Multirate Digital Signal ProcessingUp-SamplerSlide 3Slide 4Slide 5Down-SamplerSlide 7Slide 8Basic Sampling Rate Alteration DevicesSlide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Cascade EquivalencesSlide 34Slide 35Slide 36Filters in Sampling Rate Alteration SystemsSlide 38Filter SpecificationsSlide 40Interpolation Filter SpecificationsInterpolation Filter SpecificationsSlide 43Slide 44Slide 45Decimation Filter SpecificationsFilter Design MethodsFilters for Fractional Sampling Rate AlterationSlide 49Slide 50Computational RequirementsSlide 52Slide 53Slide 54Slide 55Slide 56Slide 57Sampling Rate Alteration Using MATLABSlide 59Slide 60Slide 61Slide 62Slide 63Slide 64Copyright © 2001, S. K. Mitra1Multirate Digital Signal Multirate Digital Signal ProcessingProcessingBasic Sampling Rate Alteration DevicesBasic Sampling Rate Alteration Devices•Up-samplerUp-sampler - Used to increase the sampling rate by an integer factor•Down-samplerDown-sampler - Used to decrease the sampling rate by an integer factorCopyright © 2001, S. K. Mitra2Up-SamplerUp-SamplerTime-Domain CharacterizationTime-Domain Characterization•An up-sampler with an up-sampling factorup-sampling factor L, where L is a positive integer, develops an output sequence with a sampling rate that is L times larger than that of the input sequence x[n]•Block-diagram representation][nxuLx[n]][nxuCopyright © 2001, S. K. Mitra3Up-SamplerUp-Sampler•Up-sampling operation is implemented by inserting equidistant zero-valued samples between two consecutive samples of x[n] •Input-output relation1Lotherwise,0,2,,0],/[][LLnLnxnxuCopyright © 2001, S. K. Mitra4Up-SamplerUp-Sampler•Figure below shows the up-sampling by a factor of 3 of a sinusoidal sequence with a frequency of 0.12 Hz obtained using Program 10_10 10 20 30 40 50-1-0.500.51Input SequenceTime index nAmplitude0 10 20 30 40 50-1-0.500.51Output sequence up-sampled by 3Time index nAmplitudeCopyright © 2001, S. K. Mitra5Up-SamplerUp-Sampler•In practice, the zero-valued samples inserted by the up-sampler are replaced with appropriate nonzero values using some type of filtering process •Process is called interpolationinterpolation and will be discussed laterCopyright © 2001, S. K. Mitra6Down-SamplerDown-SamplerTime-Domain CharacterizationTime-Domain Characterization•An down-sampler with a down-sampling down-sampling factorfactor M, where M is a positive integer, develops an output sequence y[n] with a sampling rate that is (1/M)-th of that of the input sequence x[n]•Block-diagram representationMx[n]y[n]Copyright © 2001, S. K. Mitra7Down-SamplerDown-Sampler•Down-sampling operation is implemented by keeping every M-th sample of x[n] and removing in-between samples to generate y[n]•Input-output relation y[n] = x[nM]1MCopyright © 2001, S. K. Mitra8Down-SamplerDown-Sampler•Figure below shows the down-sampling by a factor of 3 of a sinusoidal sequence of frequency 0.042 Hz obtained using Program 10_20 10 20 30 40 50-1-0.500.51Input SequenceTime index nAmplitude0 10 20 30 40 50-1-0.500.51Output sequence down-sampled by 3AmplitudeTime index nCopyright © 2001, S. K. Mitra9Basic Sampling Rate Basic Sampling Rate Alteration DevicesAlteration Devices•Sampling periods have not been explicitly shown in the block-diagram representations of the up-sampler and the down-sampler •This is for simplicity and the fact that the mathematical theory of multirate systemsmathematical theory of multirate systems can be understood without bringing the sampling period T or the sampling frequency into the pictureTFCopyright © 2001, S. K. Mitra10Down-SamplerDown-Sampler•Figure below shows explicitly the time-dimensions for the down-samplerM)(][ nMTxnya)(][ nTxnxaInput sampling frequencyTFT1Output sampling frequency'1'TMFFTTCopyright © 2001, S. K. Mitra11Up-SamplerUp-Sampler•Figure below shows explicitly the time-dimensions for the up-samplerInput sampling frequencyTFT1otherwise0,2,,0),/( LLnLnTxaL)(][ nTxnxay[n]Output sampling frequency'1'TLFFTTCopyright © 2001, S. K. Mitra12Basic Sampling Rate Basic Sampling Rate Alteration DevicesAlteration Devices•The up-samplerup-sampler and the down-samplerdown-sampler are linearlinear but time-varying discrete-time time-varying discrete-time systemssystems•We illustrate the time-varying property of a down-sampler•The time-varying property of an up-sampler can be proved in a similar mannerCopyright © 2001, S. K. Mitra13Basic Sampling Rate Basic Sampling Rate Alteration DevicesAlteration Devices•Consider a factor-of-M down-sampler defined by•Its output for an input is then given by•From the input-output relation of the down-sampler we obtainy[n] = x[nM]][1ny][][01nnxnx ][][][011nMnxMnxny )]([][00nnMxnny ][][10nyMnMnx Copyright © 2001, S. K. Mitra14Up-SamplerUp-SamplerFrequency-Domain CharacterizationFrequency-Domain Characterization•Consider first a factor-of-2 up-sampler whose input-output relation in the time-domain is given byotherwise,,,,],/[][04202 nnxnxuCopyright © 2001, S. K. Mitra15Up-SamplerUp-Sampler•In terms of the z-transform, the input-output relation is then given byeven]/[][)(nnnnnuuznxznxzX 22 2[ ] ( )mmx m z X z  Copyright © 2001, S. K. Mitra16Up-SamplerUp-Sampler•In a similar manner, we can show that for a factor-of-factor-of-LL up-sampler up-sampler•On the unit circle, for , the input-output relation is given by)()(LuzXzX jez )()(LjjueXeXCopyright © 2001, S. K. Mitra17Up-SamplerUp-Sampler•Figure below shows the relation between and for L = 2 in the case of a typical sequence x[n])(jeX)(jueXCopyright © 2001, S. K. Mitra18Up-SamplerUp-Sampler•As can be seen, a factor-of-2 sampling rate expansion leads to a compression of by a factor of 2 and a 2-fold repetition in the baseband [0, 2]•This process is called imagingimaging as we get an additional “image” of the input spectrum)(jeXCopyright © 2001, S. K. Mitra19Up-SamplerUp-Sampler•Similarly in the case of a factor-of-L sampling rate expansion, there will be additional images of