标签: recursive

Anewrecursivealgorithmforlinearequationanditsapplicationtoadaptivesignalprocessing

Recursivefiltersareanefficientwayofachievingalongimpulseresponse,withouthavingtoperformalongconvolution.Theyexecuteveryrapidly,buthavelessperformanceandflexibilitythanotherdigitalfilters.RecursivefiltersarealsocalledInfiniteImpulseResponse(IIR)filters,sincetheirimpulseresponsesarecomposedofdecayingexponentials.Thisdistinguishesthemfromdigitalfilterscarriedoutbyconvolution,calledFiniteImpulseResponse(FIR)filters.Thischapterisanintroductiontohowrecursivefiltersoperate,andhowsimplemembersofthefamilycanbedesigned.Chapters20,26and33presentmoresophisticateddesignmethods.CHAPTERRecursiveFilters19Recursivefiltersareanefficientwayofachievingalongimpulseresponse,withouthavingtoperformalongconvolution.Theyexecuteveryrapidly,buthavelessperformanceandflexibilitythanotherdigitalfilters.RecursivefiltersarealsocalledInfiniteImpulseResponse(IIR)filters,sincetheirimpulseresponsesarecomposedofdecayingexponentials.Thisdistinguishesthemfromdigitalfilterscarriedoutbyconvolution,calledFiniteImpulseResponse(FIR)filters.Thischapterisanintroductiontohowrecursivefiltersoperate,andhowsimplemembersofthefamilycanbedesigned.Chapters20,26and33presentmoresophisticateddesignmethods.TheRecursiveMethodTostar……

ThemovingaverageisthemostcommonfilterinDSP,mainlybecauseitistheeasiestdigitalfiltertounderstandanduse.Inspiteofitssimplicity,themovingaveragefilterisoptimalforacommontask:reducingrandomnoisewhileretainingasharpstepresponse.Thismakesitthepremierfilterfortimedomainencodedsignals.However,themovingaverageistheworstfilterforfrequencydomainencodedsignals,withlittleabilitytoseparateonebandoffrequenciesfromanother.RelativesofthemovingaveragefilterincludetheGaussian,Blackman,andmultiplepassmovingaverage.Thesehaveslightlybetterperformanceinthefrequencydomain,attheexpenseofincreasedcomputationtime.CHAPTERMovingAverageFilters15ThemovingaverageisthemostcommonfilterinDSP,mainlybecauseitistheeasiestdigitalfiltertounderstandanduse.Inspiteofitssimplicity,themovingaveragefilterisoptimalforacommontask:reducingrandomnoisewhileretainingasharpstepresponse.Thismakesitthepremierfilterfortimedomainencodedsignals.However,themovingaverageistheworstfilterforfrequencydomainencodedsignals,withlittleabilitytoseparateonebandoffrequenciesfromanother.RelativesofthemovingaveragefilterincludetheGaussian,Blackman,andmultiple-passmovingaverage.Thesehaveslightlybetterperformanceinthefrequencydomain,attheexpenseofincreasedcomputationtime.Im……

JustasanalogfiltersaredesignedusingtheLaplacetransform,recursivedigitalfiltersaredevelopedwithaparalleltechniquecalledthez-transform.Theoverallstrategyofthesetwotransformsisthesame:probetheimpulseresponsewithsinusoidsandexponentialstofindthesystem'spolesandzeros.TheLaplacetransformdealswithdifferentialequations,thes-domain,andthes-plane.Correspondingly,thez-transformdealswithdifferenceequations,thez-domain,andthez-plane.However,thetwotechniquesarenotamirrorimageofeachother;thes-planeisarrangedinarectangularcoordinatesystem,whilethez-planeusesapolarformat.Recursivedigitalfiltersareoftendesignedbystartingwithoneoftheclassicanalogfilters,suchastheButterworth,Chebyshev,orelliptic.Aseriesofmathematicalconversionsarethenusedtoobtainthedesireddigitalfilter.Thez-transformprovidestheframeworkforthismathematics.TheChebyshevfilterdesignprogrampresentedinChapter20usesthisapproach,andisdiscussedindetailinthischapter.CHAPTERThez-Transform33JustasanalogfiltersaredesignedusingtheLaplacetransform,recursivedigitalfiltersaredevelopedwithaparalleltechniquecalledthez-transform.Theoverallstrategyofthesetwotransformsisthesame:probetheimpulseresponsewithsinusoidsandexponentialstofindthesystem'spolesandzeros.TheLaplacetransformdealswithdifferentialequations,thes-domain,andthes-plane.Correspondingly,thez-transformdealswithdifferenceequations,thez-domain,andthez-plane.However,thetwotechniquesarenotamirrorimageofeachother;thes-planeisarrangedinarectangularcoordinatesystem,whilethez-planeusesapolarformat.Recursivedigitalfiltersareoftendesignedbystartingwithone……

TraditionalDSPisbasedonalgorithms,changingdatafromoneformtoanotherthroughstep-bystepprocedures.Mostofthesetechniquesalsoneedparameterstooperate.Forexample:recursivefiltersuserecursioncoefficients,featuredetectioncanbeimplementedbycorrelationandthresholds,animagedisplaydependsonthebrightnessandcontrastsettings,etc.Algorithmsdescribewhatistobedone,whileparametersprovideabenchmarktojudgethedata.Theproperselectionofparametersisoftenmoreimportantthanthealgorithmitself.Neuralnetworkstakethisideatotheextremebyusingverysimplealgorithms,butmanyhighlyoptimizedparameters.Thisisarevolutionarydeparturefromthetraditionalmainstaysofscienceandengineering:mathematicallogicandtheorizingfollowedbyexperimentation.Neuralnetworksreplacetheseproblemsolvingstrategieswithtrial&error,pragmaticsolutions,anda"thisworksbetterthanthat"methodology.ThischapterpresentsavarietyofissuesregardingparameterselectioninbothneuralnetworksandmoretraditionalDSPalgorithms.CHAPTERNeuralNetworks(andmore!)26TraditionalDSPisbasedonalgorithms,changingdatafromoneformtoanotherthroughstep-by-stepprocedures.Mostofthesetechniquesalsoneedparameterstooperate.Forexample:recursivefiltersuserecursioncoefficients,featuredetectioncanbeimplementedbycorrelationandthresholds,animagedisplaydependsonthebrightnessandcontrastsettings,etc.Algorithmsdescribewhatistobedone,whileparametersprovideabenchmarktojudgethedata.Theproperselectionofparametersisoftenmoreimportantthanthealgorithmitself.Neuralnetworkstakethisideatotheextremebyusingverysimplealgorithms,butmanyhighlyoptimizedparameters.Thisisarevolutionarydeparturefromthetraditionalmai……

recursivemerge-filteralgorithmforcomputingthediscreteRecursiveMerge-FilterAlgorithmforComputingtheDiscreteWaveletTransformKunalMukherjeeandAmarMukherjeeAbstractWepresentanewwavelettransformalgorithmwithadataflowthatcanfullyexploitthelocalitypropertyofwavelets.Thisleadstohighlyoptimizedfinegrainedwaveletcodingalgorithms,intermsofpipeliningperformance,flexibledatagranularityandreliabilityoftransmission.Itcanbeusedbyallwaveletcodingmethods,andhasbeendemonstratedtoimprovetheperformanceofthemostsuccessfulones.Weproposeanewbottom-upEmbeddedZerotreeWavelet(EZW)imagecodingalgorithm,anddemonstratea5-10%speedupoverEZW,bymeansofclosecouplingbetweenthenewwavelettransformalgorithmandtheEZWencoding.TheRecursiveMergeFilter(RMF)operatorint……