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Unformatted text preview: MODULE 14 MULTIRATE PROCESSING AND SUBBAND FILTER BANKS Sampling Rate Conversion Narrowband Fourier Transform ShortTime Fourier Transform and Spectrogram Subband Decompositions and Wavelets Page 14.1 INDEX MultiRate Processing Decimation FrequencyDomain Analysis of Decimator Interpolation Frequency Domain Analysis of Interpolation Sampling Rate Conversion Narrowband Fourier Transform ShortTime Fourier Transform and Spectrogram Window Function Selection Discrete STFT (DSTFT) Sampled DSTFT Subband Decomposition AnalysisSynthesis Filter Banks Subband Design TwoBand Decomposition Quadrature Mirror Filters (QMFs) Perfect Reconstruction QMFs Conjugate Quadrature Filters (CQFs) Perfect Reconstruction CQFs CQF PR Design Subband Tree Structures Linear Versus Octave Bands Relationship to Wavelets MAIN INDEX Page 14.2 14. MULTIRATE PROCESSING AND SUBBAND FILTER BANKS index READ : Sections 4.6 4.7, 10.0  10.5 of Oppenheim & Schafer. Ref : Crochiere/Rabiner, Multirate Digital Signal Processing. Decimation Let D 1 be an integer rational D will be considered later. The process of reducing the sampling rate of a sampled signal by a factor D is termed decimation . A decimator is depicted: H ( e j ) LPF x ( n ) y ( n ) w ( n ) Decimator subsampler or downsampler D where D y ( n ) w ( n ) takes every D th sample from w ( n ): y ( n ) = w ( nD ). Page 14.3 FrequencyDomain Analysis of Decimator index Clearly Y ( e j ) =  = n y ( n ) ej n =  = n w ( nD ) ej n  = m w ( m ) ej m / D !! Instead, Y ( e j ) =  = m s ( m ) w ( m ) ej m / D where s ( m ) = = else ; ; 1 kD m =  = k ( m kD ) But any periodic s ( m ) has a discrete Fourier series : s ( m ) = 1 D  = 1 D p e j 2 pm / D so Y ( e j ) = 1 D  = 1 D p  = m w ( m ) ej (  2 p ) m / D = 1 D  = 1 D p W [  D p j e 2 ] Page 14.4 Spectrum Expansi on index Hence: Y ( e j ) = 1 D  = 1 D p H [  D p j e 2 ] X [  D p j e 2 ] Decimation stretches the spectrum of the signal by a factor of D  not surprising, since we are sampling at a lower rate. Thus, the unit D is sometimes called an expander . Aliasing must be avoided somehow  the role of the LPF H ( e j ). Page 14.5 Example of Decimation index Digital signal spectrum:  A X ( e j ) LPF (ideal LPF dotted): )  1 H ( e j  c c Prefiltered result: ) X ( e j H ( e j )  A c c Decimated signal (showing aliasing ):  A /D Y ( e j ) c D cD Page 14.6 AntiAliasing Filter Bandwidth index If the cutoff frequency of the antialiasing LPF is not chosen properly, severe aliasing can occur....
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This note was uploaded on 05/06/2010 for the course ECE 539 taught by Professor Alanc.bovik during the Spring '04 term at University of New Brunswick.
 Spring '04
 AlanC.Bovik
 Digital Signal Processing, Frequency, Signal Processing

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