Jacob Benesty Yiteng (Arden) Huang Adaptive Signal Processing

39 Pages·2002·0.3 MB·English

Jacob Benesty
Yiteng (Arden) Huang
Adaptive Signal Processing:
Application to Real-World Problems
SPIN Springer’s internal project number, if known
Springer
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Contents
1 Algorithms for Adaptive Equalization in Wireless
Applications .................................................. 1
Markus Rupp, Andreas Burg
1.1Introduction................................................. 1
1.2Criteria for Equalization ...................................... 3
1.3Channel Equalization......................................... 5
1.3.1Infinite Filter Length Solutions for Single Channels .......... 6
1.3.2FiniteandInfiniteFilterLengthSolutionsforMultipleChannels 7
1.3.3Finite Filter Length Solutions for Single Channels ........... 10
1.3.4Decision Feedback Equalizers ............................. 13
1.4Adaptive Algorithms for Channel Equalization................... 17
1.4.1Adaptively Minimizing ZF................................ 17
1.4.2Adaptively Minimizing MMSE ............................ 19
1.4.3Training and Tracking ................................... 19
1.5Channel Estimation .......................................... 21
1.5.1Channel Estimation in MIMO Systems..................... 22
1.5.2Estimation of Wireless Channels .......................... 23
1.5.3Channel Estimation by Basis Functions .................... 23
1.5.4Channel Estimation by Predictive Methods ................. 24
1.6Maximum Likelihood Equalization ............................. 25
1.6.1Viterbi Algorithm ....................................... 26
1.7Blind Algorithms ............................................ 27
1.8Conclusions ................................................. 29
References ..................................................... 29
Index......................................................... 35

1 Algorithms for Adaptive Equalization in
Wireless Applications
Markus Rupp1 and Andreas Burg2
1 TU Wien
Institutefor Communication and RFEngineering
Gusshausstr. 25/389
A-1040 Vienna, Austria
E-mail: [email protected]
2 ETH Zurich
Integrated Systems Laboratory
Gloriastr. 35
CH-8092 Zurich,Switzerland
E-mail: [email protected]
Abstract. Since theintroduction of adaptiveequalizers in digital communication
systems by Lucky [1], much progress has been made. Due to their particular con-
straints many new and different concepts in the wireless domain have been pro-
posed. The wireless channel is typically time and frequency dispersive, making it
difficult tousestandard equalizertechniques.Also, duetoitstimevaryingnature,
longtransmissionburstsmaygetcorruptedandrequireacontinuoustrackingoper-
ation.Thus,transmissionisoftenperformedinshortbursts,allowingonlyalimited
amount of training data. Furthermore, quite recently, advantages of the multiple-
input multiple-output character of wireless channels have been recognized. This
chapterpresents an overview of equalization techniquesin use and emphasizes the
particularities of wireless applications.
1.1 Introduction
Consider the following simplified problem:a linear channel,characterizedby
a time-discrete FIR filter function with L coefficients
C
L(cid:1)C−1
C(q−1)= c q−l, (1.1)
l
l=0
whereq−1denotesthedelayoperator1andc ∈Cl thecoefficients.Itisfedby
l
transmitted symbols s(k) ∈ A from a finite alphabet2 A ∈ Cl . The received
1 Note that the operator style is utilized with shift operator q−1 throughout the
chaptersinceitdoesnotrequiretheexistenceofaz-transformandthuscanalso
be applied to noise sequences. Note also that the description of signals and sys-
temsispurelydiscrete-timeassuming thatequivalentdiscrete-timecounterparts
to continuous-timesignals and systems exist.
2 Thetransmittedsignals(k)isassumedtobenormalizedsuchthatE[|s(k)|2]=1.
2 M. Rupp,A. Burg
sequence r(k) ∈ Cl is given by the convolution of the transmitted symbols
s(k) with the channel filter and additive noise v(k):
L(cid:1)C−1
r(k)=C(q−1)[s(k)]+v(k)= c s(k−l)+v(k). (1.2)
l
l=0
Clearly,thecoefficientsofthechanneldonotonlyprovideameansfortrans-
mitting the symbols, but also cause a linear distortion due to the time-
dispersive nature of the wireless channel, called inter-symbol-interference
(ISI).Forthemoment,itsfrequency-dispersivecharacterisneglectedandwill
be consideredlater.Ifasituationcanbe establishedinwhichonlyonecoeffi-
cientexists,the decodingprocesscanbesimplified.Thus,the questionarises
whetherthereexistsalinearfilterthatcanguaranteethere-establishmentof
F (q−1)C(q−1)=g q−D (1.3)
D D
with a finite delay D and g (cid:4)= 0. Following such an approach, the additive
D
noise is neglected and a filter F (q−1) of length L is desired. Such a crite-
D F
rion is called zero-forcing (ZF) since it forces all coefficients but one of the
resulting filter to zero. The remaining coefficient g can be set to one due
D
to the linear nature of the equalization filter F (q−1) and will no longer be
D
usedexplicitlyinthecontextofZF.Sofar,thenoisecomponenthasbeenne-
glected.However,itimpacts thedecisionbythe filteredvalue F (q−1)[v(k)],
D
which can result in a noise enhancement. Therefore, it appears a better ap-
proach to take the noise into account when searching for the optimal filter
F (q−1), which can be obtained by minimizing the so called mean square
D
error (MSE), i.e.,
(cid:2)(cid:3) (cid:3) (cid:4)
MSE=∆ E (cid:3)q−Ds(k)−F (q−1)r(k)(cid:3)2 . (1.4)
D
Minimizing (1.4)leadsto the minimum mean square error(MMSE). Neither
solution, (1.3) or (1.4) is straightforwardto obtain. Depending on the equal-
izer structure and its length, different solutions may occur. The following
Sect. 1.2 discusses criteria for optimal equalizer performance. For a time-
invariant channel it is sufficient to compute the equalizer solution just once
at the initialization phase of a transmission. Algorithms for this case will be
addressedinSect.1.3includingfractionallyspacedequalizers,multiple-input
multiple-output(MIMO)channelequalizersanddecisionfeedbackstructures.
Adaptive algorithms based either on estimating the channel and noise vari-
anceorwithoutexplicitlyknowingitsimpulseresponsecanachievetheinitial
training of such structures. The most famous least-mean-square LMS algo-
rithm is introduced for ZF and MMSE solutions and its implications are
discussedin Sect. 1.4.With some modificationsuch adaptivefilters alsoper-
form well for time-variantchannels as long as the rate of change is relatively
slow and training is performed periodically.
1 AdaptiveEqualization in Wireless Applications 3
OthertechniquesarepresentedinSect.1.6thattrytofindthemost likely
transmittedsequencebyaso-calledmaximumlikelihood(ML)technique.This
requiresa-prioriknowledgeofthechannelimpulseresponseandthusSect.1.5
gives an overview of estimation techniques for time-variant channels and in
MIMO settings. Finally, Sect. 1.7 presents a short overview of blind tech-
niques as they are being applied today in wireless communications.
1.2 Criteria for Equalization
Indatacommunicationssystemsandinparticularinwirelesstransmissionthe
bestcriteriontocomparetheequalizerperformanceiseithertheresultingbit-
error-rate (BER) or symbol-error-rate (SER). Unfortunately, such measures
are usually not available and other performance metrics need to be utilized
instead.
Nexttothesignal to interference plus noise ratio(SINR) attheequalizer
output, the MMSE measure alreadymentionedin the previoussectionis the
most common measure and will be considered first. Substituting (1.2) into
the definition of the MSE (1.4) yields for the equalizer output
MMSE= (1.5)
(cid:2)(cid:3)(cid:5) (cid:6) (cid:3) (cid:4) (cid:2)(cid:3) (cid:3) (cid:4)
min E (cid:3) q−D−F (q−1)C(q−1) s(k)(cid:3)2 +E (cid:3)F (q−1)v(k)(cid:3)2 .
D D
FD(q−1) (cid:7) (cid:8)(cid:9) (cid:10) (cid:7) (cid:8)(cid:9) (cid:10)
ISI part noise part
The first part presents the remaining ISI while the second is the additive
noise. Without equalization the SINR at the detector is given by
|c |2
SINR= (cid:11) D , (1.6)
LC−1 |c |2+σ2
i=0,i(cid:2)=D i v
where the index 0 ≤ D < L indicates the channel coefficient on which the
C
signalisdetected(typicallythe strongestone).Clearly,thereissignalenergy
(cid:11)
in the term LC−1 |c |2 but without additional means it cannot be used
i=0,i(cid:2)=D i
as suchandappearsas a disturbance to the signal.Define the convolutionof
the channelanda finite lengthfilter function bya new polynomialG(q−1)=
F (q−1)C(q−1)withthecoefficientsg ; i=0,...,L +L −2=0,...,L −1.
D i C F G
The impact of ISI after equalization can be described by either of the two
following measures (note that 0≤D <L and in general g (cid:4)=1)
G D
(cid:11)
LG−1|g |2
ISI = i=0 i −1, (1.7)
|g |2
(cid:11) D
LG−1|g |
PD= i=0 i −1. (1.8)
|g |
D
4 M. Rupp,A. Burg
The second metric is called peak distortion (PD) measure. The convolution
of channel and equalizer filter results in a new SINReq
|g |2 1
SINReq = (cid:11)Li=G0−,i1(cid:2)=D|gi|2+Dσv2(cid:11)Li=F0−1|fi|2 = ISI+ |gσDv2|2 (cid:11)Li=F0−1|fi|2.(1.9)
Theproblemistofindequalizerfiltervalues{f }suchthat(1.9)ismaximized.
i
Both criteria, MSE and SINR are related by
|g |2
MSE= D +|1−g |2, (1.10)
D
SINReq
which simply becomes MSEZF = 1/SNReq in the ZF case. As mentioned
before, the BER measure is the best criterion, however,it is very difficult to
obtain. In the simple case of binary signaling (BPSK) or quadrature phase
shiftkeying(QPSK)withGraycodingoveranAWGNchannelwithconstant
gainc anddelayD,theBERcanbedeterminedbyevaluatingtheexpression
D
(cid:12)(cid:13) (cid:14)
1 |c |2
D
BERAWGN,B/QPSK = 2erfc 2σ2 . (1.11)
v
For other modulation schemes at least an upper bound of the form
(cid:15)√ (cid:16)
BERAWGN ≤Kerfc δSNR (1.12)
exists[2],whereK andδ areconstantsdependingonthemodulationscheme.
SNRstandsforsignaltonoiseratio,i.e.,theSINRwithoutinterference.Once
ISI is present, the BER measure can be modified to
(cid:12)(cid:13) (cid:14)
BERISI = 12(cid:1)I pi(si)erfc |cD+2cσIS2I(si)|2 , (1.13)
i=1 v
whereforallI =PLC−1 possibilitieswithprobabilitypi(si)signalcorruption
is caused by cISI(si). The vectors si contain all possible combinations of I
transmitted symbols.The formulais for QAM transmissionwith equidistant
symbolsandisonlycorrectaslongastheISIissmallenough(max|cISI(si)|<
|cD|).For example,withBPSKthe valuescD+cISI(si)mustremainpositive
andforQPSKtheymustremaininthefirstquarterofthecomplexplane.The
valueofP is2forBPSKand4forQPSKandcanbeverylarge,dependingon
the size of the symbol alphabet. Clearly, with a large number of coefficients
the complexity of such an expression becomes very high. Applying a linear
equalizer, the BER reads
(cid:12)(cid:13) (cid:14)
BERISI,eq = 21(cid:1)I pi(si)erfc 2|gσD2(cid:11)+LgFIS−I(1s|if)||22 . (1.14)
i=1 v i=0 i
1 AdaptiveEqualization in Wireless Applications 5
Optimizing such an expression is quite difficult. Approximations can be ob-
tainedforsmallISI,i.e.,max|gISI(si)|(cid:10)|gD|.Thenthecorruptionresulting
fromISIcanberegardedasaGaussianprocessandcanbeaddedtothenoise
term:
 
(cid:20) (cid:12)(cid:26) (cid:14)
(cid:21)
BERISI,eq ≈ 12erfc(cid:21)(cid:22)2(cid:2)|gσDv2|2 (cid:11)Li=F01−1|fi|2+ISI(cid:4)= 12erfc SIN2Req .
(1.15)
Inthe caseofsmallPD <1 anotheroptionforapproximationis to derivean
upper bound by the worst case
(cid:20) 
(cid:21)(cid:3) (cid:3)
(cid:21)(cid:3) (cid:3)2
(cid:21)(cid:3)g −|g |PD(cid:3) 
BERISI ≤ 12erfc(cid:22)2σD2(cid:11)LFD−1|f |2, (1.16)
v i=0 i
with the above defined peak distortion measure (1.8). Note that (perfect)
ZF solutions always result in simpler expressions of the form (1.11) with
gD =1 and gISI =0, while MMSE will result in the much more complicated
expression (1.13) with the need to use approximate results. The erfc(·) links
the BER to the SNReq obtained from the ideal ZF solution and the BER to
the SINReq from the MMSE solution. However, in general it remains open
which criterion leads to smaller BER3. Note also that there exist unbiased
MMSE solutions as well. In this case g = 1, simply obtained by dividing
D
the MMSE solutionbyg . In [4]it is arguedthat unbiasedMMSE solutions
D
give lower BER than standard MMSE.
1.3 Channel Equalization
Channel equalization tries to restore the transmission signal s(k) by means
of linear or non-linear filtering. Such anapproachseems straightforwardand
abundant literature is available, see for example [3], [4], [5], [6] to name a
few. An overview of such techniques will be given in the following section
where a trade-off has been made between a detailed description and suffi-
cientinformationforwirelessapplications.Channelequalizationasdescribed
in this section is not specific to wireless systems and can also be (and has
successfully been) applied to other fields where time-invariant channels are
common.
3 For a non ISI channel, as it appears, for example, in OFDM (also called DMT)
and PAM transmission, it can be shown that minimizing BER is equivalent to
theZF solution, i.e., fD =1/cD.
6 M. Rupp,A. Burg
1.3.1 Infinite Filter Length Solutions for Single Channels
It is quite educational to assume the equalizer filter solution to be of infinite
length. Minimizing only the ISI part in (1.6), the ZF criterion is obtained,
leading to the following expression:
C∗(q−1) q−D
FZF,∞(q−1)= |C(q−1)|2q−D = C(q−1). (1.17)
This solution is typically of infinite length as can be shown by a simple
example. Assume the channel impulse response to be of the form
−1 −1
C(q )=c0+c1q . (1.18)
Then the ZF solution (for D =0) requires inversion of the channel, i.e.,
1 1
−1
FZF(q )= c0 1+ c1q−1, (1.19)
c0
which is the structure of a first order recursive filter. If |c1/c0| < 1, a
causal,stable filter solutionwith infinite length exists. On the other hand, if
|c1/c0|>1, a stable, anti-causal filter exists with impulse response spanning
from−∞to0.Inpracticesuchanti-causalsolutioncanbe handledbyallow-
ingadditionaldelaysD >0sothatthepartoftheimpulseresponsecarrying
most of its energy canbe representedas a causalsolution andthe remaining
anti-causaltailisneglected.Inthefollowingpartofthissectionthedifference
of causal and anti-causal filters will not be considered and instead a general
equalizer filter of double infinite length will be assumed. Substituting the
general solution (1.17) into the MMSE expression above clearly zeroes the
ISI part. The remaining term is controlled by the noise obtaining
(cid:27)
1 π σ2
MSEZF,∞ = 2π |C(e−vjΩ)|2dΩ. (1.20)
−π
In contrastto the ZF solution, the MMSE solutionminimizing ISI andnoise
simultaneously, reads
C∗(q−1)
FMMSE,∞(q−1)= |C(q−1)|2+σ2q−D. (1.21)
v
The MMSE for this solution results in
(cid:27)
1 π σ2
MMSELIN,∞ = 2π |C(e−jΩv)|2+σ2dΩ, (1.22)
−π v
which is for σ2 > 0 always smaller than the MSE of the ZF solution. Note
v
that this advantagerequiresexactknowledgeofthe noise poweras indicated
in (1.21). Especially in wireless applications, due to time and frequency
dispersive channels, such knowledge is usually not available and it can be