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Digital Signal Processing II Lecture 6: Maximally Decimated Filter Banks

Digital Signal Processing II Lecture 6: Maximally Decimated 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 Applications

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Digital Signal Processing II Lecture 6: Maximally Decimated Filter Banks

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  1. Digital Signal Processing IILecture 6: Maximally Decimated 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 • Applications • Intro perfect reconstruction filter banks (PR FBs) : Maximally decimated FBs • Multi-rate systems review • 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 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  3. PART-II : Filter Banks LECTURE-6 : Maximally decimated FBs • `Interludium’: Review of multi-rate systems • Perfect reconstruction (PR) FBs • 2-channel case • M-channel case • `Interludium’: Paraconjugation & paraunitary functions • Paraunitary PR FBs Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  4. N u[0], u[N], u[2N]... u[0],u[1],u[2]... u[0], u[1], u[2],... N u[0],0,..0,u[1],0,…,0,u[2]... Review of Multi-rate Systems (1) • Decimation : decimator (downsampler) example : u[k]: 1,2,3,4,5,6,7,8,9,… 2-fold downsampling: 1,3,5,7,9,... • Interpolation : expander (upsampler) example : u[k]: 1,2,3,4,5,6,7,8,9,… 2-fold upsampling: 1,0,2,0,3,0,4,0,5,0... Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  5. 3 N N `images’ u[0], u[1], u[2],... u[0],0,..0,u[1],0,…,0,u[2]... Review of Multi-rate Systems (2) • Z-transform (frequency domain) analysis of expander `expansion in time domain ~ compression in frequency domain’ expander mostly followed by interpolation filter to remove all images Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  6. N N u[0], u[N], u[2N]... u[0],u[1],u[2]... Review of Multi-rate Systems (3) • Z-transform (frequency domain) analysis of decimator decimation introduces ALIASING if input signal occupies frequency band larger than , for hence decimation mostly preceded by anti-aliasing (decimation) filter i=2 i=0 i=1 3 Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  7. N Review of Multi-rate Systems (4) • Z-transform analysis of decimator (continued) - Note that is periodic with period while is periodic with period the summation with i=0…N-1 restores the periodicity with period ! - Example: Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  8. 3 subband processing 3 G1(z) H1(z) OUT IN 3 subband processing 3 G2(z) H2(z) + 3 subband processing 3 G3(z) H3(z) 3 subband processing 3 G4(z) H4(z) PS: Filter bank set-up revisited - analysis filters Hi(z) are also decimation (anti-aliasing) filters, to avoid aliased contributions in subband signals - synthesis filters Gi(z) are also interpolation filters, to remove images after expanders (upsampling) Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  9. u1[k] N N N N + u2[k] N = N x x a a Review of Multi-rate Systems (5) • Interconnection of multi-rate building blocks : identities also hold if all decimators are replaced by expanders u1[k] N = + u2[k] u1[k] N = x u1[k] x u2[k] u2[k] Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  10. u[k] y[k] u[k] y[k] = N N Review of Multi-rate Systems (6) • `Noble identities’ (I) : (only for rational functions) Example : N=2 h[0],h[1],0,0,0,… Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  11. Review of Multi-rate Systems (7) • `Noble identities’ (II) : (only for rational functions) Example : N=2 h[0],h[1],0,0,0,… u[k] y[k] u[k] y[k] N = N Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  12. Review of Multi-rate Systems (8) Application of `noble identities : efficient multi-rate filter implementations through… • Polyphase decomposition: example :(2-fold decomposition) example :(3-fold decomposition) general: (N-fold decomposition) Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  13. 2 2 2 Review of Multi-rate Systems (9) • Polyphase decomposition: example : Efficient implementation of a decimation filter i.e. all filter operations performed at the lowest rate H(z) u[k] + u[k] = + Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  14. u[k] H(z) 2 2 2 + Review of Multi-rate Systems (10) • Polyphase decomposition: example : Efficient implementation of an interpolation filter i.e. all filter operations performed at the lowest rate u[k] = + Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  15. 3 OUT 3 + 3 3 Refresh General `subband processing’ set-up: - 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 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  16. + 4 4 4 u[k] 4 u[k-3] 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. PS: Perfect reconstruction property as such is easily satisfied, if there aren’t any (analysis) filter specs, e.g. (see Lecture-5) …but this is not very useful/practical. Stringent filter specs. necessary for subband coding, etc. This lecture : Maximally decimated FB’s : Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  17. y[k] u[k] 2 2 F0(z) H0(z) + 2 2 F1(z) H1(z) Perfect Reconstruction : 2-Channel Case It is proved that...(try it!) • U(-z) represents aliased signals, hence the `alias transfer function’ A(z) should ideally be zero • T(z) is referred to as `distortion function’ (amplitude & phase distortion). For perfect reconstruction, T(z) should ideally be a pure delay Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  18. y[k] u[k] 2 2 F0(z) H0(z) + 2 2 F1(z) H1(z) Perfect Reconstruction : 2-Channel Case • Requirement for `alias-free’ filter bank : • Requirement for `perfect reconstruction’ filter bank (= alias-free + distortion-free): i) ii) • PS: if A(z)=0, then Y(z)=T(z).U(z), hence the complete filter bank behaves as a linear time invariant (LTI) system (despite up- & downsampling). Remarkable !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  19. y[k] u[k] 2 2 F0(z) H0(z) + 2 2 F1(z) H1(z) Perfect Reconstruction : 2-Channel Case • An initial choice is ….. : so that For the real coefficient case, i.e. which means the amplitude response of H1 is the mirror image of the amplitude response of Ho with respect to the quadrature frequency hence the name `quadrature mirror filter’ (QMF) Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  20. y[k] u[k] 2 2 F0(z) H0(z) + 2 2 F1(z) H1(z) Ho H1 Perfect Reconstruction : 2-Channel Case) `quadrature mirror filter’ (QMF) : hence if Ho (=Fo) is designed to be a good lowpass filter, then H1 (=-F1) is a good high-pass filter. Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  21. y[k] u[k] 2 2 F0(z) H0(z) + 2 2 F1(z) H1(z) Perfect Reconstruction : 2-Channel Case • A 2nd (better) choice is: [Smith & Barnwell 1984] [Mintzer 1985] i) so that (alias cancellation) ii) `power symmetric’ Ho(z) (real coefficients case) iii) so that (distortion function) ignore the details! This is a so-called`paraunitary’ perfect reconstruction bank (see below), based on a lossless system Ho,H1 : This is already pretty complicated… Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  22. 4 4 F0(z) H0(z) u[k] y[k] 4 4 F1(z) H1(z) + 4 4 F2(z) H2(z) 4 4 F3(z) H3(z) Perfect Reconstruction : M-Channel Case It is proved that...(try it!) • 2nd term represents aliased signals, hence all `alias transfer functions’ Al(z) should ideally be zero(for all l ) • H(z) is referred to as `distortion function’ (amplitude & phase distortion). For perfect reconstruction, H(z) should ideally be a pure delay Sigh !!… Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  23. 4 4 4 4 + 4 4 4 4 Perfect Reconstruction : M-Channel Case • A simpler analysis results from a polyphase description : i-th row of E(z) has polyphase components of Hi(z) i-th column of R(z) has polyphase components of Fi(z) u[k] u[k-3] Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks Do not continue until you understand how formulae correspond to block scheme!

  24. u[k] u[k-3] 4 4 4 4 + 4 4 4 4 Perfect Reconstruction : M-Channel Case • with the `noble identities’, this is equivalent to: Necessary & sufficient conditions for i) alias cancellation ii) perfect reconstruction are then derived, based on the product Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  25. u[k] u[k-3] 4 4 4 4 + 4 4 4 4 Perfect Reconstruction : M-Channel Case Necessary & sufficient condition for alias-free FB is…: a pseudo-circulant matrix is a circulant matrix with the additional feature that elements below the main diagonal are multiplied by 1/z, i.e. ..and first row of R(z).E(z) are polyphase cmpnts of `distortion function’ T(z) read on-> Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  26. u[k] 4 4 4 4 + + 4 4 4 4 Perfect Reconstruction : M-Channel Case PS: This can be explained as follows: first, previous block scheme is equivalent to (cfr. Noble identities) then (iff R.E is pseudo-circ.)… so that finally.. 4 4 4 4 T(z)*u[k-3] 4 4 u[k] 4 4 Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  27. u[k-3] u[k] 4 4 4 4 + 4 4 4 4 Perfect Reconstruction : M-Channel Case Hence necessary & sufficient condition for PR (where T(z)=pure delay): In is nxn identity matrix, r is arbitrary (Obvious) example : Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  28. u[k-3] u[k] 4 4 4 4 + 4 4 4 4 Perfect Reconstruction : M-Channel Case For conciseness, will use this from now on : - Procedure: 1. Design all analysis filters (=DSP-II/Part-I). 2. This determines E(z) (=polyphase matrix). 3. Assuming E(z) can be inverted (?), choose synthesis filters - Example : DFT/IDFT Filter bank (Lecture-5) : E(z)=F, R(z)=F^-1 - FIR E(z) generally leads to IIR R(z), where stability is a concern… Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  29. Perfect Reconstruction : M-Channel Case Question: Can we find polynomial (FIR) matrices E(z) that have a FIR inverse? Answer: YES, so-called `unimodular’ matrices (where determinant=constant*z^-d) Example: where the Ei’s are constant (=not a function of z) invertible matrices procedure : optimize Ei’s to obtain analysis filter specs (ripple, etc.) Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  30. Perfect Reconstruction : M-Channel Case Question: Can we avoid direct inversion, e.g. through the usage of (again FIR) E(z) matrices with additional `special properties’ ? (compare with (real) orthogonal or (complex) unitary matrices) Answer: YES, `paraunitary’ matrices (=special class of FIR matrices with FIR inverse) See next slides…. In the sequel, will focus on paraunitary E(z) leading to paraunitary PR filter banks Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  31. Paraunitary PR Filter Banks Interludium : `PARACONJUGATION’ • For a scalar transfer function H(z), paraconjugate is i.e it is obtained from H(z) by - replacing z by 1/z - replacing each coefficient by its complex conjugate Example : On the unit circle, paraconjugation corresponds to complex conjugation paraconjugation = `analytic extension’ of unit-circle conjugation Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  32. Paraunitary PR Filter Banks Interludium : `PARACONJUGATION’ • For a matrix transfer function H(z), paraconjugate is i.e it is obtained from H(z) by - transposition - replacing z by 1/z - replacing each coefficient by is complex conjugate Example : On the unit circle, paraconjugation corresponds to transpose conjugation paraconjugation = `analytic extension’ of unit-circle transpose conjugation Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  33. Paraunitary PR Filter Banks Interludium : `PARAUNITARY matrix transfer functions’ • Matrix transfer function H(z), is paraunitary if (possibly up to a scalar) For a square matrix function A paraunitary matrix is unitary on the unit circle paraunitary = `analytic extension’ of unit-circle unitary. PS: if H1(z) and H2(z) are paraunitary, then H1(z).H2(z) is paraunitary Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  34. u[k-3] u[k] 4 4 4 4 + 4 4 4 4 Paraunitary PR Filter Banks - If E(z) is paraunitary hence perfect reconstruction is obtained with If E(z) is FIR, then R(z) is also FIR !! (cfr definition paraconjugation) Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  35. Paraunitary PR Filter Banks • Example: paraunitary FIR E(z) with FIR inverse R(z) where the Ei’s are constant unitary matrices Procedure : optimize unitary Ei’s to obtain analysis filter specs. ps: 2-channel case with real coefficients, then hence optimize phi’s... (=lossless lattice, see lecture-2) paraunitary Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  36. Paraunitary PR Filter Banks Properties of paraunitary PR filter banks: • If polyphase matrix E(z) (and hence E(z^N)) is paranunitary, and then vector transfer function H(z) (=all analysis filters) is paraunitary • If vector transfer function H(z) is paraunitary, then its components are power complementary (lossless 1-input/N-output system) Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  37. Paraunitary PR Filter Banks Properties of paraunitary PR filter banks (continued): • Synthesis filter coefficients are obtained by conjugating the analysis filter coefficients + reversing the order : • Magnitude response of synthesis filter Fk is the same as magnitude response of corresponding analysis filter Hk: • Analysis filters are power complementary (cfr. supra) • Synthesis filters are power complementary • example: DFT/IDFT bank, Lecture-5 example: 2-channel case, page 21 • Great properties/designs.... (proofs omitted) Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

  38. Conclusions • Have derived general conditions for perfect reconstruction, based on polyphase matrices for analysis/synthesis bank • Seen example of general PR filter bank design : Paraunitary PR FBs • Sequel = other (better) PR structures Lecture 7: Modulated filter banks Lecture 8: Oversampled filter banks, etc.. • Reference: `Multirate Systems & Filter Banks’ , P.P. Vaidyanathan Prentice Hall 1993. Version 2005-2006 Lecture-6 Maximally Decimated Filter Banks

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