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1、2504IEEETRANSACTIONSONSIGNALPROCESSING,VOL.42,NO.9,SEPTEMBER1994DesignofEquirippleNonrecursiveDigitalDifferentiatorsandHilbert'kansformersUsingaWeightedLeast-SquaresTechniqueS.SunderandV.RamachandranAbslmet-Aprocedureforthedesignofequirippledigitaldifferen-tiators(DD's)andHilbert
2、transformers@IT'S)usingaweightedleast-squarestechniqueisdescribed.Thisprocedureinvolvesaniterativealgorithminwhichtheappropriatefrequency-dependentweightingfunc-tionthatyieldsanequirippledesignisdetermined.OurprocedureisusedincoqjunctionwithpredictionteehniquesfortheDDandHTlength
3、stodesignDD'sandHT's,respectively,satbeingprescribedspecifications.I.INTRODUCTIONDigitaldifferentiators(DD's)andHilberttransformers(HT's)aregenerallydesignedusingtheminimaxcriterion[11,theFourierseriesmethodinconjunctionwiththeKaiserwindowfunction[2],[3],theFourierseriesmethodinc
4、onjunctionwithaccuracyconstraints[4],theeigenfiltermethod[5],[6],andamethodbasedonleast-squaresoptimization[7].a1a1a3a4a5a6am8winrd/aRecently,aweightedleast-squares(WLS)approachforthedesignoflinear-phasenonrecursivefiltershasbeenproposedin[8]asanFig.1.Variationofatypicalerrorfunc
5、tioninthepassband.alternativetotheminimaxmethodbasedontheRemezexchangealgorithm[l].Inthispaper,weextendtheWLSapproachtothedesignoflinear-phaseDD'sandHT's.Inaddition,theapproachis111.ERRORFUNCTIONMINIMIZATIONusedinconjunctionwithpredictiontechniquesfortheDDandHTTheweightedmean-squ
6、aredifferencebetweenD(w)andM(d)lengthstodesignDD'sandHT's,respectively,satisfyingprescribedwithrespecttotheDDpassbandcanbeexpressedasspecifications.KEmse=W(wr)E:(d1)(3)1=111.PROBLEMFORMULATIONFORDIFFEWNTIATORSwhere&(U)=D(w)-M(d)istheerrorfunction,W(w)isaForthedesignoflinear-phase
7、DD's,theimpulseresponse,h(n),isrequiredtobeantisymmetrical,i.e.,h(n)=-h(N-1-n),withfrequency-dependentweightingfunctionandKisthenumberofh(F)=0whenNisodd.Thefrequencyresponseofthefilterpointsatwhichtheerrorfunctionissampled.TheamplitudefunctionoftheDDcanbewrittenasM(w)=withanantis
8、ymmetricalimpulseresponsecanbeexpresseda
