Efficient Nonlinear Adaptive Filters: Design, Analysis and Applications

271 Pages·2023·12.701 MB·English

by  Haiquan Zhao, Badong Chen

Haiquan Zhao
Badong Chen
Efficient
Nonlinear
Adaptive
Filters
Design, Analysis and Applications
fi
Ef cient Nonlinear Adaptive Filters
(cid:129)
Haiquan Zhao Badong Chen
fi
Ef cient Nonlinear Adaptive
Filters
Design, Analysis and Applications
HaiquanZhao BadongChen
SchoolofElectricalEngineering InstArtificialIntelligence&Robo
SouthwestJiaotongUniversity Xi'anJiaotongUniversity
Chengdu,China Xi'an,China
ISBN978-3-031-20817-1 ISBN978-3-031-20818-8 (eBook)
https://doi.org/10.1007/978-3-031-20818-8
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Preface
Inrecentyears,signal-processingtechnologytakengreataleapforward.Especially
with the development of digital circuit technology, the efficiency of digital signal
processing(DSP)hasbeengreatlyimproved.Digitalfilteringtechnology,animpor-
tant branch of DSP, has been widely studied and applied in many fields, which
mainly aims to extract the useful information contained in the received signal. In
practice,thedevicethatachievesfilteringfunctionisgenerallycalledafilter,which
canextractthedesiredinformationfromtheinputsignal.
Digitalfilterisusedtoprocessdiscrete-timesignal.Forlineartimeinvariant(LTI)
filter,itsinternalparametersandstructurearefixed,andtheoutputsignalisthelinear
mapping of the input signal. However, when the statistical characteristics of the
signal to be processed are unknown, the LTI filter cannot provide good signal
processing capability. At this time, adaptive filter is a very attractive solution,
which can optimize its internal free parameters according to the input signal to
provideeffectiveperformance.Strictlyspeaking,adaptivefilterisakindofnonlinear
filter (itscharacteristics depend onthe inputsignal), soit does notsatisfy superpo-
sition andhomogeneity.However,atacertain moment, theparameters ofthefilter
arefixed,andtheoutputofthefilterisalinearmappingoftheinputsignal.
Atthecruxofadaptivefiltersisthedesignofthefilteringalgorithm,i.e.howthe
parametersofthefilterareadaptivelyadjustedtomeettheperformancerequirements
inresponsetochangesintheenvironment(inputanddesiredsignal).Thealgorithms
discussed in this book are all based on discrete-time signals, because the rapid
development of VLSI technology makes the processing of discrete-time signals
morerapidandconvenient.
An adaptive filter generally consists of three parts: (1) Application. Adaptive
filteringtechnologyhasbeenappliedinmanyaspects,suchaschannelequalization,
signalprediction,echocancellation,beam-forming,systemidentification,andsignal
enhancement.(2)Structure.Adaptivefiltercanbecomposedofmanystructures,and
differentstructurecorrespondstodifferentcomputationalcomplexity.Accordingto
the form of impulse response, adaptive filter can be divided into finite impulse
response (FIR) filter and infinite impulse response (IIR) filter. The most widely
v
vi Preface
usedFIRfilteristransversefilter,itstransferfunctionhasnopolepoint,sothereis
no system stability issue. For this structure, the output of the filter is a linear
combinationoftheinputsignals.However,mostoftheactualsystemsarenonlinear,
thelinearadaptivefilterisnotsuitabletodealwiththiskindofsituationbecauseof
itsinherentdefects,sothenonlinearadaptivefilterisproposedtoovercomeabove-
mentioned problem, such as Volterra filter, function link artificial neural network
(FLANN), spline filter, and kernel function–based filter. (3) Algorithm. The algo-
rithmadaptivelyadjuststhecoefficientsofthefiltertominimizeacertainoptimiza-
tioncriterion.
In fact, the theory of linear adaptive filtering is mature enough, and a large
number of journals and books have summarized it in detail. However, there are
veryfewbooksonnonlinearadaptivefilters.Therefore,thecorecontentofthisbook
is to introduce some nonlinear adaptive filters with complete theoretical systems,
includingsomeclassicalapplications,nonlinearfilterstructures,andalgorithms.The
first chapter of this book briefly introduces the basic knowledge of classical linear
adaptivefiltering.Theunderstandingofthisbasicknowledgeisthebasisforfurther
studyofnonlinearadaptivefilteringmethodsinthefollowingchapters.
Themaincontentsofthisbookconsistoffivechapters,whicharesummarizedas
follows:
Chapter 1 mainly introduces the linear adaptive filter and several classical
adaptive filtering algorithms. Finally, a brief introduction is given to the nonlinear
filterthatwillbedescribedinthefollowingchapters.
Chapter2introducestheVolterrafilterfornonlinearsystems,mainlyincludesthe
pipelined Volterrafilter,convexcombined Volterra filterandrobustVolterrafilter,
andtheircorrespondingnonlinearfilteringalgorithms.Moreover,arobustdiffusion
Volterra(DV)algorithmfordistributednonlinearnetworkisalsodescribedindetail.
Finally,computersimulationsareprovided.
Chapter3describesthefunctionallinkartificialneuralnetwork(FLANN)-based
nonlinearfilter,mainlyincludesthestructure,principle,andsomeimprovedmodels
of the FLANN-based filter. The nonlinear property and modelling ability of the
FLANN-basedfilterareverifiedbycomputersimulations.
InChap.4,thenonlinearsplinefilterandadaptivealgorithmsareintroduced.In
addition,theconvergencebehaviorofarobustsplinefilteringalgorithmisanalyzed,
andthevalidityofanalysisresultsareverifiedbycomputersimulations.Finally,the
applicationofsplinefilterinactivenoisecontrolisgiven.
In Chap. 5, we introduce the kernel adaptive filter and several classical kernel
adaptive filtering algorithms. In particular, in order to reduce the high computing
consumptionandstoragespacecausedbythelarge-scalehiddenlayernodesofthese
algorithms,severalnetworkoptimizationmethodsarepresented.Finally,computer
simulationsareprovidedtoverifythevalidityoftheseoptimizationmethods.
This book provides a reference for researchers and students in the field of
developing and researching advanced signal processing of adaptive filters, and
alsoprovidesaconvenientwayforpracticalengineersinrelatedfieldstounderstand
effective algorithms. The readers of this book need to understand some basic
principles of digital signal processing, random processes, and matrix theory,
Preface vii
including finite impulse response (FIR) digital filter realization, random variables,
and first-order and second-order statistics. Assuming that the readers have such a
background,theywillhavenoproblemreadingthisbook.Inaddition,anumberof
referencesaregivenattheendofeachchaptertofacilitatethereaders’furtherstudy
ofachapter.
Chengdu,China HaiquanZhao
Xi’an,China BadongChen
Acknowledgments
We would like to thank some of my former and current graduate students. In
particular, we would like to thank PhD. Yingying Zhu, PhD. Shaohui Lv, PhD.
WenjingXu,PhD.DongxuLiu,PhD.PengfeiLi,Dr.ChuangLiu,Ms.YuanGao,
Ms.BoyuTian,Ms.JinweiLou,Ms.XinhaoXu,Dr.ZhengdaQin,andDr.LeiXing
withwhomwehaveworkedonthetopicofthisbookandwhocontributedtosome
oftheresultsreportedhere.ThisworkwaspartiallysupportedbyNationalNatural
ScienceFoundationofChina(grant:62171388,61871461),FundamentalResearch
Funds for the Central Universities (grant: 2682021ZTPY091), and Southwest
Jiaotong University Graduate Teaching Materials (Monograph) Funding Construc-
tionProject(grant:SWJTU-ZZ2022-017).
ix
Contents
1 AdaptiveFilter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 LinearAdaptiveFilters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.1 LMSAlgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.2 AffineProjectionAlgorithm. . . . . . . . . . . . . . . . . . . . . . . 2
1.2.3 RecursiveLeast-SquaresAlgorithm. . . . . . . . . . . . . . . . . 4
1.2.4 SubbandAlgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.5 KalmanFilter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 NonlinearAdaptiveFilters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 VolterraFilter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2 FLANNAdaptiveFilter. . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.3 SplineAdaptiveFilter. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.4 KernelAdaptiveFilter. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 VolterraAdaptiveFilter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 VolterraFilterModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 PipelinedVolterraFilter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 ConvexCombinationofVolterraFilter. . . . . . . . . . . . . . . . . . . . 24
2.4.1 TheAlgorithmI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.2 TheAlgorithmII. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 RobustVolterraFilteringAlgorithm. . . . . . . . . . . . . . . . . . .. . . . 33
2.6 TheVolterraExpansionModelBased
Filtered-xLogarithmicContinuousLeastMeanp-Norm
(VFxlogCLMP)AlgorithmforActiveNoiseControl
Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6.1 VFxlogLMPAlgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6.2 VFxlogCLMPAlgorithm. . . . . . . . . . . . . . . . . . . . . . . . . 40
xi
xii Contents
2.6.3 PerformanceAnalysisoftheVFxlogCLMP
Algorithm. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6.4 EMSEAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6.5 ConvergenceConditionoftheVFxlogCLMP
Algorithm. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.7 DiffusionVolterraNonlinearFilteringAlgorithm. . . . . . . . . . . . . 48
2.7.1 DiffusionLeastMeanSquare(DLMS)Algorithm. . . . . . . 49
2.7.2 ProblemFormulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.7.3 TheDVFilteringAlgorithm. . . . . . . . . . . . . . . . . . . . . . . 51
2.8 SimulationResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.8.1 PipelinedVolterraFilter. . . . . . . . .. . . . . . . . . . . . . . . . . 56
2.8.2 ConvexCombinationofVolterraFilter. . . . . . . . . . . . . . . 60
2.8.3 RobustVolterraFilteringAlgorithm. . . . . . . . . . . . . . . . . 64
2.8.4 TheVFxlogCLMPAlgorithmforANCApplication. . . . . . 66
2.8.5 DiffusionVolterraFilteringAlgorithm. . . . . . . . . . . . . . . 71
2.9 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3 FLANNAdaptiveFilter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2 NeuralNetworkStructures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.2.1 MLP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.2.2 ChNN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.3 FLANN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.2.4 LeNN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3 RecursiveFLANN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.3.1 FeedbackFLANNFilter. . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.3.2 ReducedFeedbackFLANNFilter. . . . . . . . . . . . . . . . . . . 91
3.3.3 RecursiveFLANNStructure. . . . . . . . . . . . . . . . . . . . . . . 95
3.4 ConvexCombinationofFLANNFilter. . . . . . . . . . . . . . . . . . . . 100
3.5 RandomFourierFilter. . . . . . . . . .. . . . . . . . . .. . . . . . . . . . .. . 107
3.5.1 RandomFourierFeature. . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.5.2 RF-LMSAlgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.5.3 CascadedRF-LMS(CRF-LMS)Algorithm. . . . . . . . . . . . 110
3.5.4 MeanConvergenceAnalysis. . . . . . . . . . . . . . . . . . . . . . 113
3.5.5 ComputationalComplexity. . . . . . . . . . . . . . . . . . . . . . . . 115
3.6 NonlinearActiveNoiseControl. . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.6.1 RobustControlAlgorithmsforNANC. . . . . . . . . . . . . . . 116
3.7 NonlinearChannelEqualization. .. . . . . .. . . . . .. . . . .. . . . . .. 126
3.7.1 CommunicationChannelEqualization. . . . . . . . . . . . . . . . 126
3.7.2 ChannelEqualizationUsingaGeneralizedNNModel. . . . 127
3.7.3 FLNNEqualizer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.8 ComputerSimulationExamples..... ..... ..... ...... ..... 135