现代数字信号处理参考.pdf
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