Frequency Synchronization Techniques in
Wireless Communication
Thesis subm itted to the U niversity o f C a rd iff in candidature for the degree of D octor of Philosophy.
Qiang Yu
Cardiff
UNIVERSITY PRIFYSCOl Ca'R Dy#
Centre of Digital Signal Processing
Cardiff University
UMI Number: U584985
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S T A T E M E N T 2
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Abstract
In this thesis various ite ra tive channel estim ation and data detection tech niques for tim e-va rying frequency selective channels w ith m u ltiple fre quency offsets are proposed.
F irstly, a m axim um likelihood approach for the estim ation of complex m u ltip a th gains (M G s) and real Doppler shifts (DSs) for a single in p u t single o u tp u t (SISO) frequency selective channel is proposed. In a tim e di vision m u ltip le access (T D M A ) system, for example the third-generation global system, or m obile GSM communications, the p ilo t symbols are generally inadequate to provide enough resolution to estimate frequency offsets. Therefore, our approach is to use the p ilo t sequence for the esti m ation and equalization of the channel w ith o u t consideration to frequency offsets, and then to use the soft estimates o f the tra n sm itte d signal as a long p ilo t sequence to determine ite ra tiv e ly the m u ltip le frequency offsets and refine the channel estimates. Inter-sym bol interference (ISI) is re moved w ith a linear structure tu rb o equalizer where the filte r coefficients are chosen based on the m inim u m mean square error (M M S E ) criterion. The detection performance is verified using the b it error rate (BER ) curves and the frequency offset estim ation performance through comparison w ith appropriate Cramer-Rao lower bounds.
This work is then extended for a m ulti-user transmission system where the channel is modelled as a m u lti in p u t m u lti o u tp u t (M IM O ) T D M A system. For the ite ra tive channel estim ation, the M IM O frequency se lective channel is decoupled in to m u ltip le SISO fla t fading sub-channels through appropriately cancelling b o th inter-symbol-interference (ISI) and inter-user-interference (IU I) from the received signal. The refined channel
estimates and the corresponding frequency offset estimates are then ob tained for each resolved M IM O m u ltip a th tap. Sim ulation results confirm a superior B E R and estim ation performance.
F inally, these ite ra tive equalization and estim ation techniques are ex tended to orthogonal frequency division m ultiplexing (O F D M ) based SISO and M IM O systems. For O F D M , the equalization is performed in two stages. In the firs t stage, the channel and the frequency offsets are es tim a te d in the tim e domain, while in the second stage, the transm itted symbols are estimated in the frequency domain and the mean values and the variances o f the symbols are determined in the frequency domain. These two procedures in teract in an iterative manner, exchanging in for m ation between the tim e and frequency domains. Sim ulation studies show th a t the proposed ite ra tive scheme has the a b ility to track frequency off sets and provide a superior B E R performance as compared to a scheme th a t does not tra ck frequency offsets.
iii
A b b r e v ia tio n s and A cron ym s
A M L A p pro xim ate M axim um Likelihood
AW G N A d d itiv e W h ite Gaussian Noise
B E R B it E rro r Rate
BP SK B in a ry Phase Shift Keying
C D M A Code D ivision M u ltip le Access
C IR Channel Im pulse Response
CP C yclic Prefix
C R LB Cram er-Rao Lower Bound
CSI Channel State In fo rm a tio n
D F E Decision Feedback Equalizer
D F T Discrete Fourier Transform
DS Doppler Shift
F B F Feedback F ilte r
F F F Feedforward F ilte r
F F T Fast Fourier Transform
F IR F in ite Impulse Response
F IM Fisher In fo rm a tio n M a trix
FO Frequency Offset
GSM G lobal System for M obile Communications
IC I In te r-C a rrie r Interference
IS I Inter-S ym bol Interference
iff I f and on ly if
IF F T Inverse Fast Fourier Transform
LE Linear Equalizer
LS Least Squares
L L R Log-likelihood Ratios
LOS Line of Sight
L T E Linear Transversal Equalizer
L T V Linear T im e Variant
M A P M axim um A Posteriori
M C M M u lti C arrier M odulation
M IM O M u ltip le In p u t and M u ltip le O u tp u t
M G M u ltip a th Gain
M L M axim um Likelihood
M L E M axim um Likelihood E stim ator
M LS E M axim um Likelihood Sequence E stim ation
M M S E M in im u m Mean Square E rror
MS M obile S tation
MSE Mean Square E rro r
M V U E M in im u m Variance Unbiased E stim ator
O F D M O rthogonal Frequency D ivision M u ltip le x in g
P D F P ro b a b ility D ensity Function
SISO Single In p u t and Single O u tp u t
SOVA Soft O u tp u t V ite rb i A lgorithm
T D M A T im e D ivision M u ltip le Access
V A V ite rb i A lg o rith m
v ii
L ist o f Sym bols
|.| Absolute value
© Schur-Hadamard product
(.)H H e rm itia n transpose operator
(.)T Transpose operator
(.)t Pseudo-inverse
= D e fin itio n
6 unknown parameter
A Diagonal m a trix
b feedback tap weight vector
cd d th element o f coordinate vector
fd D oppler shift
f forw ard filte r tap weight
Hd dth. colum n vector of convolution m a trix
H Channel convolution m a trix
Im { .) Im ag ina ry p a rt o f a complex number
In ( .) N a tu ra l lo garithm
V l l l
d ia g (v ) D iagonal m a trix w ith vector v on its m ain diagonal
Cov(.) Covariance operator
E { - } E xp ectation operator
I
Id e n tity m a trixL Length of the channel
M Equalizer length
n Num ber of sample
u> Zero mean w hite Gaussian noise
p(.) P ro b a b ility density function
R Covariance m a trix
R e {.} Real p a rt
S T ra in in g signal m a trix
U Received signal vector
V a r(.) Variance
w Equalizer tap weight vector
ix
S ta te m e n t o f O riginality
The co n tribu tions in this thesis are on the proposal and analysis of ite ra tive channel and frequency offset estim ation and detection algorithms for m u ltip a th channels w ith m ultiple frequency offsets. New techniques have been proposed for both SISO and M IM O systems and for T D M A and O F D M access schemes. The novel contributions are supported by one IE E E jo u rn a l, four IE E E conference papers. The summary of these contribu tions are given below.
1. In chapter 3, and [I, II] an ite ra tive approximate m axim um likelihood (A M L ) estim ator for a SISO m u ltip a th channel w ith d istin ct frequency offsets (FOs) has been proposed. The p ilo t symbols are generally inad equate to ob ta in an accurate estimate of the FOs due to lim ita tio n on the frequency resolution o f the estimator. Hence the soft estimate of the tra n sm itted signal are treated as a long p ilo t sequence to determine m ul tip le FOs and to refine channel estimates iteratively. The performance of the proposed scheme has been studied for both flat-fading and frequency selective fading channels in a GSM system. The performance of the pro posed scheme has been investigated w ith and w ith o u t error control coding, and compared w ith appropriate Cramer-Rao Lower bounds (C R LB).
2. In chapter 4, the param eter estim ation and equalization proposed for a SISO channel has been extended to M IM O m u ltip a th channels w ith dis tin c t FOs. The soft estimates of the tran sm itted signal has been used as a p ilo t signal to resolve m u ltip le users and m ultipaths and estimate and correct FOs in each resolved paths. The sim ulation results include BE R
X
curves and comparison of the variance of the FO estimates w ith the cor responding C R LBs. The results have been published in [III, V].
3. In chapter 5 and [IV ], iterative FO estim ation and correction tech niques for an O F D M system have been proposed to track m ultiple FOs due to d is tin c t Doppler shifts associated w ith m ultipaths. The long tra in ing signal available in an O F D M symbol has been used to obtain in itia l estimates o f the channel and FOs. Then the soft estimates of the data symbols are obtained in an iterative manner to track carrier frequency offsets and m u ltip a th channels coefficients in the subsequent packets. The proposed ite ra tive technique has the a b ility to resolve m ultipaths, thereby converting the jo in t m u ltip le FO estim ation problem in to estim ation of d istin ct FOs. The technique has also been extended to a M IM O O FD M system [VI].
P u b lic a tio n s
Journal Papers
I) Q. Yu, S. Lam botharan, “ Ite ra tive (Turbo) E stim ation and Detection Techniques for Frequency Selective Channels w ith M u ltip le Frequency Off sets,” IE E E Signal Process. Lett., vol. 14, pp. 236-239, A p r. 2007.
x i Conference Papers
I I) Q. Yu, S. Lam botharan, “ Iterative Channel E stim ation and Equaliza tio n Techniques for M u ltip a th Channels w ith M u ltip le Frequency Offsets” ,
The Tenth IE E E In te rn a tio n a l Conference on Communications Systems (ICCS), Singapore, Nov. 2006.
I I I ) Q. Yu, S. Lam botharan, “Iterative (Turbo) E stim ation and Detec tio n Techniques for Frequency Selective Channels w ith M u ltip le Frequency Offsets in M IM O System” , IE E E 65th Vehicular Technology Conference (V T C ), D u b lin , Ireland, A p ril 2007.
IV ) Q. Yu, S. Lam botharan, “ A n Iterative E stim ation and Detection Technique for O F D M Systems w ith M u ltip le Frequency Offsets (Doppler Shifts),” The 18th IE E E personal, indoor and mobile radio communica tions (P IM R C ), Athens, Greece, Sept. 2007.
V ) K. Cumanan, R. Krishna, Q. Yu and S. Lam botharan, “ M axim um likelihood frequency offset estim ation for frequency selective M IM O chan nels: A n average periodogram in te rpretation and ite ra tive techniques,” The 15th European Signal Processing Conference (EU SIPC O ), Poznan, Poland, Sept. 2007.
V I) Q. Yu, S. Lam botharan, “ Ite ra tive E stim ation and Detection Tech nique for a M IM O O F D M channel w ith M u ltip le Frequency Offsets” , Sub m itted to IE E E 67th Vehicular Technology Conference (V T C ), Singapore, M ay 2008.
A ck n ow led gm en t
I w ould like to th a n k especially my respected supervisor Dr. S. Lam botharan for his invaluable suggestions, and constant support in the de velopment o f th is research. His invaluable expertise, advice, and encour agement made th is work possible.
I am also extrem ely grateful to Prof. Jonathon A. Chambers for his help, invaluable discussions, and comments on this research. I am also grateful to him for p ro o f reading this thesis.
I also wish to th a n k Dr. Sajid Ahmed for his advice on ite ra tive methods, help in using LaTex and various practical problems.
I am greatly indebted to m y parents for th e ir constant support and encour agement o f pursuing m y education. I could not have succeeded w ith o u t th e ir support.
M y study here at C a rd iff U n iversity was valuable and enjoyable due to the excellent faculty, staff, and students. I wish to thank all my col leagues, in pa rticu la r, I would like to thank Dr. Wenwu Wang, Dr. Zhuo Zhang, Dr. Lay Teen Ong, D r. Rab Nawaz, Dr. Thomas Bowles, Dr. Yonggang Zhang, Dr. Clive Cheong Took, V im a l Sharma, T ariq Qaisir- ani, Loukianos Spyrou, L i Zhang, M in Jing, Cheng Shen, Lei Lei L i and Andrew Aubrey, for th e ir in sig h tfu l discussions and support.
Finally, I am grateful to m y girlfriend, Yue Zheng, and her fa m ily for th e ir caring support.
CONTENTS
LIST OF FIG U R E S
xvi
1 IN T R O D U C T IO N
1
1.1 Signal fading 1
1.1.1 Doppler sh ift 3
1.1.2 Channel Classification 4
1.2 O u tline o f the Thesis 5
2 P A R A M E T E R E ST IM A T IO N A N D EQUALIZATIO N
8
2.1 Channel param eter estim ation techniques 9
2.1.1 M axim um likelihood estim ation 10
2.1.2 Least squares estim ation 14
2.1.3 Joint estim ation of channel impulse response (C IR )
and frequency offset (FO) 14
2.2 Frequency estim ation techniques 17
2.2.1 Classical techniques based on Fourier transform 18
2.2.2 Subspace methods 18
2.2.3 B lin d parameter estim ation 19
2.3 Cramer Rao lower bound 20
2.3.1 The estim ation problem 20
2.3.2 M in im u m variance unbiased estim ation 21
2.3.3 Cramer Rao lower bound 22
x iv
2.4 E q ualization techniques 24
2.4.1 Linear transversal M MSE equalizer 25
2.4.2 Decision feedback equalization 27
2.4.3 Ite ra tiv e equalization 30
2.5 Sum m ary 34
3 IT E R A T IV E P A R A M E T E R ESTIM ATIO N A N D EQUAL
IZATIO N FO R SISO C H A N N ELS W ITH M ULTIPLE FR E
Q U E N C Y O FFSETS
36
3.1 Problem statement 38
3.2 E stim a tio n of m u ltip a th gains and frequency offsets 40
3.3 M M S E equalizer design 42
3.3.1 Equalizer for channels w ith o u t frequency offsets 42 3.3.2 Equalizer for channels w ith frequency offsets 44
3.4 Ite ra tive channel estim ation 47
3.4.1 Ite ra tive channel estim ation 47
3.4.2 Ite ra tive channel estim ation w ith M A P decoder 50
3.5 Sim ulations 52
3.6 Summ ary 59
3.7 A p pend ix 1 61
4 ITER A T IV E P A R A M E T E R EST IM A T IO N A N D EQUAL
IZATION FO R M IM O C H A N N E LS W IT H M ULTIPLE
F R E Q U E N C Y O FFSETS
62
4.1 Problem statem ent 64
4.2 E stim ation of m u ltip a th gains and frequency offsets 66
4.3 M IM O M M S E equalizer design 67
4.4 Ite ra tive M IM O M M S E equalizer design 69
4.5 Ite ra tive Channel E stim ation 71
X V
4.7 Sum m ary 76
4.8 Appendices 2 78
5 F R E Q U E N C Y SYNCH RO NIZATIO N A N D C H ANNEL
E ST IM A T IO N TE C H N IQ U E S FOR MIMO O FDM SYS
T E M S
83
5.1 A b rie f overview of an O F D M system 84
5.2 Problem statement (SISO case) 86
5.2.1 T im e domain estim ation 87
5.2.2 Ite ra tive channel and FO estim ation in SISO O F D M 89 5.2.3 Sim ulations results for SISO channels 92
5.3 Problem statem ent (M IM O case) 96
5.3.1 Ite ra tiv e channel and FO estim ation 97 5.3.2 Sim ulations results for M IM O channels 99
5.4 Sum m ary 100
6 C O N C L U SIO N A N D F U T U R E W O RK
102
6.1 Conclusion 102
6.2 Future w ork 104
List of Figures
2.1 Block diagram of a baseband wireless com munication sys tem u tiliz in g channel estim ator and equalizer. 8 2.2 GSM burst structure; channel estim ator utilizes the known
tra in in g bits. 10
2.3 Block diagram o f a noise-corrupted baseband communica
tio n system. 11
2.4 Baseband representation of a channel and an equalizer. 25 2.5 A linear equalizer implemented as a transversal filte r. 26 2.6 The baseband model of a channel and a decision feedback
equalizer (D F E ). 28
2.7 A n M M S E ite ra tiv e equalization scheme. 31
3.1 The block diagram describing iterative channel estim ation
and equalization at the receiver. 47
3.2 The block diagram describing the tra n s m itte r as well as the ite ra tive channel estim ation, equalization and decoding at
the receiver. 51
3.3 Comparison o f the variance of the iterative channel gain estimates for h0 and the CRLBs assuming 26 and 142 p ilo t
symbols. 53
LIST O F FIGURES x v ii
3.4 Comparison o f the variance of the iterative channel gain estimates fo r h\ and the CRLBs assuming 26 and 142 p ilo t
symbols. 54
3.5 Comparison of the variance of the iterative channel gain estimates fo r h2 and the CRLBs assuming 26 and 142 p ilo t
symbols. 54
3.6 Comparison of the variance of the iterative frequency offset estimates for fo and the CRLBs assuming 26 and 142 p ilo t
symbols. 55
3.7 Com parison of the variance of the iterative frequency offset estimates for f \ and the CRLBs assuming 26 and 142 p ilo t
symbols. 56
3.8 Com parison of the variance of the iterative frequency offset estimates for f 2 and the CRLBs assuming 26 and 142 p ilo t
symbols. 56
3.9 The B E R performance o f the proposed ite ra tive equalizer and an equalizer which ignores the effect of frequency offsets. 57 3.10 The uncoded B E R performance of the proposed iterative
equalizer and an equalizer which ignores the effect of fre quency offsets. A h a lf rate convolution coding scheme has
been used. 59
4.1 Ite ra tiv e M IM O receiver. 71
4.2 Comparison of the variance of the iterative channel gain estimate for h u and the CRLBs assuming 26 and 142 p ilo t symbols. hqt is the channel gain between the first receive antenna and the tra n s m it antenna t for path k. The esti m ation performance for hqt is also sim ilar to th a t of h u ,
LIS T O F FIGURES X V 1 1 1
4.3 Comparison o f the variance of the iterative frequency off set estim ate fo r f n and the CRLBs assuming 26 and 142 p ilo t symbols. f q t is the frequency offset between the first
receive antenna and the tran sm it antenna t for path k . The
estim ation perform ance for f q t is also sim ilar to th a t of f n ,
hence not depicted. 76
4.4 The B E R perform ance of the proposed M IM O iterative equalizer and a M IM O equalizer which ignores the effect
of frequency offsets. 77
5.1 Baseband O F D M system model. 86
5.2 O F D M tra in in g structure. 90
5.3 Comparison o f the variance of the iterative channel gain estim ate for h 0 and the CRLBs assuming 64 p ilo t symbols.
The estim ation performance for h i is also sim ilar to th a t of
h 0 , hence n o t depicted. 93
5.4 Comparison of the variance of the iterative frequency offset estim ate for fo and the CR LBs assuming 64 p ilo t symbols. The e stim ation performance for f \ and is also sim ilar to
th a t o f f o , hence n ot depicted. 94
5.5 Tracking frequency offsets for m u ltip a th channel. 94 5.6 The B E R perform ance of the proposed ite ra tive method
and the one w hich does not track the frequency offsets. 95
5.7 2 x 2 M IM O O F D M baseband system. 96
5.8 Tracking frequency offsets for m u ltip a th channel for a M IM O
O F D M system. 101
5.9 The B E R performance of the proposed iterative method and the one which does not track frequency offsets for a
Chapter 1
INTRODUCTION
The demand for high data rate applications such as m ultim edia and inter active services over wireless links is steadily increasing [1]. The research ac tiv itie s in wireless com m unication systems such as 3G [2], 4G, W L A N [3], D A B [4] and D V B [5] are focussed on supporting high ban dw idth and high data rate services. In order to meet these requirements, development of advanced signal processing algorithm s for m itig a tin g channel im pairments becomes an im p o rta n t task.
In this thesis, various tra n s m itte r and receiver signal processing tech niques are proposed in order to improve the system performance, specif ically through the use o f m u ltip le antennas at the tra n s m itte r and the receiver. H igh data rate com munications are lim ite d by several factors, among which intersym bol interference (IS I), which results from the signal spreading characteristic of the channel, plays an im p o rta n t role. More over, the effect o f frequency offset (FO) introduced by local oscillator mismatch or D oppler shifts (DS) w ill become more severe as the data rate is increased.
1.1 Signal fading
In a wireless system, a com m unication channel is often no longer a single line of sight path, p a rtic u la rly in an urban environment. The received sig
Section 1.1. Signal fading 2
nal consists of a large number of reflected, refracted and scattered waves. The signal from the tra n s m itte r to the receiver arrives via more than one path. Due to m u ltip a th propagation, different attenuated versions of the tra n sm itte d signal arrive sequentially at the receiver. The differential tim e delay introduces a relative phase shift between different components. These signals could add constructively or distractively at the receiver so th a t the received signal could vary significantly. In a dynamic environ ment, due to the m o tio n between the tra n sm itte r and the receiver, there is a continuous change in p a th lengths. The am plitude and the phase of the signal could continuously vary w ith tim e whereby there are construc tive additions in some locations and distractive additions in certain other locations. T h is phenomenon is known as signal fading.
LA RG E-SC ALE F A D IN G A N D SM ALL-SCALE FA DING
Large-scale fading represents the average signal power attenuation or path loss due to m otion over large areas [6]. This phenomenon is affected by prom inent te rra in such as hills, forests and buildings between the trans m itte r and receiver. Small-scale fading is used to describe the rapid fluc tu a tio n of the am p litu d e o f a radio signal over a short period of tim e or travel distance, so th a t large-scale path loss effects may be ignored. Small- scale fading is caused by interference between two or more versions of the tra n sm itted signal w hich arrive at the receiver at slig h tly different times. For mobile radio applications, the channel is tim e-variant because m otion between the tra n s m itte r and receiver results in propagation path changes. M u ltip a th in the radio channel creates small-scale fading effects. The three most im p o rta n t effects are [6]:• Rapid changes in signal strength over a small travel distance or tim e interval.
Section 1.1. Signal fading 3
• Random frequency m odulation due to varying Doppler shifts on d if ferent m u ltip a th signals.
• Tim e dispersion (echoes) caused by m u ltip a th propagation delays. This thesis is aimed at pro vid in g various signal processing techniques to m itigate the effect o f sm all scale fading, in p a rticular for high data rate transmission over wireless links. The work covers b oth single in p u t and single o u tp u t (SISO) and m u ltip le in p u t and m u ltip le ou tp u t (M IM O ) systems as well as tim e division m ultip le access (T D M A ) and orthogonal frequency division m u ltip le (O F D M ) access schemes.
1.1.1 Doppler shift
Due to relative m o tio n between the mobile and the basestation, each m ul tip a th wave experiences an apparent sh ift in frequency. The shift in the received signal frequency due to this m otion is named “ Doppler shift” (DS). In order to im prove the lin k layer performance of the receiver, the effects of the tim e sele ctivity of the channel due to DS must be cancelled. In most of the available lite ra tu re [7-10], all m u ltipa th s have been assumed to have identical DSs. In th is case, the DS can be compensated for prior to equalization. The phase change in the received signal is defined is [6]
. , 2 n v A t . . .
A(f> = — -— cosO (1.1.1)
A
where v is the velocity of the tra n s m itte r relative to the receiver in me ters/second: positive when m oving towards one another, negative when moving away. A t is the tim e required for the signal to travel from the tra n sm itte r to the receiver. A is the wavelength, and 9 is the angle of arrival in radians. Therefore, the DS is given is [6]
Section 1.1. Signal fading 4
The equation in (1.1.2) shows th a t if the relative speed between the trans m itte r and receiver is constant, then the DS is a function of the angle of arrival. Therefore, each m u ltip a th which arrives w ith different angle of arrival could have different frequency offsets, and this effect should be accounted for in the receiver design. This is one of the main m otivation of this thesis.
1.1.2 Channel Classification
A channel is said to be non -d istortin g if, w ith in the bandwidth, B W, occupied by the tra n s m itte d signal, the am plitude response, A(cj), of the channel is constant, tim e -in va ria n t and the phase response, 6(uj), is a linear function of cu. However, if the am plitude of the spectrum is tim e varying, the fading in the signal is called frequency non-selective or fla t fading. In the tim e domain, it can be said th a t the delay spread denoted as Td, is less than the sym bol period, Ts [11,12]. F la t fading does not produce ISI. Therefore, for fla t fading, the channel satisfies
Td < Ts (1.1.3)
On the other hand, m u ltip a th propagation can spread the transm itted signal over an in terval of tim e which is longer than the symbol period, which can cause ISI, th a t lim its the data rate of a com munication system and increases the associated b it error rate. In the frequency domain, it means th a t the frequency response is not fla t for the entire bandw idth of the signal. Hence, each frequency component o f the signal is amplified and phase shifted differently. The fading in the received signal in this case is called frequency selective fading. Therefore, for frequency selective fading,
Section 1.2. Outline o f the Thesis 5
Frequency selective fading occurs in the so-called ‘small scale’ m u ltip a th fading which dictates the instantaneous behaviour of the underlying chan nel conditions; whereas “ Large scale” refers to long term effects of the channel over a longer period of tim e. The design of a relatively low com ple xity receiver th a t can provide significant improvement in BE R perfor mance over a conventional receiver in a frequency selective environment is the focus of th is thesis.
1.2 Outline of the Thesis
C h a p te r 2: A detailed lite ra tu re survey is provided together w ith nec essary theoretical background. Various FO and channel estim ation tech niques are reviewed followed by the in tro d u ctio n of m inim um variance unbiased estim ation techniques. A detailed discussion on the existing equalization techniques is also provided.
C h a p te r 3: Param eter estim ation and iterative equalization techniques for a SISO system are proposed. The FOs associated w ith each path of the channel have been assumed to be d istinct. Unlike an identical DS problem as in [7,13,14], the d is tin c t DSs can not be compensated p rio r to equaliza tion. Hence a sophisticated equalizer is required. In order to design such an equalizer, the complex m u ltip a th gains (MGs) and FOs are required. A m axim um likelihoo d (M L ) estim ation approach is used to obtain the complex MGs and FOs. In a T D M A based com munication system, such as GSM, the p ilo t symbols are generally inadequate to obtain an accurate estimate of the FOs due to lim ita tio n on the frequency resolution of the es tim ato r. Therefore, in the proposed technique, an in itia l channel estimate is obtained using a very short p ilo t (train ing ) signal and the soft estimate of the tra n sm itte d signal is then treated as a p ilo t signal to determine
Section 1.2. O utline o f th e Thesis 6
m ultiple FOs and to refine channel estimates iteratively. The proposed iterative technique has the a b ility to resolve m ultipaths, thereby m odify ing the m u ltip le FO problem in to one in which distin ct frequency offsets are estimated. The performance of the estimator is assessed through com parison of the variance of estimates w ith the Cramer-Rao lower bound (C R LB).
C h a p te r 4: The w o rk presented in chapter 3 is extended to a M IM O frequency selective channel. M IM O systems are very a ttra ctive in order to boost the capacity of a wireless com munication system th a t operates in a rich m u ltip a th environm ent. In this chapter, com munication over a M IM O system, allow ing for a frequency selective channel between each tran sm it and receive antenna is considered w ith each pa th having distinct DSs. The tra in in g signals tra n sm itte d from the antennas are assumed to be spatia lly and te m p o ra lly uncorrelated. A n iterative channel parameter estim ator is provided whereby the M IM O frequency selective channel is decoupled in to m u ltip le SISO fla t fading sub-channels through appropri ately cancelling b o th inter-sym bol-interference (IS I) and the inter-user- interference (IU I) from the received signal. The refined channel estimates and the corresponding FO estimates are then obtained for each resolved M IM O m u ltip a th tap. The performance o f the algorithm is compared w ith the matched filte r bound. In addition to providing superior BER performance, the proposed estim ator is also efficient in th a t it tends to a tta in the C R L B derived assuming all tran sm itted symbols in the burst are known p ilo t symbols.
C h a p te r 5: Param eter estim ation and equalization techniques for a M IM O - O F D M system w ith tim e -va ryin g channels are proposed. To combat m ul tip a th delay spread in high data rate wireless systems, wireless commu
Section 1.2. Outline o f th e Thesis 7
nication standards such as IE E E 802.11 adopted an O F D M transmission scheme which transform s a m u ltip a th channel into parallel independent flat-fading subchannels. O F D M systems are very sensitive to FOs, and an ite ra tive technique is proposal to estimate and track FOs. In itia l esti mates of the channel gains and FOs are obtained using the long trainin g signal available in O F D M . These estimates are used to obtain the soft estimates of the da ta symbols in the subsequent packets in an iterative manner to tra ck carrier FOs. The m erit of this iterative method is its a b ility to tra ck FOs. The proposed iterative technique has the a b ility to resolve m u ltip a th s, b rin g in g the m u ltip le FO problem in to the estimation of d istin ct FOs. S im u la tio n results show a superior B E R performance for the proposed a lg o rith m over a scheme th a t does not consider FO correc tion.
C h a p te r 6: Conclusions are drawn and possible fu tu re research direc tions are outlined.
Chapter 2
PARAMETER ESTIMATION
AND EQUALIZATION
The demand for wireless com m unication systems th a t can p o tentia lly sup p o rt high q u a lity and high data rate m ultim edia and interactive services is continuously increasing. The channels in mobile radio systems are usually frequency selective and m u ltip a th s may give rise to intersym bol interfer ence (IS I), which lim its high data rate transmission [15]. In order to remove the effect o f ISI, an equalizer is generally required, which in tu rn requires a channel estim ator.
Noise Channel Estimator Detector or Equalizer interleaver Signal Source Channel Decoder Receiver Filter Modulator Multipath Channel Channel Encoder Deinterleaver
Figure 2.1.
Block diagram o f a baseband wireless com munication system u tiliz in g channel estim ator and equalizer.Section 2.1. Channel param eter estimation techniques 9
Fig. 2.1 depicts a block diagram of a baseband wireless com munication system, which employs a channel estimator and equalizer at the receiver. The signal is usually firs t protected by channel coding and interleaving against fading phenomena, after th a t the binary signal is modulated and tra n sm itted over the m u ltip a th fading channel. A t the receiver, the task of the equalizer is to remove or m itigate the effect of IS I introduced by the m u ltip a th channels. A significant amount of research has been performed in the area of equalization over the last few decades and several well known techniques are available [6,15]. However, equalizers such as the maximum likelihood sequence estim ator (M LSE) or m axim um a posteriori proba b ility (M A P ) detector need to know the channel impulse response (CIR) to ensure successful equalization (removal of IS I). Usually, the channel estim ation is based on a know n sequence of bits, which is unique for a cer ta in tra n s m itte r and know n to the receiver. The channel estim ator could estimate C IR for each b u rst separately by exploiting the known tra n sm it ted bits and the corresponding received samples. Note th a t equalization w ith o u t separate channel estim ation (e.g., adaptive linear and decision- feedback equalizer) is also possible, however th e ir performance m ight not be sufficient for a ra p id ly varying wireless channel. A fte r equalization, the signal is deinterleaved and decoded to extract the original message. In this chapter, a b rie f background on channel parameter estim ation and equalization is presented.
2.1 Channel parameter estimation techniques
A n im p o rta n t p a rt o f a receiver design is to develop an accurate channel estim ation technique. Even w ith lim ite d knowledge o f the wireless channel properties, a receiver can gain insight in to the in form ation th a t was sent by the tra n s m itte r using a known p ilo t signal. The channel need to be
Section 2.1. Channel parameter estimation techniques 1 0
estimated before equalization and this is the focus of this section.
3
26
3
Tail
58
data bits
Training
58
data bits
Tail
bits
sequence
bits
F ig u re 2.2. GSM b urst structure; channel estimator utilizes the known tra in in g bits.
In a burst d ig ita l com m unication system, for example GSM [16], the 26 tra in in g bits in the m iddle o f the burst are dedicated for channel estimation as shown in Fig. 2.2. The receiver can u tilize the known tra in in g bits and the corresponding received samples to estimate C IR for each burst separately. The best know n channel estim ation methods are M axim um likelihood estim ation (M L E ) and Least-squares (LS) estim ation [17].
2.1.1 M axim um likelihood estimation
Consider a baseband com m unication system, which is assumed to be cor rupted by zero mean a d d itive w h ite Gaussian noise (AW G N ) as depicted in Fig. 2.3.
The d ig ita l signal s ( n ) from a fin ite constellation is tran sm itted over a fading m u ltip a th channel o f length L, where ‘n ’ denotes the discrete tone index. Therm al noise is generated at the receiver and it is modelled by the zero mean Gaussian d is trib u tio n . The received signal is passed through a filte r which is matched to the frequency band of the transm itter. There fore, if the sampling rate at the receiver is equal to the symbol transmission rate, then the received signal can be w ritte n in the convolution form as
Section 2.1. Channel param eter estimation techniques 1 1 Noise w(n) s(n) Receiver filter Channel estimator Signal source Detector or Equalizer Multipath channel
Figure 2.3.
Block diagram o f a noise-corrupted baseband communication system.L—1
u (n ) = '^^/ h( l ) s ( n — I) + uj(n) (2-1.1) 1—0
where h(Z) is the unknow n complex channel gain for the Zth tap, L is the length o f the channel and uj(n) is additive circu la rly sym m etric zero mean w hite (complex) Gaussian noise w ith variance of,.
The m ethod of m axim um likelihoo d determines the parameters th a t max imize the likelihood o f the available set of observed data. In principle, the M L E technique m ay be applied to most of the data models [17], however, the im plem entation could be com putation ally expensive. The M received samples in (2.1.1) can be w ritte n in a vector form as
Section 2.1. Channel param eter estimation techniques 1 2 where u = u(0) u ( l) • • • u ( M — 1) S = 5(0) 5(1) 5 ( - l ) 5(0) 8(1 - L ) s(2 - L ) s ( M - l ) s ( M — 2) ••• s ( M - L ) and h = h( 0) h{ 1) ••• h ( L - l ) U) = [ cj(0) cu(l) • • • u ( M — 1)
The likelihood fu n ctio n o f the received samples can be w ritte n as
M - l
p ( u ;h ) = p [ u ( i) ; h ] . (2.1.3)
i = 0
In many cases, it is more convenient to work w ith the n a tu ra l logarithm of the likelihood fu n ctio n , rath er than w ith the likelihood function itself. Thus M - l ln [p (u ;h )] = ^ l n p [ u ( i ) ; h ] i= 0 = In 1 - M ( u —S h ) ( u —S h ) (u - S h ) * ( u - Sh) In (7r a l ) 2 - a < Lu
The lo garithm ic fu n ctio n ln [p(u; h) ] is a m onotonically increasing func tio n of p ( u ; h ) between 0 and 1. Therefore, the parameter vector for
Section 2.1. Channel param eter estimation techniques 13
which the likelihoo d fu n ctio n
p(u;
h ) is a maximum is exactly the same as the parameter vector for which the log-likelihood function ln[p(u; h) ] is a maximum. Taking the derivative of the log-likelihood function produces2 i= |fc i! . i(s 'u -s -s h )
UJ= 4 r (S " S ){(S " S )“1S " u - h }
which upon being set to zero yields the M L Eh
= (SHS )-1SHu.
(2.1.4)To find how effective the m axim um likelihood m ethod is, a lower bound, namely the Cram er Rao Lower Bound (C R L B ), on the variance of any unbiased estimate is derived in section 2.3. I f the variance of an unbiased estim ator is equal to the C R L B [17], then, the estim ator is said to be efficient.
A n M L E has the follow ing three salient properties [17],
• M axim u m -like lih o o d estim ators are consistent, i.e., increasing the sample size o f a m axim um likelihood estim ator decreases the vari ance of the estimate. I t attains the C R LB asym ptotically.
• M axim um -likeliho od estim ators are asym ptotically unbiased; th a t is lim E { 0 } = 0.
N —h x>
• The d is trib u tio n o f the m axim um -likelihood estimators are asymp to tic a lly Gaussian.
The drawback of the M L E can be its com plexity and its d iffic u lt to apply for the signal models where the noise is not Gaussian.
Section 2.1. Channel param eter estimation techniques 14
2.1.2 Least squares estimation
Least squares e stim ation (LSE) is widely used in practice due to its ease of im plem entation and o p tim a lity in Gaussian noise. A salient feature of the m ethod is th a t no p ro b a b ilistic assumptions are made about the data, only a signal model is assumed. The main drawback of the least squares estim ator is th a t it does n ot generally guarantee the o p tim a lity of the estimator. Furtherm ore, the sta tistica l performance cannot be assessed w ith o u t some specific assumptions about the probabilistic structure of the data. A n LSE o f an unknown parameter vector, 0, minimizes the sum of the squared error between the real observed data, u(n) and the estimate s(n) as [17]
M - l
J ( ° ) = M n) - s(n)i2 71=0
where the observation in te rva l is assumed to be n = 0, 1, • • • , M — 1, and the dependence o f J on 9 is via s(n). The value of 6 th a t minimizes J(9) is the LSE. The perform ance o f the LSE w ill undoubtedly depend upon the properties of the c o rru p tin g noise as well as any m odelling errors. LSE is usually applied in situatio ns where a precise statistical characterization of the data is unknown or where an o p tim a l estim ator cannot be found or may be too com plicated to app ly in practice [17].
2.1.3 Joint estimation of channel impulse response (C IR ) and fre
quency offset (FO )
In this section, an approxim ate m axim um likelihood (A M L ) estim ator of the complex channel gains (CGs) and a single FO is considered. I t is assumed th a t the signal is propagated through L different paths and each path has the same D oppler sh ift (DS) f d. The received complex baseband
Section 2.1. Channel param eter estimation techniques 15
signal is given by
L —l
U
(
n) =
^ 2 h ( l ) s ( n—
l)e^2n^dn+
w( n) (2.1.5) 1=0and M received samples in a vector form can be expressed as
where Aw =
u — ArfSh +
uj 1 0 0 0 0 ej2* fd 0 0 0 0 ' - . 0 0 0 0 (2.1.6) (2.1.7)The tra in in g signal m a trix ,
S,
and the m u ltip a th gains (MGs) vector,h,
have been defined in the previous section. Thus, the log-likelihood function of the received signal can be expressed as (ignoring the constant terms)lnp(u; h;
f d) =- ( u - AdSh)"(u - A rfSh)
(2.1.8)where (.)H denotes conjugate transpose. M a xim iza tion of the log-likelihood function (2.1.8) is equivalent to m in im izin g the follow ing cost function [18]
J (
h;
f d)= (u - A«JSh)"(u - A<;Sh)
(2.1.9)which is a nonlinear least-squares problem. M in im iza tio n of (2.1.9) w ith respect to
h*
yields9 J (h ; h )
d h * = - S ffA ? u + S "S h ,
H,
Section 2.1. Channel param eter estimation techniques 16
Equating it to zero, then
h = u. (2.1.11)
The m a trix A d is unknow n and depends on fd- Therefore h in (2.1.11) can not be estimated in the current form . The FO is estimated by m inim izing the cost function, obtained by su bstituting (2.1.11) into (2.1.9)
J ( f d ) = u " u - u HA ,i S(SH S ) - 1SHA f u (2.1.12)
Note th a t the tra in in g samples s(n) are assumed uncorrelated. Therefore S ^S ~ k l , where k is constant over the frame considered. The m inim iza tio n of (2.1.12) is therefore equivalent to m axim ization of the second term in (2.1.12) th a t can be w ritte n as [18]
<t>(fd) = u ^ A r f S S ^ A f u
1 M - l L -1
= M
E E
- 0 | 2 . (2.1.13)71=0 1 =0
The m axim um likelihoo d solution of fd is therefore,
f d = arg max <j>(f) (2.1.14)
The FO fd can be estim ated using a grid search method. The grid search method could determ ine the frequency bin th a t maximizes (2.1.13), but it requires a very high co m putation al complexity. However, in practice [15] a less com putation ally expensive approach called the Fast Fourier Transform (F F T ) can be used. Once the FOs are estimated, the CGs, h, can be estimated using (2.1.11).
Section 2.2. Frequency estim ation techniques 17
2.2 Frequency estimation techniques
A frequency offset creates an extra burden for the equalizer in order to track channel variations. In such cases the equalizer can be simplified by estim ating the FO and rem oving it as much as possible before equaliza tio n starts. Frequency offset estim ation can be divided into two broad categories, data-aided and non-data-aided or blin d techniques.
For known tra in in g signals and channel inform ation, an FO estimator based upon the m axim um -likeliho od (M L ) criterion was presented in [19]. Likewise, [20] and [21] proposed methods where only the tra in in g signals are required to be known. In [20], the least squares (LS) criterion has been employed, while in [21], the M L crite rio n has been adopted to jo in tly es tim ate the channel and the FO. How to choose the best tra in in g sequence for the above estim ation problem has been addressed in [8,22]. When the tra in in g signals are periodic, Moose [23] provided an M L estimator for the FO based on tw o id entical O F D M symbols. The m axim um offset th a t can be handled is one subcarrier spacing. However a scheme using one tra in in g O F D M sym bol w ith two identical components, where the estim ation range is tw o subcarrier spacing has been proposed in [24]. A narrow estim ation range means th a t oscillators w ith high precision should be used, thus im proving the cost of the system. In [25], an enhanced FO estim ation scheme was proposed to extend the estim ation range to M sub carriers spacing, using one O F D M symbol w ith M identical components. The methods in [26] and [27] e xploit the redundancy associated w ith a cyclic prefix.
Various techniques are available for the estim ation of frequency compo nents. Am ong them , the follow ing two technique are very powerful.
Section 2.2. Frequency estimation techniques 18
2.2.1 Classical techniques based on Fourier transform
Spectral analysis based on the frequency domain transform ations of the received signal is a basic technique for the estim ation of FO. The received signal samples are converted in to the frequency domain by using the dis crete Fourier transform (D F T ). The FOs in the received signal samples can be estimated by determ ining the frequency at which the periodogram, or “spectrum ” , attains its m axim um [28]. The am plitude of the received samples at different frequencies can be determined by using the discrete Fourier transform [17,29-31],
1 M - l
= M E (2.2.1)
71=0
where F ( f ) is the complex am plitude of the received signal at frequency / and u( n) is the received signal sample. One drawback of this method is th a t it is d iffic u lt to determine the frequency offsets when m ultiple frequency offsets are present in the m u ltip a th channel.
2.2.2 Subspace methods
Subspace methods can provide high resolution frequency estim ation but could result in increased com putational complexity. These techniques are based on the p rinciple o f separating the noisy data in to a signal subspace and a noise subspace. The FO is determined using the fact th a t the sig nal vectors w ill be orthogonal to the noise subspace. V. F. Pisarenko, first introduced th is concept in 1973 and his technique is known as the Pisarenko harm onic decomposition [32]. This concept was later used to develop more advanced techniques such as M USIC (m ultip le signal clas sification) and E S P R IT (estim ation of signal parameters via ro ta tio n a l invariance) [33-35]. These methods require an eigenvalue decomposition
Section 2.2. Frequency estimation techniques 19
of the received sample covariance m a trix R to determine the signal and the noise subspaces
R = Q A Q H (2.2.2)
where A is a diagonal m a trix of eigenvalues o f the covariance m a trix, and
Q = [Q s C M (2.2.3)
where Q s and Qn contain the basis vectors for the signal and the noise subspaces respectively. Thus the M U SIC based spectral estim ator is formed as the inverse of the sum of inner products between the signal vector and the vectors in the noise subspace. The frequencies o f the signal components are taken to be the peaks of the M U S IC spectral estimate [34]
1 1 s " ( / ) Q ; v Q £ s ( / ) ^ 2 ’ 2
Sm U S Ic U ) = n \ H f e \ / e( ~ o ’ ol (2.2.4)
T 1T
where s( / ) = 1 e- i 2irf ... e - j 2 i r f N js ^he frequency scanning vec tor, and N is the order of the covariance m a trix . A drawback in using sub-space based algorithm s is th a t as the num ber o f frequency components in the received signal increases, the order of the associated covariance ma tr ix needs to be increased. Therefore, perform ing eigenvalue decomposi tio n of a higher order covariance m a trix , requires very high com putational complexity.
2.2.3 Blind parameter estimation
Most existing FO estim ation techniques rely on periodic transmission of p ilo t symbols, which could in e v ita b ly reduce bandw idth efficiency. A blind
Section 2.3. Cramer Rao lower bound 2 0
FO estim ation technique is a ttra ctive because it saves bandw idth, i.e., no tra in in g signals are required. However, b lin d algorithm s generally require a very large set o f data to estimate the parameters, hence they are suit able only when the channel changes slowly. Therefore, they are not good for burst communication, where only a sm all num ber o f b its is available. Various b lin d FO estim ation techniques can be found in [36-38] and the references therein. Since these techniques are n ot suitable for a highly dy namic wireless environment, this approach w ill therefore n o t be considered in this thesis.
2.3 Cramer Rao lower bound
Being able to place a lower bound on the variance o f any unbiased estima to r is extrem ely useful in practice. There are various m ethods available to determine the lower bound on the variance of the estim ators, e.g. [17,39], but the Cramer-Rao Lower Bound (C R L B ) [17] is stra ig h tfo rw a rd to de termine. A n estim ator th a t is unbiased and whose variance is always m inim um when compared to other estim ators b u t is n o t less th a n CR LB is called the M in im u m Variance Unbiased estim ator (M V U E ).
2.3.1 The estimation problem
Assume real sample u(n) contains the param eter o f interest 9 corrupted by noise cu(n) ,
u( n) = 9 + tu(n) (2.3.1)
where uj(n) is an additive w h ite Gaussian noise (A W G N ) process w ith PD F N {0, a 2). The observation o f 9 made at M intervals is given by the data set [u(0), u ( l) , • • • , u ( M — 1)]. The jo in t p ro b a b ility d istrib u tio n of data is given by p (u (0 ), u{ 1), • ■ • , u ( M — 1); 9) or sim ply in vector
Section 2.3. Cramer Rao lower bound 2 1
form as
1 1 M~l
P(U; 6) = ( 2 n < r ) W eXp(_ 2^ E , (“ ( " ) “ ( 2'3'2)
' n=0
The PD F is a function of both the data u and the unknown parameter 0. W hen the P D F is viewed as a function o f the unknow n parameter 0 it is called the likelihood function. In tu itiv e ly , the P D F describes how accurately the parameter 0 can be estimated.
I f the p ro b a b ility d is trib u tio n of the data is known, then the problem of finding an estim ator 0 is sim ply finding a fu n ctio n o f da ta w hich maximizes the likelihood function. The v a ria b ility of the estimates determines the efficiency of the estimator. The higher the variance o f the estimates the less effective (or reliable) the estimates are. Hence various estimators can be found for the data b ut the one w ith the lowest variance is the best estimator.
2.3.2 Minimum variance unbiased estimation
Efficient estim ation o f channel parameters is very im p o rta n t to decode the tran sm itted data accurately. Therefore, it is very im p o rta n t to determine whether the estim ator being designed is unbiased or biased and if it is unbiased w hat is its variance about the tru e value. A n estim ator is said to be unbiased iff
E { 6 } = 0, a < 9 < b
where a and b represent the s ta rt and end range of possible values of 0. On the other hand an estim ator is said to be biased if
Section 2.3. Cramer Rao lower bound 2 2
which is the bias of the estimate. The mean squared error (MSE) can be denoted as
M S E {0 } = E { { 9 - 9 ) 2}
= E { { { e - E { e ) ) + { E { 6 ) - e ) ] 2} = Va r( 9) + [E(9) - 9]2
= Va r{ 9) + b2(9)
I t can be seen th a t the MSE takes in to account the variance of the esti m ator as well as its bias. I f the estim ator is constrained to be unbiased, and determined as the one w ith m inim u m variance, such an estim ator is called m inim um variance unbiased estim ator (M V U E ).
There are several methods for determ ining the M V U E . The most common ones are based on the Cramer Rao Lower Bound (C R L B ).
2.3.3 Cramer Rao lower bound
The theory o f the C R LB allows the dete rm in a tio n o f the M V U E , if it exists. Therefore, if the PD F p (u; 9) satisfies the re g u la rity condition [17]
£ j a i n g u ^ ) j = ( ) f Q r a l i e (2 3 3)
where the expectation is taken w ith respect to p {u; 9) and if 9 is the estim ator o f 9, then the bound on the variance o f an unbiased estimator is given as
^ (2. 3. 4)
Section 2.3. Cramer Rao lower bound 23
d ln p ( u ; 0 ) _ 1 ,
ae
~ W r
’
{
’
where c(9) is a scalar constant whose value may depend on 6. Therefore, this equation implies th a t if the log-likelihood function can be w ritte n in this form then 9 w ill be the M V U E . By differentiating (2.3.5) again, the value of c(9) can be found as
a
d l n p ( u ; 6 ) _a (
1
ae
ae
ae
\ c ( e )d2 ln p (u ; 6 ) l _ _ o\
ae2
c($)ae ^
'
c(6) = 1
Therefore, if (2.3.5) can be w ritte n in general form
9lnpg{p e) = 1(6) (g(u) - 6) , (2.3.6)
then the efficient M V U E and its variance bounded by the C R L B are given by (2.3.7) and (2.3.8) respectively.
9 = g( u) (2.3.7)
(2.3.8)
where 1(6) is termed the Fisher inform ation. A n example is given below [17]. Consider a DC level estim ation problem for samples observed in zero mean w hite Gaussian noise.
V A R ( 6 ) = 1(6)
Section 2.4. Equalization techniques 24
where uj(n) is W G N w ith variance a2. The likelihood fu n ctio n can be w ritte n as
1 1 M ~l
P ( U; S ) = (2x<t)m/2 eXpl~ 2 ^ ~ g) 2l ( 2'3 ' 10)
Taking the first derivative
<91np(u;0)
d j
r/
2\ —
de
do
M - l G 71=0 M = - z ( u - 0 ) (2.3.11) cr^This can be compared to (2.3.5), where u is the sample mean. D ifferenti ating (2.3.11) again,
3 2 ln P( u ; * ) _ M
de2
r 2In this case, the second derivative is a constant, and the C R LB is given as
2
Var{ 6) > ~ (2.3.13)
The C R LB derived in this section can easily be extended for vector para meters and it w ill be used throughout the thesis to assess the performance of the estimators [17].
2.4 Equalization techniques
Equalization techniques have been developed from the 1960s, [40-42]. A n equalizer is required at the receiver in order to m itig a te the effect of ISI. Equalization techniques fall into two broad categories: linear and non linear. The linear equalization techniques are generally the simplest to
Section 2.4. Equalization techniques 25
implement. However, linear equalization techniques ty p ic a lly suffer from more noise enhancement than the nonlinear equalizers, and therefore they are not used in most wireless applications. Among nonlinear equalization techniques, decision-feedback equalization (D FE ) is the most common, since it is fa irly simple to implement and generally performs well.
Noise w(n)
u(n)
s(n)
s(n)
H(z)
Equalizer
F ig u re 2.4. Baseband representation of a channel and an equalizer.
For the discussion, the system model in Fig. 2.4 w ill be used. The trans m itte d signal s(n) is passed through an ISI channel modelled as a F in ite Impulse Response (F IR ) filte r w ith z-domain transfer function H ( z ) . As in (2.1.1), the distorted channel o u tput u(n) is then processed by an equalizer th a t provides an estimate of the transm itted symbol s(n).
2.4.1 Linear transversal M M SE equalizer
A linear equalizer can be implemented using an F IR filte r. The current and the past values o f the received signal are linearly weighted by the filte r coefficients and summed to produce the output, as shown in Fig. 2.5. The equalizer in p u t u(n) and the output s(n) before decision m aking can be expressed in the convolution form as [6]
p-1
s(n) = w*{p)u{n — p) = w Hu ( n ) (2-4.1)
Section 2.4. Equalization techniques 26 u(n) w*(0) A A -1 w*(l ) w*(P-3) /O s w*(P-2) ► 0
—
> 0—
s(n).F ig u r e 2.5. A linear equalizer implemented as a transversal filte r.
where P is the length of the equalizer, w = [w{0), w( 1), ••• , w ( P — 1)]T represents the equalizer tap weight vector, and the received signal vector is denoted as u = ['u(O), u( n — 1), • • • , u(n — P — 1)]T . I f the equalizer approximates the inverse of the channel, this is referred to as a zero- forcing equalizer where the coefficients are chosen based on the zero-ISI criterion [43]. Norm ally, this leads to a very long equalizer and large noise am plification when the signal is weak. Instead of using the zero-ISI criterion, the L E coefficients can be chosen such th a t s(n) becomes as closer to s(n) as possible in the mean square error sense. T his is a well- known and w idely used approach and known as a M inim u m Mean Square E rror (M M SE ) filte r. Since the complex M M SE equalizer design w ill be o f special interest in Chapter 3, 4 and 5, thus only a b rie f in tro d u c tio n to M M SE equalizer design is provided here. Suppose the equalizer is designed to retrieve the tran sm itted signal, s(n), w ith a delay d, the mean square error can be w ritte n as
Section 2.4. Equalization techniques 27
J = E { | s(n) — s(n — d) |2} = E { \ w Hu( n) — s(n — d) |2}
- w ^ { u ( ? i ) u H ( n ) } w - w H E { u { n ) s * ( n - d ) }
—E { u H (n)s(n — d )} w + E { \ s(n) |2} (2.4.2)
where the sta tistica l expectation E is taken over the statistics of the noise and the data sequence. Define a channel convolution m a trix H of dim en sions P x ( L + P — 1) as H = h0 hi . . . h L - \ 0 . . . 0 0 ho • • • h i- 2 h ^ i • • : : 0 ... 0 0 . . . 0 ho h\ . . . hi,—i (2.4.3)
D ifferentiating the cost function in (2.4.2) w ith respect to w *, the M M S E equalizer is obtained as [17,18]
2 \ —1
% ) H c d (2.4.4)
where I is an id e n tity m a trix and Cd is a coordinate vector such th a t Hcd chooses the d th column of H .
w = ( H H ^ +
2.4.2 Decision feedback equalization
The D F E has been of considerable attention due to its im proved perfor mance over a linear equalizer and reduced im plem entation com ple xity as compared to a nonlinear m axim um -likelihood receiver. The basic idea be hind decision feedback equalization is th a t once an in fo rm a tio n symbol has been detected and decided upon, the ISI th a t it induces on the futu re
Section 2.4. Equalization techniques 28
symbols can be estimated and subtracted out before detection of subse quent symbols [15]. This is done by a feedforward filte r (F F F ) A ( z ) and a supplementary feedback filte r (FB F) B(z) . The F B F is driven by the decision on the o u tp u t of the detector, and its coefficients can be adjusted to cancel the IS I on the current symbol from past detected symbols. A block diagram of the D F E structure is shown in Fig. 2.6.
w(n)
B(z)
H (z)
Decision device
F ig u r e 2.6. The baseband model of a channel and a decision feedback equalizer (D F E ).
Assuming th a t the equalizer has N f taps in the feed forw ard filte r and 7V& taps in the feedback filte r, the filters A(z) and B(z) are w ritte n as
N f - 1 A( z) = £ ^ i — 0 N b B {z ) = Y , b*kz (2.4.5) (2.4.6) k= 1
Combining the o u tp u t of the feedforward and feedback filters, the equalizer o u tp u t can be expressed as
Section 2.4. Equalization techniques 29 N f - 1 Nb
y
( n ) =
f
* u (n ~ * ) _
Y 2 b^ n ~ d ~ k)
i=0 k—1 = f ^ u — b Hs = f * ( H s + w ) - b ^ M s (2.4.7)where s(n — d) is the hard decision of the previously estimated symbols at the o u tp u t of a nonlinear decision device, and f = [ / o / i ■ • • f { N f - i ) ] H
is the forw ard filte r tap weight vector and b = [61 62 • • ■ b ^ ] 11 is the feedback tap weight vector. The vector s is related to s through the m a trix M as
SjV6xl = [OjVbXd IN bx N b ON bx ( N f + N h- l - N b- d ) ] S ( N f + N h- l ) x l = M s (2.4.8)
where Nh is the channel length. The mean squared error function can be w ritte n as
J ( f , b ) = E {| y(n) - s(n - d) |2}
= ( f " H - bHM ) (H Hf - M"b)cr2 + f" f< r2 - ( f f l H - b H M ) a 2scd
—cd ( H H f - M"b)<72 + f7g (2.4.9)
where cd is a coordinate vector. The expressions for the feedback and feedforward tap weights can be obtained as
b = M H " f (2.4.10)
2
f = ( H ( I - M " M ) H " + ^ ) - ' H c j (2.4.11)
The D F E is known to outperform the tra d itio n a l linear equalizer, p a rtic u la rly if the channel has deep spectral nulls in its frequency response [15]. However, performance degradation in the D F E occurs when in correctly
