现代数字信号处理参考.pdfVIP

Preface

•Mostadaptivefilteringalgorithmsareobtainedby

simplemodificationsofiterativemethodsforsolving

deterministicopyimizationproblems.

•Gradient-basedoptimizationmethodsprovoidethe

groundforthedevelopmentofthemostwidelyused

adaptivefiliteringalgorithms.

Preface

Theerrorperformancesurfaceofanoptimumfilter,ina

stationarySOE,isgivenby

HHH

P(c)P−cd−dc+cRc

y

Therearetwodistinctwaystofindtheminimumofthe

aboveequation:

•SolvethenormalequationsRc=d,usingadirectlinear

systemsolutionmethod.

•FindtheminimumofP(c)usinganiterativeminimization

algorithm.

Preface

Comparedwithdirectmethods,theadvantagesofiterative

methodsare:

1.requirelessnumericalprecision

putationallylespensive

3.workwhenRisnotinvertible

4.theonlychoicefornonquadraticperformancefunctions.

Inalliterativemethods,westartwithanapproximatesolution,

andkeepchanginguntilreachingtheminimum.Whatdifferentiates

variousoptimizationalgorithmsishowtochoosethedirectionand

thesizeofeachstep.

Steepest-descentalgorithm(SDA)

IfthefunctionP(c)hascontinuousderivatives,itispossibleto

approximateitsvalueatanarbitraryneighboringpointc+cby

usingtheTaylorexpansion

Ormorecompactly

where∇P(c)isthegradientvector,withelements,

andistheHessianmatrix,withelements

Steepest-descentalgorithm(SDA)

Forsimplicityweconsiderfilterswithrealcoefficients,

buttheconclusionsapplywhenthecoefficientsarecomplex.

Thenwehave

andthe