q+1 is the basis dimension that satisfies 2d?DmaxNe? Q ? N ?1 with ?Dmax the
maximum normalized Doppler bandwidth. The above model is valid over a block of N symbols over
which the BEM coefficients remain time-invariant. The BEM coefficients may change from block
to block. Therefore, we will refer to the equalizers of [45, 46, 3, 5, 68] as BEM-based equalizers.
78
More specifically, the complex exponential basis functions [26] are used in [45, 46, 3, 5, 68], where
?q(n) = ej2?(q?Q=2)n=P: (4.3)
if P = N where N is the block size in symbols, the CE-BEM is said to be critically sampled
whereas when P > N, it is said to be oversampled [37]. The oversampling reduces frequency
spacing of complex exponentials and gives a better representation of the channel impulse response.
[68] uses P = 2N whereas [5] has used P = N.
Collecting the symbols 'x(r)(n)?r=Rr=1 received by the R receivers into the vector x(n) :=
[x1(n);x2(n);:::;xR(n)]T 2CR, we can obtain the matrix-vector form of (4.1):
x(n) =
LX
l=0
Q=2X
q=?Q=2
wq(l)s(n?l)ej2?qn=P +?(n); (4.4)
where wq(l) := [w(1)q (l);:::;w(R)q (l)]T 2CT and ?(n) := [?(1)(n);:::;?(R)(n)]R 2CR.
If the input vector of a BEM-based equalizer of order L0 is defined as y(n) := [xT(n);xT(n?
1);:::;xT(n?L0)]T 2CR(L0+1), the output of the equalizer can be written as:
^s(n) = gH(n)y(n); (4.5)
where the vector g(n) 2 CR(L0+1) collects all the equalizer parameters and d is the equalization
delay.
The following model assumptions are considered throughout this chapter:
(H4.1): The information symbols s(n) consists of zero-mean, finite-alphabet of independent and
identically distributed (i.i.d.) random variables, satisfying Efs(m)s?(n)g = 2s?(m?n).
79
(H4.2): The measurement noise ?(r)(n) in Eq. (3.1) is uncorrelated with s(n), and Ef?(m)??(n)g =
2??(m?n).
4.2.2 Existing Linear Time-Varying MMSE Equalizers
In [45, 46, 3, 5, 68] both zeroforcing and MMSE linear equalizers have been considered. Since
the Kalman filter yields a linear MMSE equalizer, in this section we will restrict ourselves to exist-
ing TV MMSE equalizers. First we briefly review two linear TV MMSE equalizers from [5] and
[68], respectively. The design of these equalizers assumes availability of the CE-BEM coefficients
w(r)q (l)s of the channel.
BLM Equalizer [5]
In [45, 46, 3, 5] one seeks
^s(n) =
L0?dX
l0=d
Q0=2X
q0=?Q0=2
RX
r=1
g(r)q0 (l0)ej2?q0n=Px(r)(n?l0); (4.6)
where design parameters are d, L0, Q0 and equalizer coefficients g(r)q0 (l0), invariant over n 2f0;1;:::;
N ?1g. Notice that the structure of the equalizer is that of a CE-BEM. Define
g(r) :=
h
g(r)?Q0=2(?d);:::;g(r)?Q0=2(L0 ?d);g(r)?Q0=2+1(?d);:::;g(r)Q0=2(?d);:::;g(r)Q0=2(L0 ?d)
iT
(4.7)
and
gblm =
h
g(1)T;g(2)T;:::;g(R)T
iT
2CR(L0+1)(Q0+1): (4.8)
80
The TV MMSE equalizer proposed in [45, 46, 3, 5](henceforth referred to as BLM) provides an
explicit frequency-domain representation, by turning a TV equalization problem into a simpler time-
invariant (TI) filtering design, which only involves the TI basis expansion parameters of the doubly-
selective channel. The (TV MMSE) BLM equalizer can be expressed as [5]:
gblm = Hblm
"
HHblmHblm(n)+
2?
2sI(Q+Q0+1)(L+L0+1)
#?1
ed (4.9)
where the [R(Q0+1)(L0+1)]?[(Q+Q0+1)(L0+L+1)] matrix Hblm collects TI BEM parameters
and TV complex exponential basis functions as given in Sec. V in [5], d 2 f0;1;:::;L+L0g is the
delay of the equalizer, and ed 2 R(Q0+Q+1)(L0+L+1) is an unit vector with the element 1 in the
(d+1)st position.
Computational Complexity: The inverse in (4.9) requires O(K3) flops (one flop is roughly
one multiply-and-accumulate operation) where K = (Q + Q0 + 1)(L + L0 + 1) and O(K3) =
K3 +0:5K2 +0:5K if one uses modified Cholesky decomposition (i.e. UD-decomposition) based
approach to matrix inverse ([28] Table 6.13). This has been called design complexity (equalizer
design) in [5]. For implementation of (4.6), one needs NR(Q0+1)(L0+1) flops; this has been called
implementation complexity in [5]. For numerical comparisons in Sec. 4.2.4, we assume that the
inverse of a positive-definite matrix has been computed via the modified Cholesky decomposition
method.
FRESH Equalizer [68]
The MMSE solution to the frequency-shift TV equalizer proposed in [68] (referred to as FRESH)
gives the canonical frequency-domain representation of the optimal norm TV-MMSE equalizers,
which leads the optimal FRESH equalizer design to construct the TI Fourier coefficients of optimal
81
equalizer weights gopt(n). The discrete Fourier series (DFS) expansion relationship of gopt(n) is:
gopt(n) =
P?1X
p=0
gp;optej2?np=P: (4.10)
Estimation of gopt(n) via estimation of gp;opt is discussed in [68] for zero-foring equalization with
a specified equalization delay. It is straightforward to modify the design to obtain a linear MMSE
equalizer; for instance, add the noise variance term to eq. (26) in [68] just as has been done in (4.9).
However, the implementation complexity of the FRESH representation of the optimal TV
MMSE equalizer maybe quite large for ?large? values of N. Therefore, a low-complexity im-
plementation of the optimal FRESH equalizer (refer to as a sub-optimum FRESH method) is also
derived in [68] by evaluate only Q0+1 Fourier coefficients in (4.10), which is similar to that consid-
ered in BLM equalizer design. So the DFS expansion for sub-optimum FRESH equalizer is given
by
gsubopt(n) =
Q0=2X
p=?Q0=2
gp;suboptej2?np=P (4.11)
wheregp;subopt = gp;opt for p = 0;1;:::;Q0=2 andgp;subopt = gp+P;opt for p = ?1;?2;:::;?Q0=2.
Computational Complexity: As discussed in [68], design complexity of both optimal and
suboptimal FRESH equalizers involves P ? O[(L + L0 + 1)3] + (P log2 P)(L + L0 + 1)2 flops
where O(K3) = K3 + 0:5K2 + 0:5K when one uses a UD-decomposition approach to positive-
definite matrix inversion. For implementation, one needs NRP(L0 + 1) flops for optimal FRESH
and NR(Q0 +1)(L0 +1) flops for suboptimal FRESH equalizer.
Remark 4.1: The design of both BLM and FRESH equalizers rely on the CE-BEM repre-
sentation of the channel. They assume that the BEM parameters w(r)q (l) are known at the receiver.
It is still not clear if these design methods apply to other basis expansion models. However, the
82
simulation results in [5] illustrate that the BEM modeling errors have a significant influence on the
performance.
Remark 4.2: Three parameters have to be considered or optimized for BLM and FRESH equalizers
design, d, L0 and Q0.
4.2.3 Kalman Filter with Equalization Delay d
The Kalman filter has been widely used in channel estimation and equalization over wireless
channels; see [74] and references therein. It requires a state-space model of the underlying system.
Define the state vector s(n)
s(n) := [s(n);s(n?1);:::;s(n?d)]T 2Cd+1; (4.12)
the state transition matrix ' and the input vector ?
' :=
2
6401?d 0
Id 0d?1
3
752R(d+1)?(d+1); ? := [1;0
1?d]T;2Rd+1 (4.13)
and the observation matrix H(n)
H(n) := [h(n;0);h(n;1);:::;h(n;L);0R?(d?L)] 2CR?(d+1); (4.14)
where h(n;l) := [h(1)(n;l);:::;h(R)(n;l)]T 2 CR. Based on these definitions and the system
model in Section 4.2.1, we have the time-invariant state equation:
s(n) = 's(n?1)+?s(n); (4.15)
83
and the time-varying observation equation:
x(n) = H(n)s(n)+?(n): (4.16)
The Kalman filter algorithm is given as follows [74], [59] where ^s(njn) denotes the linear MMSE
estimate of s(n) based on observations x(k), k = 1;2;:::;n.
Initialization: At time n = 0, ^s(1j0) = Efs(1)g = 0 and V^s(1j0) = 2sId+1.
Filtering: At time n = 1;2:::
Vz(n) = H(n)V^s(njn?1)HH(n)+ 2?IR; (4.17)
K(n) = V^s(njn?1)HH(n)V ?1z (n); (4.18)
z(n) = x(n)?H(n)^s(njn?1); (4.19)
^s(njn) = ^s(njn?1)+K(n)z(n); (4.20)
V^s(njn) = [Id+1 ?K(n)H(n)]V^s(njn?1); (4.21)
V^s(n+1jn) = 'V^s(njn)'H + 2s??H; (4.22)
^s(n+1jn) = '^s(njn): (4.23)
Since ^s(njn) = [^s(njn);^s(n ? 1jn);:::;^s(n ? djn)]T , we extract its last term ^s(n ? djn) as the
desired equalized output.
Remark 4.3: For Kalman filter, the design process does not rely on a particular basis expansion
model as long as the estimated/fitted channel impulse response is known at the receiver, so it can be
applied to any doubly-selective channel model.
84
Table 4.1: Operation summary for Kalman filter
Operation flops
H(n)V^s(njn?1) (d+1)2R
Vz(n) 12(d+1)R2 + 12(d+1)R
Vz(n)?1(n) R3 + 12R2 + 12R
K(n) (d+1)R2
V^s(njn) 12(d+1)2R+ 12(d+1)R
Total C = 32(d+1)2R+ 32(d+1)R2 +(d+1)R+R3 + 12R2 + 12R
Remark 4.4: Only one parameter, equalization delay d, is needed for the Kalman filter design and
will influence its performance.
Design Complexity: Computations in (4.17),(4.18),(4.21) and (4.22) comprise the design equa-
tions for Kalman filtering. Following Table 6.5 in [28], the required number of flops C per time-step
are listed in Table 4.1. Therefore, for a transmitted block of size N symbols, one needs NC flops.
Implementation Complexity: Computations in (4.19),(4.20),and (4.23) comprise the imple-
mentation equations for Kalman filtering. The number of flops needed per time-step are 2R(d+1)
where given the nature of ', (4.23) does not require any flops. Therefore, for a transmitted block
of size N symbols, one needs 2NR(d + 1) flops. On the other hand, both BLM and suboptimal
FRESH equalizers require NR(Q0 +1)(L0 +1) flops each.
4.2.4 Numerical Examples
In this section, the BER performance and the computational complexity of the proposed Kalman
filter are investigated by means of Monte Carlo computer simulations, and compared with BEM-
based BLM [5] and FRESH [68] equalizers. For computational complexity calculations, we assume
85
that the inverse of a positive-definite matrix has been computed via the modified Cholesky decom-
position method.
A random time- and frequency- selective Rayleigh fading channel is simulated according to
[81] with channel order L = 3 (4 taps). For different l?s, h(n;l)?s are mutually independent, satisfy
Jakes? model, and each tap is generated via the method of [81] given the symbol duration Ts and the
Doppler spread fd. It is important to point out that each channel tap follows the Jakes? spectrum,
rather then the assumed BEM representation. The data were generated using the double-selective
channel described above. However, for equalizer design one needs CE-BEM representation of the
true channel; this was obtained by a least-squares fit of the assumed BEM to the true channel in
each Monte Carlo run, just as in [5] and [68], to obtain the BEM coefficients (which vary from
run-to-run). These BEM coefficients were used in the designs of [5] and [68], as well as in Kalman
filtering where h(n;l) in (4.14) is generated via (4.2) with fitted BEM parameters (i.e. h(n;l) in
(4.14) is not the true channel, but its BEM approximation). Note that one could have directly used
the Jakes? channel h(n;l) in the Kalman filter implementation of the linear MMSE equalizer unlike
the approaches of [5] and [68]. However, this would not be a fair comparison with the approaches of
[5] and [68]. Most importantly, this would not illustrate the effects of modeling errors since practical
channels are rarely Jakes? channels although their use in simulations and analysis is widespread.
In all simulations, the transmitter transmits binary phase shift keying (BPSK) and quadrature
phase shift keying (QPSK) modulated symbols. The SNR refers to the energy per bit over one-sided
noise spectral density.
86
Table 4.2: Computation complexity; N = 50, P = 2N = 100
Equalizer Design Complexity(flops) Implementation Complexity(flops)
FRESH-OPT [68] P ?O[(L+L0 + 1)3] + (Plog2P)(L+L0 + 1)2 = 171;939 NRP(L0 + 1) = 70;000
FRESH-SUBOPT [68] P ?O[(L+L0 + 1)3] + (Plog2P)(L+L0 + 1)2 = 171;939 NR(Q0 + 1)(L0 + 1) = 13;300
BLM [5] O(K3) = 12;193;565 NR(Q0 + 1)(L0 + 1) = 13;300
Kalman filter (proposed) NC = 8;350 2NR(d+ 1) = 1;200
Example 1: BER versus SNR under Simulation Parameters in [68]
We consider the same simulation parameters as in [[68], Experiment 3], i.e., the block size
(number of information symbols) is N = 50, P = 2N (oversampled CE-BEM with a factor of
2), the number of receive antennas R = 2, symbol duration Ts = 160us, the maximum Doppler
spread fd = 100Hz, the equalization delay d = 5 symbols and Q = 2dfdPTse = 4. For BEM-
based equalizers (both BLM and FRESH), the equalizer order L0 = 6 and the number of Fourier
coefficients (equalizer BEM coefficients) Q0 = 18, as in [[68], Experiment 3]. The BER averaged
over 10;000 Monte Carlo runs versus SNR is shown in Fig. 4.1 for the four approaches: BLM [5],
optimum FRESH [68], suboptimum FRESH [68] and the proposed Kalman filter solution. Note
that the Kalman filter does not need L0 or Q0. The results for BPSK modulation are plotted in Fig.
4.1. It is seen from Fig. 4.1 that the performances of the four approaches are very close each other.
The computational complexity measured in terms of flops for entire block is shown in Table 4.2.
Notice that the Kalman filter requires significantly fewer flops than the other two approaches, both
for design as well as implementation.
Fig. 4.2 plots the BER versus SNR results for QPSK modulation, where the information se-
quences are QPSK signals. Although the performance of Kalman filter is slightly worse than the
other two equalizers at high SNR, the three approaches have similar performances in this case.
87
0 5 10 1510
?5
10?4
10?3
10?2
10?1
100
SNR(dB)
BER
FRESH?OPT
FRESH?SUBOPT
BLM
Kalman filter (proposed)
Figure 4.1: BER performance versus SNR using BPSK modulation, averaged over 10,000 runs.
BLM denotes the method of [5]; FRESH-OPT and FRESH-SUBOPT are methods of [68]. N = 50,
P = 100, Q = 4, L = 3, d = 5, Q0 = 18, L0 = 6
Table 4.3: Computation complexity; N = 800, P = 2N = 1600
Equalizer Design Complexity(flops) Implementation Complexity(flops)
FRESH-OPT [68] P ?O[(L+L0 + 1)3] + (Plog2P)(L+L0 + 1)2 = 11;130;924 NRP(L0 + 1) = 33;280;000
FRESH-SUBOPT [68] P ?O[(L+L0 + 1)3] + (Plog2P)(L+L0 + 1)2 = 11;130;924 NR(Q0 + 1)(L0 + 1) = 274;400
BLM [5] O(K3) = 37;989;672 NR(Q0 + 1)(L0 + 1) = 274;400
Kalman filter (proposed) NC = 22;000 2NR(d+ 1) = 35;200
88
0 5 10 1510
?4
10?3
10?2
10?1
100
SNR(dB)
BER
FRESH?OPT
BLM
Kalman filter
Figure 4.2: BER performance versus SNR using QPSK modulation, averaged over 10,000 runs
Example 2: BER versus SNR under Simulation Parameters in [5]
Now the simulation parameters in [[68], Fig. 11] are used where N = 800, R = 2, Ts = 25us,
fd = 100Hz, d = 10, L0 = 12, and Q0 = 12. Two cases are considered: P = 2N and P = N,
leading to Q is 8 and 4, respectively. The results are shown in Fig. 4.3 which were obtained by
carrying out 1000 Monte Carlo runs. When P = N, the performance of the Kalman filter and
BLM equalizer [5] are quite close to each other as in Fig. 4.1. The error floors at high SNR are
caused by the BEM channel modeling error. However, the Kalman filter significantly outperforms
the BLM equalizer [5] when P = 2N, the case where the channel modeling errors in approximating
a Jakes? channel with CE-BEM are significantly smaller compared the case of P = N: recall that
the data are generated via the true Jakes? channel. The computational complexity measured in terms
of number of flops for entire block is shown in Table 4.3 for the case P = 2N = 1600. Notice
89
2 4 6 8 10 12 14 1610
?7
10?6
10?5
10?4
10?3
10?2
10?1
SNR(dB)
BER
FRESH?OPT, Q=4, P=N
FRESH?SUBOPT, Q=4, P=N
BLM, Q=4, P=N
Kalman filter (proposed), Q=4, P=N
FRESH?OPT, Q=8, P=2N
FRESH?SUBOPT, Q=8, P=2N
BLM, Q=8, P=2N
Kalman filter (proposed), Q=8, P=2N
Figure 4.3: BER performance versus SNR using BPSK modulation, averaged over 1,000 runs. BLM
denotes the method of [5]; FRESH-OPT and FRESH-SUBOPT are methods of [68]. Kalman filter
is based on P = 800 when Q = 4 and on P = 1600 when Q = 8. P = N corresponds to critically
sampled CE-BEM and P = 2N corresponds to oversampled CE-BEM. N = 800, P = 800 or
1600, Q = 4 or Q = 8, L = 3, d = 10, Q0 = 12, L0 = 12
90
0 5 10 15 2010
?7
10?6
10?5
10?4
10?3
10?2
10?1
100
SNR(dB)
BER
BLM, P=N, Q=4
Kalman, Q=4
BLM, P=2N, Q=8
Kalman, Q=8
Figure 4.4: BER performance versus SNR using QPSK modulation, averaged over 1,000 runs
0 5 10 15 2010
?7
10?6
10?5
10?4
10?3
10?2
10?1
SNR(dB)
BER
CE?BEM, P=N, Q=4
DPS?BEM, Q=4
CE?BEM, P=2N, Q=8
DPS?BEM, Q=8
Figure 4.5: BER performance versus SNR For DPS-BEM and CE-BEM channel modeling using
Kalman filtering, BPSK modulation
91
0 5 10 15 2010
?7
10?6
10?5
10?4
10?3
10?2
10?1
100
SNR(dB)
BER
CE?BEM, P=N,Q=4
DPS?BEM, Q=4
CE?BEM, P=2N, Q=8
DPS?BEM, Q=8
Figure 4.6: BER performance versus SNR For DPS-BEM and CE-BEM channel modeling using
Kalman filtering, QPSK modulation
again that the Kalman filter requires significantly fewer flops than the other two approaches, both
for design as well as implementation
In Fig. 4.4 the BER performances for QPSK modulation are plotted under two schemes: BLM
and Kalman filter. The results in Fig. 4.4 are very similar to Fig. 4.3, except that the BER by using
QPSK information symbols are worse than that using BPSK symbols, as expected.
Example 3: Performance of Kalman Filter Using DPS-BEM
Now we test the performance of the Kalman filer by using Discrete Prolate Spheroidal (DPS)
BEM , in addition to CE-BEM, in (4.3), for approximating the true Jakes? channel. As we noted
earlier, the Kalman filter has the flexibility that it is not restricted to a particular BEM and can be
applied to any doubly-selective channel model. In Fig. 4.5, we compare the BER performances of
92
the Kalman filter for CE-BEM and DPS-BEM cases by using the simulation parameters in Example
2. It is known that the channel modeling error of DPS-BEM is several magnitudes smaller then that
of critically sampled (P = N) CE-BEM [77], which is the reason why the BER performance of
DPS-BEM case is better then CE-BEM for P = N, Q = 4. For the oversampling case of P = 2N
and Q = 8, the performances of DPS-BEM and CE-BEM are very close to each other since the
modeling error differences are not so pronounced any more. This example shows the flexibility of
the Kalman filter: its application is not limited to the CE-BEM channel model unlike FRESH [68]
and BLM [5] equalizers.
Without loss of generality, in Fig. 4.6 we again plot the simulation results under QPSK modu-
lation where the information symbols are QPSK signals.
4.3 Decision Feedback Equalization
4.3.1 System Model
Consider a doubly-selective finite impulse response (FIR) linear channel with single input and
multiple outputs (SIMO), where the expressions of input and output of the SIMO channel are the
same as (4.1)-(4.4) in Section 4.2.1.
Consider an FIR DFE with equalization delay d, feedforward (FF) filter of length lf and feed-
back (FB) filter of length lb. Let ^s(n) be the ?soft? estimate of symbol s(n) and let ~s(n) denote its
quantized (hard decision) value. Then the soft output of the DFE is given by
^s(n?d) =
lf?1X
m=0
fTm(n)x(n?m)?
lbX
k=1
bk(n)~s(n?d?k) (4.24)
93
where the R-column vectors fm(n)?s and scalars bk(n)?s are the tap-gains of FF and FB time-
varying filters at time n. Note that both FF and FB filters are FIR. In the minimum mean-square
error (MMSE) design of DFE, one seeks the tap-gains to minimize Efjs(n?d)?^s(n?d)j2g. As
discussed in [2] and [57, Sec. 3.3], one first designs the FB filter assuming that ~s(n) = s(n) (i.e. no
decision errors), and then one designs the FF filter given the FB filter.
4.3.2 Time-Varying FIR MMSE DFEs
In this section we first briefly review the TV FIR MMSE DFE from [46],[3]. Then our pro-
posed formulation is presented. The design of these equalizers assumes availability of the CE-BEM
coefficients w(r)q (l)s of the channel.
FIR MMSE DFE of [46],[3]
In [46],[3], one approximates time-varying FF and FB tap-gains via another sets of CE-BEMs,
leading to
^s(n?d) =
lf?1X
l0=0
Q0
2X
q0=?Q02
RX
r=1
f(r)q0 (l0)ej2?qnP x(r)(n?l0)?
lbX
l00=1
Q00
2X
q00=?Q002
bq00(l00)ej2?qnP ~s(n?d?l00)
(4.25)
where design parameters are d, lf, lb, Q0, Q00 and equalizer FF coefficients f(r)q0 (l0) and FB coef-
ficients bq00(l00), invariant over n 2 f0;1;:::;N ? 1g. Notice that the structure of the FF and FB
parts of the equalizer is that of a CE-BEM.
The design equations may be found in [46],[3]; in particular, see [46, Sec. 5.2]. The TV
FIR MMSE DFE proposed in [46],[3] (henceforth referred to as BLM: Barhumi, Leus, Moonen)
provides an explicit ?frequency-domain? representation, by turning a TV equalization problem into
94
a simpler time-invariant (TI) filtering design, which only involves the TI basis expansion parameters
of the equalizer tap-gains. For details, we refer the reader to [46, Sec. 5.2].
We now provide details from [46, Sec. 5.2] (modified to conform to the notation in this paper).
Define the [lb(Q00 +1)+1]?1 vector
~b = [bQ00
2 ;lb
;bQ00
2 ;lb?1
;:::;bQ00
2 ;1
;:::;b1;1;b0;lb;:::;b0;1;0;b?1;lb;:::;b?1;1;:::;b?Q00
2 ;1
]T;
the [lf(Q0 +1)]?1 vector
~f(r) = [f(r)
Q0
2 ;lf?1
;:::;f(r)Q0
2 ;0
;:::;f(r)?Q0
2 ;0
]T;
and the [Rlf(Q0 +1)]?1 vector
~f = [ ~f(1)T;:::; ~f(R)T]T:
Define the lf ?[lf +L] Toeplitz matrix
W(r)q :=
2
66
66
64
w(r)q (L) ??? w(r)q (0)
... ...
w(r)q (L) ??? w(r)q (0)
3
77
77
75;
the [(Q0 +1)lf]?[(Q+Q0 +1)(lf +L)] matrix
H(r) :=
2
66
66
64
?Q2 W(r)q ??? ??Q2 W(r)q
... ...
?Q2 W(r)q ??? ??Q2 W(r)q
3
77
77
75
95
and the [R(Q0 +1)lf]?[(Q+Q0 +1)(lf +L)] matrix
H :=
h
H(1)T;:::;H(R)T
iT
;
where ? := diagf1;ej2?=P;:::;ej2?(lf?1)=Pg. Define a [(Q00 + 1)lf]?[(Q + Q0 + 1)(L + lf)]
?selection? matrix
P :=
2
66
66
64
IQ00
2
?P1
0fi?fl P2 0fi?fl
IQ00
2
?P1
3
77
77
75
with fi = (Q00 +1)lb +1, fl = (Q+Q0 ?Q00)(L+lf)=2,
P1 := [0lb?[L+lf?1?lb?d)];Ilb;0lb?(d+1)];
P2 := [0(lb+1)?[L+lf?1?lb?d)];Ilb+1;0(lb+1)?d]:
Let e denote the [(Q00+1)lb +1]?1 unit vector with a one in position 1+lb +(Q00lb=2). Then the
MMSE DFE tap-gains are given by [46, Sec. 5.2]
~bMMSE = R?1MMSEe
eTR?1MMSEe ?e; (4.26)
RMMSE = P ? ?2? HHH+ ?2s I??1PH; (4.27)
~fTMMSE = ~bTMMSEP
?
HHH+
2?
2sI
!?1
HH: (4.28)
Computational Complexity: To design the feedforward and feedback filter coefficients, the
inverse in (4.28) and (4.26) require 2?O(K3) flops (one flop is roughly one multiply-and-accumulate
96
operation) where K = (Q+Q0+1)(L+lf) and O(K3) = K3+0:5K2+0:5K if one uses modified
Cholesky decomposition (i.e. UD-decomposition) based approach to matrix inverse [2, Table 6.13].
This has been called design complexity (equalizer design) in [3]. For implementation of (4.25),
one needs NR(Q0 + 1)lf flops for feedforward part and N(Q00 + 1)lb flops for feedback part; this
has been called implementation complexity in [3]. For numerical comparisons in Section 4.3.3, we
assume that the inverse of a positive-definite matrix has been computed via the modified Cholesky
decomposition method.
Remark 4.5: The design of BLM DFEs relies on CE-BEM representation of the channel. They
assume that the BEM parameters w(r)q (l) are known at the receiver. It is not clear if these design
methods apply to other basis expansion models. However, the simulation results in [5] illustrate that
the BEM modeling errors have a significant influence on the equalizer performance.
Remark 4.6: Five parameters have to be considered or optimized for BLM DFE design: d, lf,
lb, Q0 and Q00.
Proposed MMSE DFE with Time-Varying Taps
Here we follow [2] (see also [57, Sec. 3.3]) where a time-invariant channel is considered. Their
results extend to time-varying channels in a straight-forward fashion; therefore, we simply state the
final results instead of repeating the entire derivation with obvious changes.
Using the estimated channel, the symbol decisions are made by an FIR MMSE-DFE [2]. Given
the lengths of the feedforward (FF) and the feedback (FB) filters as lf and lb respectively, the
estimate of the information symbol at time n with equalization delay d is given by (4.24). Stack the
97
inputs of the FF filter at time n into a ?tall? vector
xf (n) :=
?
xT (n) xT (n?1) ??? xT (n?lf +1)
?T
;
and also define ?f (n) likewise. By (4.4), we have
xf (n) = H(n)sf (n)+?f (n) (4.29)
where
H(n) :=
2
66
66
64
h(n;0) ??? h(n;L)
... ... ...
h(n?lf +1;0) ??? h(n?lf +1;L)
3
77
77
75 (4.30)
and
sf (n) :=
?
s(n) s(n?1) ??? s(n?lf ?L+1)
?T
:
We further define
sb (n) :=
?
s(n?d) s(n?d?1) ??? s(n?d?lb)
?T
:
By the assumption that fs(n)g are i.i.d. with variance 2s, and based on (4.29), we have
Rss (n) := E'sb (n)sHb (n)? = 2sIlb+1;
Rsx (n) := E'sb (n)xHf (n)? = 2s'HH (n);
Rxx (n) := E'xf (n)xHf (n)? = 2sH(n)HH (n)+ 2vIRlf
98
where ' :=
?
0(lb+1)?d Ilb+1 0(lb+1)?(lf+L?d?lb?1)
?
.
Let f (n) and b(n) denote the vectors of time-varying taps of FF and FB filters,
f (n) :=
?
fT0 (n) fT1 (n) ??? fTlf?1 (n)
?T
;
b(n) :=
?
1 b1 (n) b2 (n) ??? blb (n)
?T
:
Assuming the decisions f~s(n)g are correct and equal to fs(n)g, the FF and the FB time-varying
filters of the MMSE-DFE are given by [2]
bMMSE (n) = R
?1
? e0
eT0R?1? e0; (4.31)
fMMSE (n) = R?1xx (n)RHsx (n)bMMSE (n) (4.32)
where e0 :=
?
1 0 0 ??? 0
?T
and
R? := Rss (n)?Rsx (n)R?1xx (n)RHsx (n): (4.33)
Computational Complexity: To design the proposed MMSE DFE, we need to compute the
inverse of a Rlf ?Rlf matrix to obtain the feedforward filter coefficients (as in eq. (4.32)), and we
need to compute the inverse of a (lb +1)?(lb +1) matrix to obtain the feedback filter coefficients
(as in eq.(4.31)). Therefore, the required number of flops is O((Rlf)3)+O((lb +1)3).
The implementation complexity associated with eq. (4.24) requires NRlf flops for the feed-
forward part and Nlb flops for the feedback part.
99
0 5 10 15 2010
?7
10?6
10?5
10?4
10?3
10?2
10?1
100
SNR(dB)
BER
BLM LE, R=1
BLM LE, R=2
BLM DFE, R=1
BLM DFE, R=2
Proposed DFE, R=1
Proposed DFE, R=2
Figure 4.7: BER performance versus SNR, averaged over 1,000 runs, BPSK signal. N = 800,
P = 1600, Q = 8, L = 3, d=5, Q0=12, lf=12, Q00 = 4, lb = 3. R denotes the number of receive
antennas.
Remark 4.7: For the proposed MMSE DFE, the design process does not rely on a particular
basis expansion model as long as the estimated/fitted channel impulse response is known at the
receiver, so it can be applied to any doubly-selective channel model.
Remark 4.8: Only three parameters, equalization delay d and filter lengths lf and lb, are
needed for the proposed filter design and will influence its performance.
4.3.3 Numerical Examples
In this section, the BER performance of the proposed DFE solution are investigated by means
of Monte Carlo computer simulations, and compared with BEM-based BLM DFE solution of
[46],[3].
100
0 5 10 15 2010
?6
10?5
10?4
10?3
10?2
10?1
100
SNR(dB)
BER
BLM LE, R=1
BLM LE, R=2
BLM DFE, R=1
BLM DFE, R=2
Proposed, R=1
Proposed, R=2
Figure 4.8: BER performance versus SNR, averaged over 1,000 runs, BPSK signal. N = 800,
P = 1600, Q = 8, L = 3, d=5, Q0=12, lf=12, Q00 = 4, lb = 3. R denotes the number of receive
antennas.
A random time- and frequency- selective Rayleigh fading channel is simulated according to
[81] with channel order L = 3 (4 taps). For different l?s, h(n;l)?s are mutually independent, satisfy
Jakes? model, and each tap is generated via the method of [81] given the symbol duration Ts and the
Doppler spread fd. It is important to point out that each channel tap follows the Jakes? spectrum,
rather then the assumed BEM representation. The data were generated using the double-selective
channel described above. However, for equalizer design one needs CE-BEM representation of the
true channel; this was obtained by a least-squares fit of the assumed BEM to the true channel in
each Monte Carlo run, just as in [46], [5] and [68], to obtain the BEM coefficients (which vary from
run-to-run). These BEM coefficients were used in the designs of [46],[3] and [5], as well as in our
proposed DFE solution.
101
Table 4.4: Computation complexity
Equalizer Design Flops Implementation Flops
BLM DFE [46],[3] 62,611,290 261,600
Proposed DFE 14,198 21,600
In our simulations, the transmitter transmits binary or quaternary phase shift keying (BPSK/
QPSK) modulated symbols. The SNR refers to the energy per symbol over one-sided noise spectral
density.
We consider the simulation parameters quite similar to those in [46],[3] and [5, Fig. 11], i.e.,
the block size (number of information symbols) is N = 800, P = 2N (oversampled CE-BEM with
a factor of 2), the number of receive antennas R = 1;2, symbol duration Ts = 25 ?s, the maximum
Doppler spread fd = 100Hz, the equalization delay d = 5 symbols and Q = 2dfdPTse = 8.
For BEM-based equalizers (both BLM LE and BLM DFE), the equalizer lengths lf = 12 and
lb = 3 with corresponding number of Fourier coefficients (equalizer BEM coefficients) Q0 = 12
and Q00 = 4. The BER averaged over 1,000 Monte Carlo runs versus SNR is shown in Fig. 4.7 for
the three approaches when using BPSK signal: BLM DFE [46],[3], BLM LE [5], and our proposed
DFE solution. Note that our formulation does not need Q0 or Q00. It is seen from Fig. 4.7 that the
DFE solutions outperform the LE solution, and the performance improves with increasing number
of receive antennas R. Furthermore, our proposed solution outperforms the BEM-based DFE of
[46],[3]. The computational complexity measured in terms of number of flops for the entire block is
shown in Table 4.4. Notice that our DFE formulation requires far fewer flops than the BEM-based
DFE of [46],[3].
Finally, Fig. 4.8 shows the BER results for QPSK signals for the same set of parameters as for
Fig. 4.7. We see comparative performance similar to Fig. 4.7 in Fig. 4.8.
102
4.4 Conclusion
In this chapter, we exploited the conventional Kalman filter and DFEs as time-varying mini-
mum mean-square-error equalizer for doubly-selective channels modeled via basis expansion mod-
els (BEM). Recently there has been much interest in designing time-variant serial FIR (finite im-
pulse response) linear equalizers and FIR DFEs using complex exponential (CE) BEMs for equal-
izers in addition to using CE-BEM for modeling the channel itself. We showed that a Kalman filter
formulation of the linear equalizer and an alternative formulation of the FIR DFE based on a CE-
BEM channel model yields same or improved BER at a much lower computational cost, without
incurring the approximation error inherent in CE-BEM modeling of equalizers.
103
CHAPTER 5
A MULTIPLE MODEL APPROACH TO DOUBLY-SELECTIVE CHANNEL ESTIMATION USING
EXPONENTIAL BASIS MODELS
5.1 Introduction
In order to ?accurately? model the underlying doubly-selective channel, the number of BEM
coefficients used to model the doubly-selective channels for channel estimation has been based on
an upper bound on the channel Doppler spread. The higher the Doppler spread, the more the number
of BEM coefficients, which leads to a higher channel estimation variance. This, in turn, leads to
higher bit error rate (BER) when the estimated channel is used for data detection and the actual
Doppler spread is (much) less than the upper bound.
In this chapter we propose to use a multiple model framework where several candidate Doppler
spread values are used to cover the range from zero to an upper bound, leading to multiple CE-BEM
channel models, each corresponding to an assumed value of the Doppler spread. Subsequently the
well known interacting multiple model (IMM) algorithm [11] is used for symbol detection based on
multiple state-space models corresponding to the multiple estimated channels.
The multiple model (or hybrid system) approach assumes the system to be in one of a finite
number of models (i.e., that is described by one out of a finite number of models). Each model
is characterized by its parameters ? the models differ in Doppler spread here. For each model a
filter ?matched? to its parameters is yielding model-conditioned estimates and covariances. A mode
probability calculator ? a Bayesian model comparator ? updates the probability of each mode
using the likehood function (innovations) of each filter and the prior probability of each model. Then
an estimate combiner computes the overall estimate and the associated covariance as the weighted
104
sum of the model-conditioned estimates and the corresponding covariance ? via the (Gaussian)
mixture equations. Among all the multiple model approaches, the interacting multiple model is
considered to be the best compromise between complexity and performance. The IMM approach
computes the state estimate that accounts for each possible current model using a suitable mixing
of the previous model conditioned estimate depending on the current model.
In this chapter an adaptive channel estimation scheme, exploiting the over-sampled complex
exponential basis expansion model (CE-BEM), is presented for doubly-selective channels where we
track the BEM coefficients via the multiple model approach. The chapter is organized as follows.
First, the system model and objectives are provided. Second, the IMM algorithm and configura-
tion are briefly summarized. The performances of the proposed design are finally illustrated by
simulation examples.
5.2 System Model and Objectives
5.2.1 System Model
Consider a doubly-selective (time- and frequency-selective) FIR (finite impulse response) lin-
ear channel. Let fs(n)g denotes a scalar sequence which is the input to time-varying channels with
the discrete-time response fh(n;l)g (the channel response at time n to a unit input at time n?l).
Then the symbol-rate noisy channel output at the rth receive antenna is given by (n = 0;1;:::;
r = 1;2;:::;R)
y(r)(n) =
LX
l=0
h(r)(n;l)s(n?l)+?(r)(n) (5.1)
105
where v(r)(n) is the zero-mean white complex-Gaussian noise with variance 2v. We assume that
'h(r)(n;l)? (r = 1;:::;R) represents a wide-sense stationary uncorrelated scattering (WSSUS)
vector channel [54].
In CE-BEM [26, 48, 37], over the k-th block consisting of an observation window of TB
symbols, the channel is represented as (?nk := (k?1)TB)
h(r)(n;l) =
QX
q=1
w(r)q (l)ej!qn; n = ?nk;:::;?nk +TB ?1; (5.2)
where one chooses (l = 0;1;:::;L, and K is an integer)
T := KTB; K ? 1; Q ? 2dfdTTse+1; (5.3)
!q := 2?T [q ?(Q+1)=2]; q = 1;2;:::;Q; (5.4)
L := b?d=Tsc; (5.5)
?d and fd are respectively the delay spread and the Doppler spread, and Ts is the symbol duration.
The BEM coefficients w(r)q (l)?s remain invariant during this block, but are allowed to change at the
next block, and the Fourier basis functions 'ej!qn? (q = 1;2;:::;Q) are common for each block.
If the delay spread ?d and the Doppler spread fd of the channel (or at least their upper-bounds)
are known, one can infer the basis functions of the CE-BEM [48]. Treating the basis functions as
known, estimation of a time-varying process is reduced to estimating the invariant coefficients over
a block of length TB symbols. Note that the BEM period is T = KTB whereas the block size is TB
symbols. If K > 1 (e.g. K = 2 or K = 3), then the Doppler spectrum is said to be over-sampled
[37] compared to the case K = 1 where the Doppler spectrum is said to be critically sampled. In
106
[26, 48] only K = 1 (henceforth called CE-BEM) is considered whereas [37] considers K ? 2
(henceforth called over-sampled CE-BEM).
5.2.2 Block-Adaptive Channel Estimation [48]
Here we summarize the time-multiplexed training approach of [48]. In Sec. 5.4 we provide
simulation comparisons with results of [48]. In [48] each transmitted block of symbols fs(n)gTB?1n=0
is segmented into P subblocks of time-multiplexed training and information symbols. Each sub-
block is of equal length lb symbols with ld information symbols and lt training symbols (lb = ld+lt).
If s denotes a column-vector composed of fs(n)gTB?1n=0 , then s is arranged as
s :=
?
bT0 cT0 bT1 cT1 ??? bTP?1 cTP?1
?T
(5.6)
where bp (p = 0;1;???P ? 1) is a column of ld information symbols and cp is a column of lt
training symbols. We clearly have TB = Plb. Given (5.1) and CE-BEM (5.2), [48] has shown that
(5.6) is an optimum structure for K = 1 with lt = 2L+1, P ? Q and
cp :=
?
0TL 0TL
?T
; > 0: (5.7)
Thus, given a transmission block of size TB, (2L+1)P symbols have to be devoted to training and
the remaining TB ?(2L + 1)P are available for information symbols. This design has been used
by others for oversampled CE-BEM also [37].
107
Let np := plb+ld+L, (p = 0;1;???P?1), denote the location of (nonzero) ?s in the optimum
cp?s in the P subblocks. Then by design, received signal (assuming timing synchronization)
y(r)(np +l) = h(r)(np +l;l)+v(r)(np +l) (5.8)
for l = 0;1;??? ;L. Using (5.2) in these y(r)(np + l)?s, one can uniquely solve for w(r)q (l)?s via a
least-squares approach. The channel estimates are given by the CE-BEM (5.2) using the estimated
BEM coefficients.
5.2.3 Objectives
Suppose that we collect the received signal over a time interval of ?T symbols. We wish to
estimate the time-variant channel using a channel model and time-multiplexed training (such as
that discussed in Sec. 5.2.2 and [48]), and subsequently using the estimated channel, estimate the
information symbols. For CE-BEM, if we choose ?T as the block size, then in general Q value will
be very high requiring estimation of a large number of parameters, thereby degrading the channel
estimation performance. If we divide ?T into blocks of size TB, and then fit CE-BEM block by
block, we need smaller Q. This is the solution considered in this chapter (and also [48]). In practical
situations, over a large ?T, the actual Doppler spread fd is likely to vary. Absent any prior knowledge,
a commonly used solution [48, 37] is to use an upper bound on the anticipated fd (based on the
maximum vehicle speed, e.g.) and pick Q accordingly. In this chapter we investigate a multiple
model framework where several candidate Doppler spread values are used to cover the range from
zero to an upper bound, leading to multiple CE-BEM channel models, each corresponding to an
assumed value of the Doppler spread. Multiple model approach has been extensively used in target
tracking applications [11, 7, 64] and more recently, has been used for tracking dispersive DS-CDMA
108
channels using multiple autoregressive (AR) models in [30]. In this chapter we propose to use such
an approach in conjunction with BEM?s.
5.3 Multiple Model Approach
5.3.1 Multiple Models
Let fd;u denote an upper bound on the anticipated Doppler spread fd. Let fd;1, fd;2,..., fd;M
denote our M candidate Doppler spreads and let Qm, 1 ? m ? M, denote the corresponding
values of Q from (5.3). Then we have M candidate channel impulse responses indexed by m over
the k-th block consisting of an observation window of TB symbols,
h(m;r)(n;l) =
QmX
q=0
w(m;r)q (l)ej!qn;n = ?nk;?nk +1;:::;?nk +TB ?1: (5.9)
We will use a Kalman filter with equalization delay d for data detection using the estimated channel.
Define
y(n) := [y(1)(n); y(2)(n); :::; y(R)(n)]T;
sd(n) := [s(n);s(n?1);:::;s(n?d)]T;
?s(n) := Efs(n)g; ~s(n) := s(n)? ?s(n);
' :=
2
6401?d 0
Id 0d?1
3
75;
? := [1;01?d]T;
Hd(n) := [h(m)(n;0);h(m)(n;1);:::;h(m)(n;L);0R?(d?L)];
h(m)(n;l) := [h(m;1)(n;l);:::;h(m;R)(n;l)]T; (5.10)
109
where ?(n) is defined just as y(n) and integer d ? L. Assume data symbols are zero-mean and
white. If s(n) is a data symbol, we have ?s(n) := 0, ~s(n) := s(n); if s(n) is a training symbol,
?s(n) := s(n), ~s(n) := 0. Then the underlying state-space model corresponding to the mth channel
is given by the state and the measurement equations
sd (n) = 'sd (n?1)+??s(n)+?~s(n); (5.11)
y(n) = H(m)d (n)sd (n)+v(n): (5.12)
In (5.11) ?s(n) and ~s(n) are defined just as sd(n).
Consider a set of TB received symbols divided up into P subblocks as in Sec. 5.2.2. For model
m, we estimate the BEM coefficients w(m;r)q (l) via the least-squares approach of Sec. 5.2.2 using
the training symbols. Then the estimated channel for the mth model is given by ^h(m;r)(n;l) =
PQm
q=1 ^w
(m;r)q (l)ej!qn.
5.3.2 Interacting Multiple Model Data Detection
Using the M estimated channels from each block of received symbols, we obtain the M models
with state equation (5.11) and measurement equation
y(n) = ^H(m)Td (n)sd (n)+?(n); (5.13)
where ^H(m)d (n) is as in Section 5.3.1 with h(m) (n;l) replaced with estimated ^h(m) (n;l). Now our
task is to estimate sd(n) given y(k), k ? n, and the M models specified by (5.11) and (5.13). In
(5.13) we treat ^H(m)d (n) as true H(m)d (n).
110
Table 5.1: Summary of the IMM algorithm (one cycle)
Interaction (i;j = 1;2;???M):
predicted mode probability: ??j (k) = Pi pij?i(k?1)
mixing probability: ?ijj = pij?i(k?1)=??j (k)
^s0dj(k?1jk?1) = Pi ^sdi(k?1jk?1)?ijj
V0dj(k?1jk?1) = PiVdi(k?1jk?1)?ijj +Xj
where the ?spread-of-the-means? term in the mixing is
Xj = Pi[^sdi(k?1jk?1)? ^s0dj(k?1jk?1)]
?[^sdi(k?1jk?1)? ^s0dj(k?1jk?1)]H?ijj
Filtering (i;j = 1;2;???M):
^sdj(kjk?1) = '^s0dj(k?1jk?1)+??s(k)
Vdj(kjk?1) = 'V0dj(k?1jk?1)'H + 2~s??T
measurement residual: zj = y(k)?Hdj^sdj(kjk?1)
residual cov.: Dj = H(j)d Vdj(kjk?1)H(j)Hd + 2vIR
fliter gain: Gj = Vdj(kjk?1)H(j)Hd D?1j
^sdj(kjk) = ^sdj(kjk?1)+Gjzj
Vdj(kjk) = Vdj(kjk?1)?GjDjGHj
likelihood function: ?j = [det(?Dj)]?1e?zHj D?1j zj
mode probability: ?j(k) = ?
?
j ?jP
i ?
?
i ?i
Combination:
^sd(kjk) = Pj ^sdj(kjk)?j
Vd(kjk) = Pj Vdj(kjk)?j +X
where the ?spread-of-the means? term in combination is
X = Pi[^sdi(kjk)? ^sd(kjk)][^sdi(kjk)? ^sd(kjk)]H?i(k)
111
We propose to use the IMM algorithm [11] to estimate sd(n). In order to do this, in keeping
with [11], we allow transitions among the M models (this also allows consideration of time-varying
fd) where these transitions are governed by a first-order homogeneous Markov chain with transition
probabilities pij, i;j 2 f1;2;:::;Mg, PMj=1 pij = 1. The data symbols input to the channel ~s(n)
are treated as Gaussian random variables. The operation of IMM algorithm in one cycle is summa-
rized in Table 5.1 where 2~s = 2s = Efjs(n)j2g for information symbol, = 0 for training symbol.
Table 1 provides one-cycle (one time sample update) of the IMM algorithm. The required initial-
ization for the algorithm is as follows: at time k = 0, ^s(1j0) = Efs(1)g = 0 and its covariance
V^s(1j0) = 2sId+1. Having obtained the IMM estimate ^sd(njn) of sd(n), we estimate s(n) with
equalization delay d by quantizing the (d+1)st component of ^sd(njn).
5.4 Numerical Examples
A random time- and frequency-selective Rayleigh fading channel is considered. We take L = 2
(3 taps) in (5.1), number of receive antennas R = 2, and h(r)(n;l) are zero-mean, complex Gaussian
with variance 2h = 1=(L+1). For different l?s and r?s, h(r)(n;l)?s are mutually independent and
satisfy the Jakes? model. To this end, we simulated each single tap following [81] (with a correction
in the appendix of [77]).
We consider a communication system with carrier frequency of 2GHz, data rate of 40kBd
(kilo-Bauds), therefore Ts = 25?s, and a varying Doppler spread fd in the range of 0Hz to 200Hz,
or the normalized Doppler spread fdTs from 0 to 0:005 (corresponding to a maximum mobile ve-
locity from 0 to 108km=h). The additive noise was zero-mean complex white Gaussian. The
(receiver) SNR refers to the average energy per symbol over one-sided noise spectral density. The
time-multiplexed training scheme of [48] described in Sec. 5.2.2 is adopted, where during data
112
0 5 10 15 2010
?3
10?2
10?1
SNR(dB)
BER
BPSK Signal
Q upperbound, linear shape channel
Q upperbound, step shape channel
Multiple model, linear shape channel
Multiple model, step shape channel
Figure 5.1: IMM with three models, BER vs SNR with BPSK information symbols.
sessions the information sequences is modulated by BPSK or QPSK with unit power. The train-
ing session is described by (5.7) with = p2L+1 so that the average symbol power of training
sessions is equal to that of data sessions.
5.4.1 Example 1: IMM with Three Models
We generated a random doubly-selective channel as discussed earlier but with two different
profiles of varying fd?s as follows:
1. fd=0 Hz for 1 ? n ? 420, fd=100 Hz for 421 ? n ? 840, fd=200 Hz for 841 ? n ? 1260,
fd=100 Hz for 1261 ? n ? 1680, fd=0 Hz for 1681 ? n ? 2100. We picked K = 2,
TB = 175 and P = 5. Each subblock has 35 symbols with 30 information symbols in the
beginning and 5 training symbols at the end (see Sec. 5.2.2). This channel is named as the
Step Shape time-varying channel.
113
0 5 10 15 2010
?3
10?2
10?1
100
SNR(dB)
BER
QPSK Signal
Q upperbound, linear shape channel
Q upperbound, step shape channel
Multiple model, linear shape channel
Multiple model, step shape channel
Figure 5.2: IMM with three models, BER vs SNR with QPSK information symbols.
2. Now fd varies linearly from 0Hz to 200Hz over 1 ? n ? 1050, and fd varies linearly from
200Hz to 0Hz over 1051 ? n ? 2100. This channel is named as the Linear Shape time-
varying channel.
Two variations on channel estimation schemes are compared using an equalization delay d = 5:
1. Q Upper bound: We used a fixed Q for all blocks with Q=5= upper bound (denoted by ?Q
upper bound? in the figs.). With 5 subblocks per non-overlapping block (total 60 blocks), we
estimated the channel for each block via the approach of Sec. 5.2.2. Then we used Kalman
filtering with d = 5 (no IMM) to detect the information symbols.
2. Proposed Multiple Model: Here we used overlapping blocks by shifting blocks by one
subblock. We used three models M = 3 with Q1=1, Q2=3 and Q3=5. The channels are
estimated over one block, then we shifted to the right by one subblock (35 symbols), and
114
estimated the 3 candidate channels again, and so on. For transition probability matrix we
picked 2
66
66
64
0:9 0:1 0
0:05 0:9 0:05
0: 0:1 0:9
3
77
77
75
which reflects the fact that transitions in fd do not jump over an intermediate value. The three
models had equal initial probabilities of 1=3.
The bit error rate (BER) of each scheme was studied by averaging over 200 runs where in each
run, a symbol sequence of length 2100 is generated and fed into a random doubly-selective channel
generated with specified fd?s. The first 70 symbols were discarded in evaluations. In Figs. 5.1 and
5.2, the performances of the two schemes under different SNR?s are compared for BPSK and QPSK
information sequences, respectively. It is readily seen that overestimating Doppler spread leads to
a performance deterioration compared to the proposed IMM approach relying on a multiple model
formulation.
5.4.2 Example 2: IMM with Two Models
In this example, the varying Doppler spread fd is in the range of 0Hz to 100Hz. Again two
variations on channel estimation schemes are compared:
1. Q Upper bound: We used a fixed Q for all blocks with Q=3= upper bound (denoted by ?Q
upper bound? in the figures). With 5 subblocks per non-overlapping block (total 50 blocks),
we estimated the channel for each block via the approach of Sec.5.2.2. Then we used Kalman
filtering with d = 5 (no IMM) to detect the information symbols.
115
2. Proposed Multiple Model: Here we used overlapping blocks by shifting blocks by one sub-
block. We used two models M = 2 with Q1=1 and Q2=3. The channels are estimated over
one block, then we shifted to the right by one subblock (35 symbols), and estimated the 2
candidate channels again, and so on. For transition probability matrix we picked
2
640:9 0:1
0:1 0:9
3
75
The two models had equal initial probabilities of 1=2.
In all numerical results, the bit error rate (BER) of each scheme was studied by averaging over
200 runs where in each run, a symbol sequence of length 1750 (total 50 blocks) is generated and
fed into a random doubly-selective channel generated with specified fd?s.
Step Shape Time-Varying Channel
We generate the Step Shape random doubly-selective channel with varying fd as follows:
fd=0 Hz for 1 ? n ? 584, fd=100 Hz for 585 ? n ? 1168, and fd=0 Hz for 1169 ? n ? 1750.
In Figs. 5.3 and 5.4, the performances of the two schemes under different SNR?s are compared
for BPSK and QPSK information sequences, respectively. It is readily seen that overestimating
Doppler spread leads to a performance deterioration. The proposed IMM approach relying on a
multiple model formulation provides a good performance improvement.
Fig. 5.5 plots how the average mode probabilities ? (in Table 5.1) change with time. It can be
seen from the figure that model 2 (Q2 = 3) that is supposed to capture larger Doppler frequency
can also capture model 1 (Q1 = 1 for fd = 0Hz), while model 1 can not capture model 2.
116
Linear Shape Time-Varying Channel
In this example the Linear Shape random doubly-selective channel is generated with varying
fd linearly: fd is from 0Hz to 100Hz for 1 ? n ? 875, and fd is from 100Hz to 0Hz for 876 ? n ?
1750.
Again the performances of the two schemes under different SNR?s are compared in Fig. 5.6
and Fig. 5.7 for BPSK and QPSK information sequences, respectively. Fig. 5.8 plots how the
average mode probabilities ? change with time.
The Effect of Training Sequence on Mode Probability
How does the training sequence structure in (5.7) influence the mode probability is tested in
this example. The random channel is generated by fixing fd = 0 for 1 ? n ? 1750 (frequency-
selective and time-invariant channel). Model 1 with Q = 1 and Model 2 with Q = 3 are exploited
for symbol detection. Suppose the exact channel information regarding Model 1 and Model 2 is
available at the receiver. Clearly Model 1 should be the right model being chosen with higher mode
probabilities than Model 2.
In one case, the training symbols with structure as (5.7) are used. The resulting average mode
probability (averaged over 200 iterations) is plotted in Fig. 5.9. Similar to the plottings in Fig. 5.5
and Fig. 5.8, we see that the mode probabilities between blocks are inconsistent.
In the other case, no training symbols are inserted in the transmitted signal. The average mode
probability result is shown in Fig. 5.10. It is seen that the mode probabilities under this case are
consistently smooth. The comparison between Fig. 5.9 and Fig. 5.10 illustrates that the inserted
zeros in training structure (5.7) result in the inconsistency of mode probability.
117
0 5 10 15 2010
?4
10?3
10?2
10?1
SNR(dB)
BER
BPSK Signal
Q upperbound
Multiple Model
Figure 5.3: IMM with two models. BER vs SNR with BPSK information symbols. Step shape
time-varying Channel
0 5 10 15 2010
?3
10?2
10?1
SNR(dB)
BER
QPSK Singal
Q upperbound
Multiple Model
Figure 5.4: IMM with two models. BER vs SNR with QPSK information symbols. Step shape
time-varying Channel
118
0 200 400 600 800 1000 1200 1400 16000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Symbol Index
AAverage Mode Probability
Model 1, Q=1
Model 2, Q=3
Figure 5.5: Average mode probability of IMM. Step shape time-varying channel. SNR=20dB
0 5 10 15 2010
?4
10?3
10?2
10?1
SNR(dB)
BER
BPSK Signal
Q upperbound
Multiple Model
Figure 5.6: IMM with two models. BER vs SNR with BPSK information symbols. Linear Shape
Time-Varying Channel
119
0 5 10 15 2010
?3
10?2
10?1
SNR(dB)
BER
QPSK Signal
Q upperbound
Multiple Model
Figure 5.7: IMM with two models. BER vs SNR with QPSK information symbols. Linear Shape
Time-Varying Channel
0 200 400 600 800 1000 1200 1400 16000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Symbol Index
Average Mode Probability
Mode 1, Q=1
Mode 2, Q=3
Figure 5.8: Average mode probability of IMM. Linear shape time-varying channel. SNR=20dB
120
0 50 100 150 200 250 300 350
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Symbol Index
Average Mode Probability
Wrong Model
Correct Model
Figure 5.9: Average mode probability of IMM. Time-invariant channel. With training insertion.
SNR=20dB
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Symbol Index
Average Mode Probability
Wrong Model
Correct Model
Figure 5.10: Average mode probability of IMM. Time-invariant channel. Without training insertion.
SNR=20dB
121
5.5 Conclusion
An adaptive channel estimation scheme, exploiting the oversampled complex exponential ba-
sis expansion model (CE-BEM), was presented for doubly-selective channels where we tracked the
BEM coefficients via a multiple model approach. In the past work the number of BEM coefficients
used to model the doubly-selective channels for channel estimation has been based on an upper
bound on the channel Doppler spread. The higher the Doppler spread, the more the number of BEM
coefficients, which leads to a higher channel estimation variance. In this chapter we proposed to use
a multiple model framework where several candidate Doppler spread values were used to cover the
range from zero to an upper bound, leading to multiple CE-BEM channel models, each correspond-
ing to an assumed value of the Doppler spread. Subsequently the well known interacting multiple
model (IMM) algorithm was used for symbol detection based on multiple state-space models cor-
responding to the multiple estimated channels. Numerical examples were presented to illustrate the
proposed approach. The results showed that the IMM approach relying on multiple models had a
better performance than traditional overestimating Doppler spread scheme.
122
CHAPTER 6
DOUBLY-SELECTIVE CHANNEL ESTIMATION FOR OFDM SYSTEMS USING DPS-BEM AND
TIME-MULTIPLEXED TRAINING
6.1 Introduction
Wireless multicarrier (MC) communication systems utilize multiple complex exponentials as
information-bearing carriers. MC transmissions thus retain their shape and orthogonality when
propagating through linear time-dispersive media. They were first conceived and implemented with
analog oscillators in the 60s [42, 83], but it was not until their all-digital implementation with the
Fast Fourier Transform (FFT), that their attractive features were unravelled and sparked widespread
interest for adoption in various single user and multiple access (MA) communication standards
[9]. Nowadays, MC systems such as the Orthogonal Frequency Division Multiplexing (OFDM)
are included in the Digital Audio/Video Broadcasting (DAB/DVB) standards in Europe while the
Discrete Multi-Tone (DMT), its wireline counterpart in the US, has been applied to high-speed
Digital Subscriber Line (DSL) modems over twisted pairs [10].
The pilot-aided doubly-selective channel estimation for OFDM systems is considered in [62],
where the channels are approximated by BEMs. The transceiver block diagram is shown in Fig.
6.1. Since the channel estimation is based on frequency-domain training, the receiver can find
no subcarrier that solely depends on pilots and thus is not contaminated by information symbols.
Due to time-variation, the resulting channel matrix (include IFFT and FFT) is no longer diagonal
IFFT FFTCP /ZP insertion CP /ZP removalchannel Channel estimationTraining insertion Symbol detection
Figure 6.1: OFDM transceiver block diagram in [62]
123
IFFT FFTCP /ZP insertion CP /ZP removalchannel Channel estimationTraining insertion Symbol detection
Figure 6.2: Proposed OFDM transceiver block diagram
matrix, which results in the very complicated channel estimation or equalization procedure [6]. In
[62], a receiver window is used to suppress the out of band interference, and it is assumed that the
channel matrix is approximately banded after windowing. It is also not clear what is the ?optimum?
strategy to place the frequency-domain training symbols. The linear MMSE channel estimator
for DPS-BEM-based MIMO-OFDM (multiple input multiple output-orthogonal frequency division
multiplexing) doubly-selective channels is introduced in [49], but their channel estimation suffers
from the frequency-domain training problem, too.
In this chapter, we will apply the optimum time-multiplexed training based channel estima-
tion introduced in Chapter 2 to OFDM systems under doubly-selective channels. The time-varying
channel is described by CE-BEM or DPS-BEM. The time-domain training clusters are inserted
after IFFT at the transmitter and the channel estimation proceeds before FFT at the receiver (see
Fig. 6.2). In this way, the ?optimum? training strategy and the corresponding channel estima-
tion schemes proposed before can be directly applied to MIMO-OFDM systems. Compared to
frequency-domain training design, the main advantage of time-domain training for OFDM system is
that the information symbols are not contaminated by the training symbols as the frequency-domain
training. The performance of frequency-domain training-based channel estimation and time-domain
training-based channel estimation are compared by simulations.
124
6.2 System Model
Fig. 6.2 depicts the proposed baseband block diagram of a cyclic prefix (CP)-OFDM system,
where the jth Nc ?1 information block
~cj := [c((j ?1)Nc);c((j ?1)Nc +1);:::;c((j ?1)Nc +Nc ?1)]T (6.1)
is first precoded by the IFFT matrix FHNc with (m;k)th entry
?FH
Nc
?
m;k =
1p
Nce
j2?mk=Nc; (6.2)
to yield the so-called ?time domain? block vector
cj = FHNc~cj: (6.3)
Then the training sequence bj of length Nb and cyclic prefix (CP) cpj of length Ncp is inserted be-
tween each cj. CP is inserted at the transmitter and discarded at the receiver to avoid the interblock
interference (IBI) [72]. The entries of the resulting redundant block sj := [cpTj ;bTj ;cTj ] are finally
sent sequentially through the channel. Suppose there are totally J blocks for transmission. Then the
general structure of transmitted signal s with length N := Ncp +Nb +Nc is
s := [cpT1 ;bT1 ;cT1 ;:::;cpTJ;bTJ;cTJ]T: (6.4)
125
Consider the doubly-selective single input single output FIR linear channel, the same as in
Section 2.2. Then the matrix-vector input-output relationship is given by
x =
QX
q=0
D?qWqs+?; (6.5)
where x := [x(0);x(1);:::;x(N ?1)]T , ? is the zero-mean white complex-Gaussian noise defined
similarly to x, D?q = diag[?q] with ?q := [?q(0);?q(1);:::;?q(N ?1)]T , and Wq?s are N ?N
lower triangular Toeplitz matrices with 1st column [wq(0);wq(1);:::;wq(L);0;:::;0]T .
This system model in (6.5) is exactly the same as in (2:9). Comparing (2:10) with (6.4), and
(2:9) with (6.5), we notice that the IBI free received signal is the same as that in Chapter 2 after CP
removal at the receiver. Therefore, the LS/MMSE channel estimator and optimum training design
presented in Chapter 2 can be directly applied here without any change. We will skip the theoretical
expressions which would be the same as in Chapter 2.
6.3 Doubly-Selective Channel Estimation for OFDM Systems Using Frequency-Domain Train-
ing [62]
Consider an OFDM system with M subcarriers. The frequency domain transmitted signal is
defined as:
~x := [tT1 ;uT1 ;tT2 ;uT2 ???tTP;uTP]T
wheretp (p = 1;2;???P) is a column of Mt training symbols anduj is a column of Mu information
symbols. Let M = (Mt +Mu)P. As illustrated in Fig 6.1, the OFDM symbol ~x is first modulated
by the IFFT operation
x = FHM ~x: (6.6)
126
We ignore the symbol index in (6.6) since only one OFDM symbol is considered.
The modulated symbol x is then concatenated by a CP and sent over the channel. If the channel
is assumed to be time-varying (TV) and approximated by a BEM, as in [62], the received signals at
the receiver after CP removal and FFT operation is given by
y = FMRcpHTcpx+v
=
QX
q=0
Dq?q~x+v; (6.7)
where Tcp := [INcp?M;IM] and Rcp := [0M?Ncp;IM] are CP insertion and CP removal matrices,
respectively; H represents the channel matrix in time-domain; v is the zero-mean white complex-
Gaussian noise; Dq is a circulant matrix whose first column is the frequency response of the qth
basis function
Dq := FMdiagf?qgFHM (6.8)
and ?q is a diagonal matrix whose diagonal is the frequency response of the BEM coefficients
corresponding to the qth basis function [62]
?q = diagfFL[wq;0;:::;wq;L]Tg: (6.9)
FL stands for the first L + 1 columns of the matrix FM. The least squares (LS) channel estimator
can be achieved based on (6.7) as in [[62], Section IV. B].
127
Due to the time-variation, the channel matrix in the frequency domain is no longer diagonal,
but approximately banded. Therefore, the channel estimation approach based on (6.7) suffers a large
estimation error especially with a high Doppler spread.
6.4 Numerical Examples
In this section, the numerical results of the proposed time-domain (TD) training channel es-
timation (Sec. 6.2, referred to as TDE) are given and compared with the frequency-domain (FD)
channel estimation (Sec. 6.3, referred to as FDE). We use binary phase shift keying (BPSK) modu-
lation in all examples.
To make a relatively fair comparison between TDE and FDE cases, we want to keep the same
transmission rates in both schemes. For TDE scheme, the OFDM system has Nc = 30 subcarriers.
There are Nb = 2L + 1 training symbols with optimal structure [0L;b;0L], b > 0 in every OFDM
symbol and J = 10 OFDM symbols are transmitted sequentially. For FDE scheme, an OFDM
system with M = (Mt + Mu)P subcarriers is considered. The subcarriers that are reserved for
pilots are grouped in P = 10 equidistant clusters, each containing Mt = 2L + 1 pilot tones. We
pick Mu = 30. Inside each cluster, the scheme referred to as ?frequency-domain Kronecker delta?
(FDKD) in [35] are exploited, where a nonzero pilot is located in the middle of the cluster with zero
guard bands on both sides. This equidistant pilot cluster scheme finds its practical advantage in [60]
although the channel follows the bathtub-shaped Doppler spectrum in that case. Another important
reason of using FDKD training pattern here is that it is convenient to keep the same transmission
rate with TDE scheme.
The doubly-selective Rayleigh fading channel is simulated according to [77, 81] with channel
order L = 2, carrier frequency of 2GHz, data rate of 40 kbps, and thus, symbol duration Ts = 25?s.
128
0 5 10 15 20 25 3010
?4
10?3
10?2
10?1
100
101
SNR(dB)
MSE
FDE, fd=4, ?Dmax=fd*Ts=1e?4
FDE, fd=40, ?Dmax=fd*Ts=1e?3
TDE, fd=4, ?Dmax=fd*Ts=1e?4
TDE, fd=40, ?Dmax=fd*Ts=1e?3
Figure 6.3: LS Channel estimation MSE comparison between TDE and FDE with varying ?Dmax
Therefore, each tap of the generated time-variant channel has a Jakes? spectrum; it is not gener-
ated using the assumed BEM modeling. Also, the 3 taps of the channel are mutually independent.
Depending on different maximum Doppler spread fds, a varying maximum normalized one-sided
Doppler bandwidth ?Dmax = fdTs can be derived, where Ts is the symbol duration. A Kalman
filter formulation is used for information detection after the channel estimation. The SNR refers
to 1= 2? where the information sequence power is normalized to one and the channel power is also
normalized to unity.
6.4.1 Example 1: Channel Estimation Performance
In this example, the LS channel estimator is used in both TDE and FDE schemes to estimate
w, and then the channel is estimated as in (2:29). The channel estimation MSE is calculated as in
(2:65). Fig. 6.3 plots the channel estimation MSE (averaged over 200 Monte Carlo runs) versus
129
SNR under different normalized maximum Doppler bandwidths, where fd = 4 or 40Hz. It is
clear that the MSE of time-domain training channel estimation is consistently smaller than that of
frequency-domain training channel estimation.
6.4.2 Example 2: CE-BEM versus DPS-BEM
The LS channel estimator in Example 1 is based on the DPS-BEM channel model. In this
example we compare the LS channel estimation MSE of TDE under CE-BEM and DPS-BEM. Fig.
6.4 plots the MSE (2:65) versus SNR under different maximum Doppler bandwidths (?(F)Dmax is for
CE-BEM, and ?(S)Dmax is for DPS-BEM). Especially, we take QF = 2d?DmaxNe for CE-BEM and
QS = d2?DmaxNe DPS-BEM. It is seen from Fig. 6.4 that the channel estimation performances
of DPS-BEM outperform that of CE-BEM for all ?Dmax. Fig. 6.5 takes BER (average over 2000
Monte Carlo runs) as a performance measure to compare the performance between DPS-BEM and
CE-BEM for TDE. Comparing with Fig. 6.4 makes it clear that the significantly reduced MSE of the
DPS-BEM channel estimation leads to a pronounced reduction in BER compared to the CE-BEM
case.
6.5 Conclusion
In this chapter, we applied the optimum time-multiplexed training based channel estimation in-
troduced in Chapter 2 to OFDM systems under doubly-selective channels. The time-varying channel
was described by CE-BEM or DPS-BEM. The time-domain training clusters are inserted after IFFT
at the transmitter and the channel estimation proceeds before FFT at the receiver. Compared to
frequency-domain training design, the main advantage of time-domain training for OFDM system
130
0 5 10 15 20 25 3010
?3
10?2
10?1
100
101
SNR(dB)
MSE
?Dmax(S) =0.001
?Dmax(S) =0.005
?Dmax(F) =0.001
?Dmax(F) =0.005
Figure 6.4: LS Channel estimation MSE comparison between DPS-BEM and CE-BEM with varying
?Dmax, TDE
0 5 10 15 20 25 3010
?3
10?2
10?1
100
SNR(dB)
BER
?Dmax(S) =0.001
?Dmax(S) =0.005
?Dmax(F) =0.001
?Dmax(F) =0.005
Figure 6.5: BER comparison between DPS-BEM and CE-BEM with varying ?Dmax, TDE
131
is that the information symbols are not contaminated by the training symbols as in the frequency-
domain training case. The simulation results confirmed the claims.
132
CHAPTER 7
SUMMARY AND FUTURE WORK
With the emergence of next-generation wireless mobile communications, multimedia services
have placed increasing demands for high data rates and high mobility. The high data rates give
rise to frequency selectivity, while the mobility and carrier offset introduce time selectivity. In
this dissertation, the channel estimation and equalization for frequency-selective and time-selective
channel were considered.
7.1 Summary of Original Work
In Chapter 2, the channel estimation for single-input single-output frequency- and time- selec-
tive channels was considered using time-multiplexed training. The time-varying channel was as-
sumed to be well-described by a basis expansion model using discrete prolate spheroidal sequences
as bases (DPS-BEM). Both linear least squares and minimum mean-square-error approaches were
exploited to estimate the basis expansion coefficients. Training designs for time-multiplexed train-
ing based on minimization of channel estimation mean-square-error were investigated. The issue
of training power allocation was addressed. Then the proposed channel estimation approaches in
Chapter 2 was extended to multiuser multiple-input multiple-output doubly-selective channels in
Chapter 3.
The linear equalization and decision feedback equalization of doubly-selective channels mod-
eled via BEMs were introduced in Chapter 4. There has been much interest in designing time-
variant serial finite impulse response linear and DFE equalizers using complex exponential BEMs
for equalizers in addition to using CE-BEM for modeling the channel itself. We showed that the
133
Kalman filter formulation of the linear equalizer and an alternative formulation of the FIR DFE
based on a CE-BEM channel model yields the same or improved BER at a lower computational
cost, without incurring the approximation error inherent in CE-BEM modeling of equalizers.
An adaptive channel estimation scheme, exploiting the oversampled complex exponential ba-
sis expansion model (CE-BEM), was presented for doubly-selective channels where we tracked
the BEM coefficients via a multiple model approach in Chapter 5. In the past work the number
of BEM coefficients used to model the doubly-selective channels for channel estimation has been
based on an upper bound on the channel Doppler spread. The higher the Doppler spread, the more
the number of BEM coefficients, which leads to a higher channel estimation variance. We proposed
to use a multiple model framework where several candidate Doppler spread values were used to
cover the range from zero to an upper bound, leading to multiple CE-BEM channel models, each
corresponding to an assumed value of the Doppler spread. Subsequently, the well known interact-
ing multiple model (IMM) algorithm was used for symbol detection based on multiple state-space
models corresponding to the multiple estimated channels.
Orthogonal Frequency-Division Multiplexing (OFDM), a digital multi-carrier modulation scheme,
has developed into a popular scheme for wideband wireless communication due to its high spec-
tral efficiency and simple equalization. In Chapter 7, we extended the optimum time-multiplexed
training-based channel estimation introduced in Chapter 2 to OFDM systems under a doubly-
selective channels. Compared to the traditional frequency-domain training design, the main ad-
vantage of time-domain training design for OFDM system is that the information symbols are not
contaminated by the training symbols as in the frequency-domain training case.
In all chapters, numerical examples based on computer simulations were presented to illustrate
the proposed approaches and confirm the conclusions.
134
7.2 Possible Future Directions
So far we have assumed that the coefficients of basis expansion model (BEM) are fixed over
data blocks. In fact, the BEM coefficients may also undergo changes.
Instead of the basis expansion model, another way to model time-varying channels is via au-
toregressive (AR) process, particularly the first order AR process [19]. Suppose h(n;l) represents a
wide-sense stationary uncorrelated scattering (WSSUS) channel. It is common to use the following
first-order AR model to describe it:
h(n;l) = fih(n?1;l)+?(n); (7.1)
where fi is the AR coefficient, and the driving noise ?(n) is zero-mean complex Gaussian with
variance 2? and statistically independent of h(n ? 1;l). Assume that h(n;l) is also zero-mean,
complex Gaussian with variance 2h. Then
fi = 1 2
h
Efh(n;l)h?(n?1;l)g; (7.2)
2? = 2h(1?jfij2): (7.3)
It is a tractable model, 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 [73].
This Markovian assumption has been verified for Rayleigh fading channels in [73], by considering
the mutual information between channel symbols. AR models describe temporal variation on a
symbol-by-symbol update basis, while BEMs depict the evolution of the channel over a period
(block) of time.
135
An AR model is not appropriate for a fast fading channels because the channel tracking depends
on the inserted training. For information sessions, channel estimates can only be obtained based on
the results from the previous training session [40]. However, it could be a good way to track the basis
expansion coefficients in BEMs. Up to this point, we suppose that the basis expansion coefficients
are fixed over data blocks. In fact, the BEM coefficients may also undergo changes, but not as fast
as the channel variations.
In the next step, we will model the doubly-selective channel by exploiting the CE-BEM for the
overall time-variant channel and an AR model for the BEM coefficients. Recall the BEM model in
(2:2). Stack the BEM coefficients into vectors
wl = [w1(l);w2(l);:::;wQ]T: (7.4)
Suppose the whole data block is divided into J sub-blocks, where the coefficient vector in (7.4) for
the jth sub-block (j = 1;2;:::;J) is denoted as wl(j) and updated every sub-block. Since a fading
channel well follows the Markov model, we further assume that the BEM coefficients of each block
are Markovian, too. The first-order AR process for BEM coefficients is then given by:
wl(j) = fiwl(j ?1)+?l(j); (7.5)
where ?l(j) defined similar to wl(j), is zero-mean complex Gaussian noise and statistically inde-
pendent of wl(j ?1). Based on the AR model in (7.5), a Kalman filter can be applied to track the
BEM coefficient for each block. After deriving the estimated BEM coefficients for jth sub-block
^wl(j), the estimated channel impulse response for the whole jth sub-block can be given, via the
136
CE-BEM, as
^h(n;l) = [ej!0n;ej!1n;:::;ej!Qn]T ^wl(j): (7.6)
It would be interesting to explore this approach and compare it with our proposed approach.
Multiple Model Approach. In applying IMM we treated the estimated channel as the true
channel. It would be interesting to incorporate estimated channel statistics, e.g., estimation error
variance, into the IMM algorithm.
137
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144
APPENDIX A
ASYMPTOTIC DPS-BEM/ SLEPIAN SEQUENCES
The Slepian sequences as solutions to (2:5) are hard to work with analytically. Following [61,
Sec. 5.3.1] (in a different context), we will use asymptotic expressions (N ?large?) based on some
heuristic considerations (as in [61, Sec. 5.3.1]). The entries in matrix C can also be expressed as
[C]y;z = 12?
Z 2??Dmax
?2??Dmax
e?i(y?z)!d!: (A:1)
Define fl := 2?Dmax and QS := Nfl. Let fl be chosen larger than 1=N so that we get QS ? 1. (To
simplify the discussion, QS and N are assumed to be even integers in what follows.) For large N,
we can approximate the integral in (A:1) as (! = 2?N p):
C ? 1N
QS=2X
p=?QS=2
a
2?
N p
?
a?
2?
N p
?
:= C0; (A:2)
where a(!) := [1;e?i!;:::;e?i!(N?1)]T , and ?ps are the QS + 1 largest eigenvalues of the matrix
C. Based on (2:24), the eigenvectors of the matrix C0 are derived as
n
1p
Na(
2?
N p)
oQS=2
p=?QS=2
with
eigenvalues ?1. Therefore, if C0 is used to approximate C, the Slepian sequences in (2:2) and (2:5)
can be expressed as
?(S)q (n) = 1pNe(?i2?(q?QS=2)n=N): (A:3)
[As noted in [61, Sec. 5.3.1], C0 does not approximate C in any rigorous sense.]
Obviously, the approximate Slepian sequences in (A:3) are complex-valued. Since the Slepian
sequences are real, we want to find a matrix with real eigenvectors that can approximate C based on
145
C0. Since C0 is a real symmetric positive semi-definite matrix with real eigenvalues ?1, it readily
follows that gr := 12pN(a+a?) and gi := 12ipN(a?a?) are also the eigenvectors of C0 and they
are real. From (A:2), we notice that
a?
2?
N p
?
= a
?2?N p
?
: (A:4)
Therefore, we can rewrite the matrix C0 as
C0 =
0X
p=?QS=2
gr
2?
N p
?
g?r
2?
N p
?
+
1X
p=?QS=2
gi
2?
N p
?
g?i
2?
N p
?
: (A:5)
With this expression of matrix C0, the normalized eigenvectors of matrix C0 are expressed as
'g
r(2?N p)
?0
p=?QS=2 and
'g
i(2?N p)
?1
p=?QS=2. Thus, the corresponding approximate Slepian se-
quences for n 2f0;:::;N ?1g are
?(S)q;r (n) =
r 2
Ncos(2?(q ?
QS
2 )n=N); ?
(S)
q;i (n) =
r 2
Nsin(2?(q ?
QS
2 )n=N): (A:6)
146
APPENDIX B
MATHEMATICAL NOTATIONS
? approximately equal to
? Kronecker product
0M?N M ?N all zeros matrix
a lower case letters for scalars
dae integer ceiling of a
bac integer floor of a
jaj magnitude of a
a lower case letters in bold face for column vectors
kak Frobenius norm of a
A upper case letters in bold face for matrices
A? complex conjugate of A
Ay Moore-Penrose pseudo-inverse operation
AH complex conjugate transpose of A
AT transpose of A
[A]n;m (n;m)-th entry of A
A upper case calligraphic letters for matrices
argmaxx f (x) value of x for which f (x) attains its maximum
argminx f (x) value of x for which f (x) attains its minimum
covf?g covariance operator
147
?(?) Kronecker delta function, defined as
?(n) =
8>
<
>:
1 if n = 0
0 if n 6= 0, t 2Z
J0(?) the zero-order Bessel function of the first kind
diag[a] diagonal matrix with a on its main diagonal
Ef?g expectation operator
EHf?g expectation operator with respect to H
IN N ?N identity matrix
max(?) maximum value operator
min(?) minimum value operator
O(?) big O notation: f (x) = O(g(x)) as x ! a (a 2R[?1),
iff jf (x)j? M jg(x)j as x ! a for some constant M > 0
R real field
trfAg trace of a square matrix A
Z integer field
j p?1
148
APPENDIX C
ABBREVIATIONS
AM amplitude modulation
AR auto-regressive
AWGN additive white Gaussian noise
BEM basis expansion model
BER bit error rate
BPSK binary phase-shift keying
CE-BEM complex exponential basis expansion model
CP cyclic prefex
CSI channel state information
DAB digital audio broadcasting
DFS discrete Fourier series DFS
DFT discrete Fourier transform
DML deterministic maximum likelihood
DMT discrete multi-tone
DPS discrete prolate spheroidal
DPS-BEM discrete prolate spheroidal basis expansion model
DS-CDMA direct sequence - code division multiple access
DSL digital subscriber line
DVB digital video broadcasting
149
FB feedback
FF feedforward
FFT fast Fourier transform
FIR finite impulse response
FM frequency modulation
IBI interblock interference
IMM interacting multiple model
ISBI inter-subblock interference
ISI inter-symbol interference
LS least squares
LTI linear time-invariant
MAC media access control
MC multicarrier
MIMO multiple-input multiple-output
ML maximum likelihood
MLSE maximum likelihood sequence estimation
MSE mean square error
MMSE minimum mean square error
m.s. mean-square
MUI multiple-user interference
150
NCMSE normalized channel mean square error
OFDM orthogonal frequency division multiplexing
PAM pulse amplitude modulation
pdf probability density function
PN pseudo-noise
PSAM pilot symbol aided modulation
QPSK quadrature phase-shift keying
RF Radio Frequency
SIMO single-input multiple-output
SISO single-input single-output
SNR signal-to-noise ratio
TI time invariant
TM time-multiplexed
TV time-varying
WSSUS wide sense stationary uncorrelated scattering
151