Deepa Kundur University of Toronto Dr Deepa Kundur University of Toronto Multirate Digital Signal Processing Part I 1 42 Chapter 11 Multirate Digital Signal Processing DiscreteTime Signals and Systems Reference Sections 111113 of John G Proakis and ID: 33331
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Chapter11:MultirateDigitalSignalProcessing11.1IntroductionImplementationofSamplingRateConversiony(mTy)=1Xn=�1x(nTx)gTxmTy Tx�n =1Xn=�1x(nTx)g(Tx(km+m�n)) letk=km�n =1Xk=�1x((km�k)Tx)g(Tx(k+m)) =1Xk=�1g((k+m)Tx)x((km�k)Tx)| {z }weightedlinearcombinationoforigsamples Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI13/42 Chapter11:MultirateDigitalSignalProcessing11.1IntroductionImplementationofSamplingRateConversiony(mTy)=1Xk=�1g((k+m)Tx)x((km�k)Tx) Im:determinesthesetofweightsIkm:speciesthesetofinputsamples Irepresentsadiscrete-timelineartime-varyingsystemIeveryoutputsamplemrequiresuseofadi erentimpulseresponse/ceocientset:gm(nTx)=g((n+m)Tx) m =mTy Tx�mTy Tx2[0;1) Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI14/42 Chapter11:MultirateDigitalSignalProcessing11.1IntroductionImplementationofSamplingRateConversiony(mTy)=1Xk=�1g((k+m)Tx)x((km�k)Tx) =1Xk=�1g((k+m)Tx)x((km�k)Tx) Igm(nTx)mayhavetoberetrievedorcomputedIingeneral,thereareasmanyweights/coecientsrequiredasinputsamplesoutputvaluestocomputeIingeneral,nosimplicationispossiblemakingcomputationofy(mTy)fromx(nTx)impractical Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI15/42 Chapter11:MultirateDigitalSignalProcessing11.1IntroductionLinearPeriodicallyTime-VaryingImplementationIsignicantsimplicationpossibleforTy Tx=Fx Fy=D IwhereD;I2Z+andGCD(D;I)=1m=mTy Tx�mTy Tx =mD I�mD I =1 ImD�mD II =1 I(mD)modI =1 I(mD)I Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI16/42 Chapter11:MultirateDigitalSignalProcessing11.1Introduction y(mTy)=1Xn=�1g((n+m)Tx)x((km�n)Tx) =1Xn=�1g((n+0)Tx)x((mD�n)Tx) =1Xn=�1g(nTx)x((mD�n)Tx)| {z }dst-timeconvolution Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI21/42 Chapter11:MultirateDigitalSignalProcessing11.1Introduction y(mTy)=1Xn=�1g(nTx)x((mD�n)Tx) =1Xn=�1sin(n) nx((mD�n)Tx) =1Xn=�1(n)x((mD�n)Tx) =x(mDTx) See Figure11.1.3oftext. Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI22/42 Chapter11:MultirateDigitalSignalProcessing11.1IntroductionInterpolation/Upsampling Tx=ITy=)Ty Tx=1 I;I2Z+ km =mTy Tx=jm Ikm=mTy Tx�mTy Tx=m I�jm Ik2f0;1=I;2=I;:::;(I�1)=Ig Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI23/42 Chapter11:MultirateDigitalSignalProcessing11.1IntroductionInterpolation/Upsamplingy(mTy)=1Xn=�1x(nTx)g(mTy�nTx) =1Xn=�1x(nTx)g(mTx I�nTx) =1Xn=�1x(nTx)sin(m I�n) m I�n =1Xn=�1x(nTx)sin( I(m�nI)) I(m�nI)| {z }sinccenteredatn=m=I See Figure11.1.4oftext. Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI24/42 Chapter11:MultirateDigitalSignalProcessing11.2DecimationbyaFactorDDownsamplingwithAnti-AlaisingFilter IDownsamplingalonemaycausealiasing,therefore,itisdesirabletointroduceananti-aliasinglterHd(!x) Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI25/42 Chapter11:MultirateDigitalSignalProcessing11.2DecimationbyaFactorD v(n)=1Xk=�1h(k)x(n�k) y(m) =v(mD)=1Xk=�1h(k)x(mD�k)| {z }lineartime-varyingsystem Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI26/42 Chapter11:MultirateDigitalSignalProcessing11.2DecimationbyaFactorDDownsampling:FrequencyDomainPerspectiveGoal :determinerelationshipbetweeninput-outputspectra Createanintermediatesignal~v(n)atrateFxbutwiththeequivalentinformationasy(m).~v(n)=v(n)n=0;D;2D;:::0otherwise =v(n)p(n)wherep(n)=1Xk=�1(n�kD) Note :y(m) =v(mD) 1= v(mD)p(mD) =~v(mD) y(m) !~v(n)(equivinfo) See Figure11.2.2oftext. Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI27/42 Chapter11:MultirateDigitalSignalProcessing11.2DecimationbyaFactorDAside:ImpulseTrainp(n)p(n)=1Xl=�1(n�lD)(periodicwithperiodD) ck =1 DD�1Xn=0p(n)ej2kn=D =D�1Xn=01Xl=�1(n�lD)| {z }zeroforl6=0ej2kn=D =1 DD�1Xn=0(n)ej2kn=D =1 D p(n) =D�1Xk=0ckej2km=D=1 DD�1Xk=0ej2km=D Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI28/42 Chapter11:MultirateDigitalSignalProcessing11.2DecimationbyaFactorDDownsampling:FrequencyDomainPerspectiveY(z)=1Xm=�1y(m)z�m=1Xm=�1~v(mD)z�m=+~v(�D)z1+~v(0)z0+~v(D)z�1+ =1Xm0=�1~v(m0)z�m0=Dsince~v(m)=0form62f0;D;2D;:::g=1Xm0=�1v(m)p(m)z�m=D=1Xm=�1v(m)1 DD�1Xk=0ej2km=Dz�m0=D =1 DD�1Xk=01Xm=�1v(m)(e�j2k=Dz1=D)�m| {z }V(e�j2k=Dz1=D) =Hd(e�j2k=Dz1=D)X(e�j2k=D) =1 DD�1Xk=0Hd(e�j2k=Dz1=D)X(e�j2k=Dz1=D) Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI29/42 Chapter11:MultirateDigitalSignalProcessing11.2DecimationbyaFactorD Intheprecedinganalysis,weemployed:v(n)Z !V(z)hd(n)Z !Hd(z)x(n)Z !X(z)V(z)=1Xm=�1v(m)z�mV(z)=Hd(z)X(z)Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI30/42 Chapter11:MultirateDigitalSignalProcessing11.2DecimationbyaFactorDLetz=ej!y:Y(z)=1 DD�1Xk=0Hd(e�j2k=Dz1=D)X(e�j2k=Dz1=D) Y(ej!y) =1 DD�1Xk=0Hd(e�j2k=Dej!y1=D)X(e�j2k=Dej!y1=D)=1 DD�1Xk=0Hd(ej(!y�2k)=D)X(ej(!y�2k)=D) Y(!y) =1 DD�1Xk=0Hd!y�2k DX!y�2k D Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI31/42 Chapter11:MultirateDigitalSignalProcessing11.2DecimationbyaFactorDY(!y)=1 DD�1Xk=0Hd!y�2k DX!y�2k D For�!y,� D!y�2k D Dfork=0�3 D!y�2k D� Dfork=1......�2+ D!y�2k D�2+3 Dfork=D�1 Note :For�!y,Hd!y�2k D=1fork=00fork=1;2;:::;D�1 Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI32/42 Chapter11:MultirateDigitalSignalProcessing11.2DecimationbyaFactorDTherefore,for�!y,Y(!y)=1 DD�1Xk=0Hd!y�2k DX!y�2k D =1 DHd!y D| {z }=1X!y D+1 DD�1Xk=1Hd!y�2k D| {z }=0X!y�2k D Y(!y)=1 DX�!y D Note:!y=Ty Tx!x=D!x. � D!x DofX(!x)isstretchedinto�!yforY(!y).See Figure11.2.3oftext. Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI33/42 Chapter11:MultirateDigitalSignalProcessing11.3InterpolationbyaFactorIInterpolationbyaFactorI IInterpolationonlyincreasesthevisibleresolutionofthesignal.INoinformationgainisachieved.AtbestHu(!y)maintainsthesameinformationiny(n)asexistsinx(n). Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI34/42 Chapter11:MultirateDigitalSignalProcessing11.3InterpolationbyaFactorIGoal :determinerelationshipbetweeninput-outputspectra Consideranintermediatesignalv(n)atrateFybutwiththeequivalentinformationasx(m). v(m)=x(m=I)m=0;I;2I;:::0otherwise V(z) =1Xm=�1v(m)z�m =+v(�I)zI+v(0)z0+v(I)z�I+ =1Xm=�1x(m)z�mI=1Xm=�1x(m)(zI)�m=X(zI) V(ej!y) =X(ej!yI)=)V(!y)=X(!yI) Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI35/42 Chapter11:MultirateDigitalSignalProcessing11.3InterpolationbyaFactorIV(!y)=X(!yI)See Figure11.3.1oftext. Hu(!y)=I0j!yj=I0otherwiseY(!y)=Hu(!y)V(!y)=IX(!yI)0j!yj=I0otherwise Y(!y)=IX(!yI)0j!yj=I0otherwise Note :!y=Ty Tx=!x I.�!xiscompressedinto�=I!y=I Dr.DeepaKundur(UniversityofToronto)MultirateDigitalSignalProcessing:PartI36/42