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:

Chapter 11: Multirate Digital Signal Processing Discrete-Time Signals and Systems Reference: Sections 11.1-11.3 of John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 4th edition, 2007. Dr. Deepa Kundur (University of Toronto)M

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