MODULE-14 - MODULE 14 MULTI-RATE PROCESSING AND SUBBAND...

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Unformatted text preview: MODULE 14 MULTI-RATE PROCESSING AND SUBBAND FILTER BANKS Sampling Rate Conversion Narrowband Fourier Transform Short-Time Fourier Transform and Spectrogram Subband Decompositions and Wavelets Page 14.1 INDEX Multi-Rate Processing Decimation Frequency-Domain Analysis of Decimator Interpolation Frequency Domain Analysis of Interpolation Sampling Rate Conversion Narrowband Fourier Transform Short-Time Fourier Transform and Spectrogram Window Function Selection Discrete STFT (D-STFT) Sampled D-STFT Subband Decomposition Analysis-Synthesis Filter Banks Subband Design Two-Band 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. MULTI-RATE 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 sub-sampler or down-sampler D where D y ( n ) w ( n ) takes every D th sample from w ( n ): y ( n ) = w ( nD ). Page 14.3 Frequency-Domain Analysis of Decimator index Clearly Y ( e j ) = - = n y ( n ) e-j n = - = n w ( nD ) e-j n - = m w ( m ) e-j m / D !! Instead, Y ( e j ) = - = m s ( m ) w ( m ) e-j 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 ) e-j ( - 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 Pre-filtered result: ) X ( e j H ( e j ) - A- c c Decimated signal (showing aliasing ): - A /D Y ( e j ) c D c-D Page 14.6 Anti-Aliasing Filter Bandwidth index If the cutoff frequency of the anti-aliasing 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.

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MODULE-14 - MODULE 14 MULTI-RATE PROCESSING AND SUBBAND...

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