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Digital Signal Processing II Lecture 7: Modulated Filter Banks

Digital Signal Processing II Lecture 7: Modulated Filter Banks. Marc Moonen Dept. E.E./ESAT, K.U.Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen/. Part-II : Filter Banks. : Preliminaries Filter bank set-up and applications

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Digital Signal Processing II Lecture 7: Modulated Filter Banks

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  1. Digital Signal Processing IILecture 7: Modulated Filter Banks Marc Moonen Dept. E.E./ESAT, K.U.Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen/

  2. Part-II : Filter Banks : Preliminaries • Filter bank set-up and applications • `Perfect reconstruction’ problem + 1st example (DFT/IDFT) • Multi-rate systems review (10 slides) : Maximally decimated FBs • Perfect reconstruction filter banks (PR FBs) • Paraunitary PR FBs : Modulated FBs • DFT-modulated FBs • Cosine-modulated FBs : Special Topics • Non-uniform FBs & Wavelets • Oversampled DFT-modulated FBs • Frequency domain filtering Lecture-5 Lecture-6 Lecture-7 Lecture-8 Version 2007-2008 Lecture-7 Modulated Filter Banks

  3. 3 OUT 3 + 3 3 Refresh General `subband processing’ set-up (Lecture-5) : - analysis bank+ synthesis bank - multi-rate structure: down-sampling after analysis, up-sampling for synthesis - aliasing vs. ``perfect reconstruction” - applications: coding, (adaptive) filtering, transmultiplexers - PS: subband processing ignored in filter bank design subband processing 3 F0(z) H0(z) IN subband processing 3 F1(z) H1(z) subband processing 3 F2(z) H2(z) subband processing 3 F3(z) H3(z) Version 2007-2008 Lecture-7 Modulated Filter Banks

  4. u[k-3] u[k] 4 4 4 4 + 4 4 4 4 Refresh Two design issues : - filter specifications, e.g. stopband attenuation, passband ripple, transition band, etc. (for each (analysis) filter!) - perfect reconstruction property (Lecture-6). PS: Lecture 6/7 = maximally decimated FB’s = Version 2007-2008 Lecture-7 Modulated Filter Banks

  5. Introduction -All design procedures so far involve monitoring of characteristics (passband ripple, stopband suppression,…) of all (analysis) filters, which may be tedious. -Design complexity may be reduced through usage of `uniform’ and `modulated’ filter banks. • DFT-modulated FBs • Cosine-modulated FBs Version 2007-2008 Lecture-7 Modulated Filter Banks

  6. H0(z) IN H1(z) H2(z) H3(z) DFT-Modulated Filter Banks Uniform versus non-uniform (analysis) filter bank: non-uniform: e.g. for speech & audio applications (cfr. human hearing) example: wavelet filter banks (next lecture) N-Channel uniform filter bank: i.e. frequency responses are uniformly shifted over the unit circle Ho(z)= `prototype’ filter (=only filter that has to be designed) Time domain equivalent is: H0 H1 H2 H3 uniform H0 H3 non-uniform H1 H2 Version 2007-2008 Lecture-7 Modulated Filter Banks

  7. H0(z) u[k] H1(z) H2(z) H3(z) i.e. DFT-Modulated Filter Banks Uniform filter banks can be realized cheaply based on polyphase decompositions + DFT(FFT) hence named `DFT modulated FBs’ 1. Analysis FB If (polyphase decomposition) then Version 2007-2008 Lecture-7 Modulated Filter Banks

  8. i.e. DFT-Modulated Filter Banks where F is NxN DFT-matrix Version 2007-2008 Lecture-7 Modulated Filter Banks

  9. u[k] i.e. DFT-Modulated Filter Banks conclusion: economy in… - implementation complexity: N filters for the price of 1, plus DFT (=FFT) ! - design complexity: design `prototype’ Ho(z), then other Hi(z)’s are automatically `co-designed’ (same passband ripple, etc…) ! Version 2007-2008 Lecture-7 Modulated Filter Banks

  10. u[k] Ho(z) H1(z) DFT-Modulated Filter Banks • Special case: DFT-filter bank, if all Ei(z)=1 Version 2007-2008 Lecture-7 Modulated Filter Banks

  11. u[k] Ho(z) H1(z) DFT-Modulated Filter Banks • PS: with F instead of F* (see Lecture-5), only filter ordering is changed Version 2007-2008 Lecture-7 Modulated Filter Banks

  12. u[k] u[k] 4 4 4 4 4 = 4 4 4 DFT-Modulated Filter Banks • Uniform DFT-modulated analysis FB +decimation (M=N) = efficient realization ! Version 2007-2008 Lecture-7 Modulated Filter Banks

  13. y[k] + + + DFT-Modulated Filter Banks 2. Synthesis FB phase shift added for convenience Version 2007-2008 Lecture-7 Modulated Filter Banks

  14. i.e. DFT-Modulated Filter Banks where F is NxN DFT-matrix Version 2007-2008 Lecture-7 Modulated Filter Banks

  15. + + + i.e. DFT-Modulated Filter Banks y[k] Version 2007-2008 Lecture-7 Modulated Filter Banks

  16. 4 4 4 4 4 4 4 + + + + + + 4 DFT-Modulated Filter Banks • Expansion (M=N) + uniform DFT-modulated synthesis FB : y[k] = = efficient realization ! y[k] Version 2007-2008 Lecture-7 Modulated Filter Banks

  17. 4 u[k] 4 4 y[k] + + + 4 4 4 4 4 DFT-Modulated Filter Banks Perfect reconstruction (PR) revisited : maximally decimated (M=N) uniform DFT-modulated analysis & synthesis… - Procedure: 1. Design prototype analysis filter Ho(z) (=DSP-II/Part-I). 2. This determines Ei(z) (=polyphase components). 3. Assuming Ei(z) can be inverted (?), choose synthesis filters Version 2007-2008 Lecture-7 Modulated Filter Banks

  18. 4 u[k] 4 4 y[k] + + + 4 4 4 4 4 DFT-Modulated Filter Banks Perfect reconstruction (PR): FIR E(z) generally leads to IIR R(z), where stability is a concern… PR with FIR analysis/synthesis bank (=guaranteed stability), only obtained with trivial choices for Ei(z)’s (next slide) Version 2007-2008 Lecture-7 Modulated Filter Banks

  19. DFT-Modulated Filter Banks • Simple example (1) is , which leads to IDFT/DFT bank (Lecture-5) i.e. Fl(z) has coefficients of Hl(z), but complex conjugated and in reverse order (hence same magnitude response) (remember this?!) • Simple example (2) is , where wi’s are constants, which leads to `windowed’ IDFT/DFT bank, a.k.a. `short- time Fourier transform’ (see Lecture-8) • Question (try to answer): when is maximally decimated PR uniform DFT-modulated FB - FIR (both analysis & synthesis) ? - paraunitary ? Version 2007-2008 Lecture-7 Modulated Filter Banks

  20. DFT-Modulated Filter Banks • Bad news: From this it is seen that the maximally decimated IDFT/DFT filter bank (or trivial modifications thereof) is the only possible uniform DFT-modulated FB that is at the same time... i) maximally decimated ii) perfect reconstruction (PR) iii) FIR (all analysis+synthesis filters) iv) paraunitary • Good news : • Cosine-modulatedPR FIR FB’s • OversampledPR FIR DFT-modulated FB’s (Lecture-8) Version 2007-2008 Lecture-7 Modulated Filter Banks

  21. H0 H1 H2 H3 P0 Ho H1 Cosine-Modulated Filter Banks • Uniform DFT-modulated filter banks: Ho(z) is prototype lowpass filter, cutoff at for N filters • Cosine-modulated filter banks : Po(z) is prototype lowpass filter, cutoff at for N filters Then... etc... Version 2007-2008 Lecture-7 Modulated Filter Banks

  22. Cosine-Modulated Filter Banks • Cosine-modulated filter banks : - if Po(z) is prototype lowpass filter designed with real coefficients po[n], n=0,1,…,L then i.e. `cosine modulation’ (with real coefficients) instead of `exponential modulation’ (for DFT-modulated bank, see page 6) - if Po(z) is `good’ lowpass filter, then Hk(z)’s are `good’ bandpass filters Version 2007-2008 Lecture-7 Modulated Filter Banks

  23. Cosine-Modulated Filter Banks Realization based on polyphase decomposition (analysis): - if Po(z) has 2N-fold polyphase expansion (ps: 2N-fold for N filters!!!) then... u[k] : : Version 2007-2008 Lecture-7 Modulated Filter Banks

  24. Cosine-Modulated Filter Banks Realization based on polyphase decomposition (continued): - if Po(z) has L+1=m.2N taps, and m is even (similar formulas for m odd) (m is the number of taps in each polyphase component) then... With ignore all details here !!!!!!!!!!!!!!! Version 2007-2008 Lecture-7 Modulated Filter Banks

  25. u[k] : : Cosine-Modulated Filter Banks Realization based on polyphase decomposition (continued): - Note that C is NxN DCT-matrix (`Type 4’) hence fast implementation (=fast matrix-vector product) based on fast discrete cosine transform procedure, complexity O(N.logN). - Modulated filter bank gives economy in * design (only prototype Po(z) ) * implementation (prototype + modulation (DCT)) Similar structure for synthesis bank Version 2007-2008 Lecture-7 Modulated Filter Banks

  26. u[k] N u[k] N N N : N N Cosine-Modulated Filter Banks Maximally decimated cosine modulated (analysis) bank : = : Version 2007-2008 Lecture-7 Modulated Filter Banks

  27. Cosine-Modulated Filter Banks Question:How do we obtain Maximal Decimation + FIR + PR + Paraunitariness? Theorem: (proof omitted) -If prototype Po(z) is a real-coefficient (L+1)-taps FIR filter, (L+1)=2N.m for some integer m and po[n]=po[L-n] (linear phase), with polyphase components Ek(z), k=0,1,…2N-1, -then the (FIR) cosine-modulated analysis bank is PARAUNITARY if and only if (for all k) are power complementary, i.e. form a lossless 1 input/2 output system Hence FIR synthesis bank (for PR) can be obtained by paraconjugation !!! = great result… ..this is the hard part… Version 2007-2008 Lecture-7 Modulated Filter Banks

  28. Cosine-Modulated Filter Banks Perfect Reconstruction (continued) Design procedure: Parameterize lossless systems for k=0,1..,N-1 Optimize all parameters in this parametrization so that the prototype Po(z) based on these polyphase components is a linear-phase lowpass filter that satisfies the given specifications Example parameterization: Parameterize lossless systems for k=0,1..,N-1, -> lattice structure (see Part-I), where parameters are rotation angles ..this is the hard part… Version 2007-2008 Lecture-7 Modulated Filter Banks

  29. Cosine-Modulated Filter Banks lossless PS: Linear phase property for po[n] implies that only half of the power complementary pairs have to be designed. The other pairs are then defined by symmetry properties. u[k] p.26 = N : : N : Version 2007-2008 Lecture-7 Modulated Filter Banks

  30. Cosine-Modulated Filter Banks PS: Cosine versus DFT modulation In a maximally decimated cosine-modulated (analysis) filter bank 2 polyphase components of the prototype filter, , actually take the place of only 1 polyphase component in the DFT- modulated case. For paraunitariness (hence FIR-PR) in a cosine-modulated bank, each such pair of polyphase filters should form a power complementary pair, i.e. represent a lossless system. In the DFT-modulated case, imposing paraunitariness is equivalent to imposing losslessness for each polyphase component separately, i.e. each polyphase component should be an `allpass’ transfer function. Allpass functions are always IIR, except for trivial cases (pure delays). Hence all FIR paraunitary DFT-modulated banks (with maximal decimation) are trivial modifications of the DFT bank. provides flexibility for FIR-design Version 2007-2008 Lecture-7 Modulated Filter Banks no FIR-design flexibility

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