Multirate Digital Signal Processing: Part I

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Chapter 11: Multirate Digital Signal ProcessingDiscrete-Time Signals and SystemsMultirate Digital Signal Processing: Part IReference:Sections 11.1-11.3 ofDr. Deepa KundurJohn G. Proakis and Dimitris G. Manolakis, Digital Signal Processing:Principles, Algorithms, and Applications, 4th edition, 2007.University of TorontoDr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part I1 / 42Chapter 11: Multirate Digital Signal ProcessingMultirate Digital Signal Processing: Part I2 / 42Chapter 11: Multirate Digital Signal ProcessingMultirate DSPSampling vs. Sampling Rate ConversionIsampling rate conversion: process of converting a givendiscrete-time signal at a given rate to a different rateImultirate digital signal processing systems: systems that employmultiple sampling ratesDr. Deepa Kundur (University of Toronto)Dr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part I3 / 42Sampling:I conversion from cts-time to dst-time by taking “samples” atdiscrete time instantsI E.g., uniform sampling: x(n) xa (nT ) where T is the samplingperiodSampling rate conversion approaches:I convert original samples to analog domain and then resample togenerate new samplesI filter original samples with a discrete-time linear time-varyingsystem to generate new samplesDr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part I4 / 42

Chapter 11: Multirate Digital Signal Processing11.1 IntroductionChapter 11: Multirate Digital Signal ProcessingSampling Rate ConversionIdeal Sampling Rate ConversionIIx(n)original/bandlimitedinterpolated signal x(t)1y(n)0II11.1 Introductionx(n): original samples at sampling rate Fx y (n): new samples at sampling rate Fy T1yoriginal/bandlimitedinterpolatedn signal x(t)1x(n): original samples at sampling rate Fxy (n): new samples at sampling rate Fy1Txx(n)0y(n)Dr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part IChapter 11: Multirate Digital Signal Processing5 / 42Dr. Deepa Kundur (University of Toronto)11.1 IntroductionMultirate Digital Signal Processing: Part IChapter 11: Multirate Digital Signal ProcessingParameter Relationships6 / 4211.1 IntroductionBridging the Parameter RelationshipsISAMPLING RATECONVERSIONrelated to the ratio:TyTxFy Tx· FxTy2πFFy2πF 2πfy FyFyTx · ωy · ωyFxTyωx 2πfx Parameter/VariableRatePeriodDst-time FrequencyCts-time FrequencyDr. Deepa Kundur (University of Toronto)x(n) x(nTx ) y (m) y (mTy )FxFyTxTyωxωyFFMultirate Digital Signal Processing: Part Iωyωx7 / 42Dr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part I8 / 42

Chapter 11: Multirate Digital Signal Processing11.1 IntroductionChapter 11: Multirate Digital Signal ProcessingImplementation of Sampling Rate ConversionImplementation of Sampling Rate ConversionWe relate the original samples x(nTx ) to the new samples y (mTy ) byassuming we convert the signal to analog and resample. Using theinterpolation formula Xy (t) 11.1 Introductiony (t) y (mTy ) {z }x(nTx )g (t nTx ) x(nTx )g (t nTx )n Xn desired samplesn Xx(nTx ) {z }original samplesg (mTy nTx ) {z}samples of g (t)wheresin(πt/Tx )g (t) πt/TxF G (F ) Tx F Fx 20 otherwiseNote: y (t) x(t) if x(t) is sampled above Nyquist.Dr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part IChapter 11: Multirate Digital Signal Processing9 / 4211.1 Introductiony (t) y (mTy ) n X10 / 4211.1 IntroductionImplementation of Sampling Rate Conversionx(nTx )g (t nTx )mTyTxx(nTx )g (mTy nTx )n Xkm mTy x(nTx )g Tx nTxn mmTy km mTxDr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part IChapter 11: Multirate Digital Signal ProcessingImplementation of Sampling Rate Conversion XDr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part I11 / 42Dr. Deepa Kundur (University of Toronto) km m {z} {z}integerremainder mTy Tx mTymTy [0, 1)TxTxMultirate Digital Signal Processing: Part I12 / 42

Chapter 11: Multirate Digital Signal Processing11.1 IntroductionChapter 11: Multirate Digital Signal ProcessingImplementation of Sampling Rate ConversionImplementation of Sampling Rate Conversion X mTyy (mTy ) x(nTx )g Tx nTxn Xy (mTy ) k XIIIx((km k)Tx )g (Tx (k m )) m : determines the set of weightskm : specifies the set of input samplesrepresents a discrete-time linear time-varying systemIevery output sample m requires use of a different impulseresponse/ceofficient set:g ((k m )Tx )x((km k)Tx )gm (nTx ) g ((n m )Tx ) mTymTy m [0, 1)TxTxk Dr. Deepa Kundur (University of Toronto){zweighted linear combination of orig samples}Multirate Digital Signal Processing: Part IChapter 11: Multirate Digital Signal Processing13 / 42 Xk XIIIg ((k m )Tx )x((km k)Tx )g ((k m )Tx )x((km k)Tx )Multirate Digital Signal Processing: Part I14 / 4211.1 IntroductionLinear Periodically Time-Varying Implementationsignificant simplification possible for TTyx where D, I Z and GCD(D, I ) 1 mgm (nTx ) may have to be retrieved or computedin general, there are as many weights/coefficients required asinput samples output values to computein general, no simplification is possible making computation ofy (mTy ) from x(nTx ) impracticalDr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part IChapter 11: Multirate Digital Signal Processingk IDr. Deepa Kundur (University of Toronto)11.1 IntroductionImplementation of Sampling Rate Conversiony (mTy ) g ((k m )Tx )x((km k)Tx )x(nTx )g (Tx (km m n))let k km n Xk n X11.1 Introduction15 / 42FxFy DI mTymTymDmD TxTxII 11mD mD I (mD) mod IIII1 (mD)IIDr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part I16 / 42

Chapter 11: Multirate Digital Signal Processing11.1 IntroductionChapter 11: Multirate Digital Signal ProcessingLinear Periodically Time-Varying Implementation11.1 IntroductionLinear Periodically Time-Varying ImplementationFurthermore, for r Z:Note:(mD)I mI11((m rI )D)I (mD rlD)III1 (mD)I mI gm r I (nTx ) gm (nTx ), r Z m rI {0, 1, 2, . . . , I 1}1 (mD)I {0, 1/I , 2/I , . . . , (I 1)/I }Igm (nTx ) g ((n m )Tx ) consists only of I distinct sets ofcoefficients!IDr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part IChapter 11: Multirate Digital Signal Processing17 / 4211.1 Introductiongm (nTx ) gm rI (nTx ) represents a discrete-time periodicallytime-varying system!Dr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part IChapter 11: Multirate Digital Signal ProcessingLinear Periodically Time-Varying Implementation18 / 4211.1 mitedinterpolated signal10nTyTy DTx D, D Z T xmTykm bmDc mD mD ZTx mTymTy m mD bmDc mD mD 0TxTxDr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part I19 / 42Dr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part I20 / 42

Chapter 11: Multirate Digital Signal Processingoriginal/bandlimitedinterpolated signal11.1 IntroductionChapter 11: Multirate Digital Signal Processingoriginal/bandlimitedinterpolated signal1y (mTy ) Xn Xn XDr. Deepa Kundur (University of Toronto)g ((n m )Tx )x((km n)Tx )y (mTy ) g ((n 0)Tx )x((mD n)Tx ) g (nTx )x((mD n)Tx ) {zdst-time convolutionMultirate Digital Signal Processing: Part ISee21 / 42Figure 11.1.3 of text Xδ(n)x((mD n)Tx ) x(mDTx ).Dr. Deepa Kundur (University of Toronto)11.1 IntroductionMultirate Digital Signal Processing: Part IChapter 11: Multirate Digital Signal Processing nTy1 ITy , I Z TI xj kmTym TxI mTymTym jmk TxTxII {0, 1/I , 2/I , . . . , (I 1)/I }Multirate Digital Signal Processing: Part I22 / 4211.1 IntroductionInterpolation/Upsampling10Dr. Deepa Kundur (University of Toronto)sin(πn)x((mD n)Tx )πnn y (mTy ) original/bandlimitedinterpolated signal mg (nTx )x((mD n)Tx )n X}Interpolation/Upsamplingkm Xn Chapter 11: Multirate Digital Signal ProcessingTxn0n 1n011.1 Introduction Xn Xn Xn Xn x(nTx )g (mTy nTx )x(nTx )g (mx(nTx )x(nTx )Tx nTx )Isin(π mI πn)π mI πnsin( πI (m nI ))π(m nI ) I {z}sinc centered at n m/ISee23 / 42Figure 11.1.4 of text.Dr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part I24 / 42

Chapter 11: Multirate Digital Signal Processing11.2 Decimation by a Factor DChapter 11: Multirate Digital Signal Processing11.2 Decimation by a Factor DDecimatorDownsampling with Anti-Alaising FilterLTI FilterDownsamplerDecimatorLTI FilterDownsampler Xv (n) IDownsampling alone may cause aliasing, therefore, it is desirableto introduce an anti-aliasing filter Hd (ωx )InterpolatorUpsamplerLTI FilterDr. Deepa Kundur (University of Toronto)LTI Filter25 / 4211.2 Decimation by a Factor DUpsamplery (m)Dr. Deepa Kundur (University of Toronto)p(n)Create an intermediate signal ṽ (n) at rate Fx but with the equivalentinformation as y (m). v (n) n 0, D, 2D, . . .ṽ (n) 0otherwise X v (n) · p(n) where p(n) δ(n kD)k }26 / 4211.2 Decimation by a Factor Dck 27 / 42(periodic with period D) D 11 Xp(n)e j2πkn/DD n 0D 1X Xδ(n lD) e j2πkn/Dn 0 l {zzero for l 6 0}D 11 X1δ(n)e j2πkn/D D n 0DD 1Xk 0.δ(n lD)l p(n) Multirate Digital Signal Processing: Part I{zlinear time-varying systemMultirate Digital Signal Processing: Part I X Note: y (m) v (mD) · 1 v (mD) · p(mD) ṽ (mD)y (m) ṽ (n) (equiv info)Dr. Deepa Kundur (University of Toronto)LTI Filterk DecimatorDownsamplerh(k)x(mD k)LTI FilterAside: Impulse Train p(n)Goal: determine relationship between input-output spectraFigure 11.2.2 of text v (mD) Chapter 11: Multirate Digital Signal ProcessingDownsampling: Frequency Domain PerspectiveSee XInterpolatorDecimatorDownsamplerMultirate Digital Signal Processing: Part IChapter 11: Multirate Digital Signal Processingh(k)x(n k)k Dr. Deepa Kundur (University of Toronto)ck e j2πkm/D D 11 X j2πkm/DeDk 0Multirate Digital Signal Processing: Part I28 / 42

Chapter 11: Multirate Digital Signal Processing11.2 Decimation by a Factor DChapter 11: Multirate Digital Signal ProcessingDecimatorDownsampling: Frequency Domain PerspectiveY (z) X y (m)z m m X11.2 Decimation by a Factor DLTI FilterDownsamplerṽ (mD)z mm 0· · · ṽ ( D)z 1 ṽ (0)z ṽ (D)z 1 · · · X0ṽ (m0 )z m /D since ṽ (m) 0 for m 6 {0, D, 2D, . . .} m0 X v (m)p(m)z m/Dm0 1D XD 11 X j2πkm/D m0 /D v (m)ezDm In the preceding analysis, we employed:k 0D 1X Xk 0m Zv (n) v (m)(e j2πk/D z 1/D ) m hd (n) {z}UpsamplerD 11 XHd (e j2πk/D z 1/D )X (e j2πk/D z 1/D )DInterpolatorx(n) LTI FilterV (z) Let z ejωyV (z)Multirate Digital Signal Processing: Part IChapter 11: Multirate Digital Signal ProcessingX (z) XLTI Filterv (m)z mDecimatorDownsamplerm k 0Dr. Deepa Kundur (University of Toronto)Hd (z)Z V (e j2πk/D z 1/D ) Hd (e j2πk/D z 1/D )X (e j2πk/D ) V (z)Z29 / 4211.2 Decimation by a Factor DDr. Deepa Kundur (University of Toronto)Hd (z) · X (z) Multirate Digital Signal Processing: Part IChapter 11: Multirate Digital Signal Processing30 / 4211.2 Decimation by a Factor D:Y (z) Y (e jωy ) 1DD 1XY (ωy )Hd (e j2πk/D z 1/D )X (e j2πk/D z 1/D )k 0k 0For π ωy π,D 11 XHd (e j2πk/D e jωy 1/D )X (e j2πk/D e jωy 1/D )Dk 0 Y (ωy ) D 11 Xωy 2πkωy 2πkHdXDDDD 11 XHd (e j(ωy 2πk)/D )X (e j(ωy 2πk)/D )Dk 0 D 1X1ωy 2πkωy 2πkHdXDDD Dπ 3πD ωy 2πkDωy 2πkD Dπ Dπ.πDωy 2πkD 2π 2π .for k 0for k 13πDfor k D 1k 0DecimatorLTI FilterDownsamplerNote: For π ωy π, ωy 2πk1Hd 0DDr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part I31 / 42Dr. Deepa Kundur (University of Toronto)for k 0for k 1, 2, . . . , D 1Multirate Digital Signal Processing: Part I32 / 42

Chapter 11: Multirate Digital Signal Processing11.2 Decimation by a Factor DChapter 11: Multirate Digital Signal ProcessingInterpolation by a Factor ITherefore, for π ωy π,Y (ωy ) D 11 Xωy 2πkωy 2πkHdXDDDk 0 ω ω 1 D 1X1ωy 2πkωy 2πkyyHd HdXXD {zD }DDDDk 1 {z} 1TyTxInterpolator1XDωy DIωx Dωx .ISee Dπ ωx πDFigure 11.2.3 of text.LTI FilterUpsampler 0Y (ωy ) Note: ωy 11.3 Interpolation by a Factor Iof X (ωx ) is stretched into π ωy π for Y (ωy ).Interpolation only increases the visible resolution of the signal.No information gain is achieved. At best Hu (ωy ) maintains thesame information in y (n) as exists in x(n).InterpolatorLTI FilterDr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part IChapter 11: Multirate Digital Signal Processing33 / 42Dr. Deepa Kundur (University of Toronto)11.3 Interpolation by a Factor ILTI FilterDecimatorDownsamplerMultirate Digital Signal Processing: Part IChapter 11: Multirate Digital Signal Processing34 / 4211.3 Interpolation by a Factor IGoal: determine relationship between input-output spectraV (ωy ) X (ωy I )Consider an intermediate signal v (n) at rate Fy but with the equivalentinformation as x(m). x(m/I ) m 0, I , 2I , . . .v (m) 0otherwise XV (z) v (m)z m · · · v ( I )z I v (0)z 0 v (I )z I · · · m Xx(m)z mI m V (e jωy ) X (e jωy I ) X mx(m)(z I )SeeFigure 11.3.1 of text. 0 ωy π/Iotherwise IX (ωy I ) 0 ωy π/IY (ωy ) Hu (ωy )V (ωy ) 0otherwiseHu (ωy ) I0 X (z I ) m Y (ωy ) V (ωy ) X (ωy I )I X (ωy I ) 0 ωy π/I0otherwiseInterpolatorUpsamplerLTI FilterNote: ωy TyTx ωxI. π ωx π is compressed into π/I ωy π/IDr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part I35 / 42Dr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part I36 / 42

Chapter 11: Multirate Digital Signal Processing11.3 Interpolation by a Factor IChapter 11: Multirate

Dr. Deepa Kundur (University of Toronto)Multirate Digital Signal Processing: Part I2 / 42 Chapter 11: Multirate Digital Signal Processing Multirate DSP Isampling rate conversion: process of converting a given discrete-time signal at a given rate to a di erent rate Imultirate digital signal processing systems: systems that employ multiple sampling rates