1. p. 1 DSP-II 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/

2. DSP-II p. 2 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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

3. DSP-II p. 3 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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

4. DSP-II p. 4 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Review of Multi-rate Systems 1/10 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... N u[0], u[N], u[2N]... u[0],u[1],u[2]... N u[0],0,..0,u[1],0,…,0,u[2]...u[0], u[1], u[2],...

5. DSP-II p. 5 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Review of Multi-rate Systems 2/10 Z-transform (frequency domain) analysis of expander `expansion in time domain ~ compression in frequency domain’ expander mostly followed by `interpolation filter’ to remove images (and `interpolate the zeros) N u[0],0,..0,u[1],0,…,0,u[2]...u[0], u[1], u[2],... N 3 `images’

6. DSP-II p. 6 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Review of Multi-rate Systems 3/10 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 N N u[0], u[N], u[2N]... u[0],u[1],u[2]... i=0i=2i=1 3

7. DSP-II p. 7 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Review of Multi-rate Systems 4/10 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: N

8. DSP-II p. 8 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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) subband processing 3 H1(z) subband processing 3 H2(z) subband processing 3 H3(z) 3 3 3 3 subband processing 3 H4(z) IN G1(z) G2(z) G3(z) G4(z) + OUT

9. DSP-II p. 9 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Review of Multi-rate Systems 5/10 Interconnection of multi-rate building blocks : identities also hold if all decimators are replaced by expanders N x a N x a = = = N + u2[k] N x u1[k] N + N u2[k] u1[k] N x N u2[k] u1[k]

10. DSP-II p. 10 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Review of Multi-rate Systems 6/10 `Noble identities’ (I) : (only for rational functions) Example : N=2 h[0],h[1],0,0,0,… =NN u[k] y[k]

11. DSP-II p. 11 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Review of Multi-rate Systems 7/10 `Noble identities’ (II) : (only for rational functions) Example : N=2 h[0],h[1],0,0,0,… =NN u[k] y[k]

12. DSP-II p. 12 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Review of Multi-rate Systems 8/10 Application of `noble identities : efficient multi-rate filter implementations through… Polyphase decomposition: example : (2-fold decomposition) example : (3-fold decomposition) general: (N-fold decomposition)

13. DSP-II p. 13 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Review of Multi-rate Systems 9/10 Polyphase decomposition: example : Efficient implementation of a decimation filter i.e. all filter operations performed at the lowest rate u[k] 2 + H(z) u[k] 2 + = 2

14. DSP-II p. 14 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Review of Multi-rate Systems 10/10 Polyphase decomposition: example : Efficient implementation of an interpolation filter i.e. all filter operations performed at the lowest rate = u[k] 2 + H(z) u[k] 2 + 2

15. DSP-II p. 15 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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 H0(z) subband processing 3 H1(z) subband processing 3 H2(z) 3 3 3 3 subband processing 3 H3(z) IN F0(z) F1(z) F2(z) F3(z) + OUT

16. DSP-II p. 16 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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 : 4 4 4 4 + u[k-3] 4 4 4 4 u[k]

17. DSP-II p. 17 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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 H0(z) H1(z) 2 2 u[k]2 2 F0(z) F1(z) + y[k]

18. DSP-II p. 18 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Perfect Reconstruction : 2-Channel Case Requirement for `alias-free’ filter bank : 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) !!!! Requirement for `perfect reconstruction’ filter bank (= alias-free + distortion-free): i) ii) H0(z) H1(z) 2 2 u[k]2 2 F0(z) F1(z) + y[k]

19. DSP-II p. 19 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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) H0(z) H1(z) 2 2 u[k]2 2 F0(z) F1(z) + y[k]

20. DSP-II p. 20 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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. H0(z) H1(z) 2 2 u[k]2 2 F0(z) F1(z) + y[k] HoH1

21. DSP-II p. 21 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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 : H0(z) H1(z) 2 2 u[k]2 2 F0(z) F1(z) + y[k] This is already pretty complicated…

22. DSP-II p. 22 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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 H2(z) H3(z) 4 4 4 4 F2(z) F3(z) y[k] H0(z) H1(z) 4 4 u[k] 4 4 F0(z) F1(z) + Sigh !!…

23. DSP-II p. 23 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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) 4 4 4 4 + u[k-3] u[k] 4 4 4 4 Do not continue until you understand how formulae correspond to block scheme!

24. DSP-II p. 24 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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 4 4 4 4 + u[k-3] u[k] 4 4 4 4

25. DSP-II p. 25 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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-> 4 4 4 4 + u[k-3] u[k] 4 4 4 4

26. DSP-II p. 26 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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 + u[k] 4 4 4 4 4 4 4 4 4 + T(z)*u[k-3] 4 4 4 u[k]

27. DSP-II p. 27 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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 : 4 4 4 4 + u[k-3] u[k] 4 4 4 4

28. DSP-II p. 28 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Perfect Reconstruction : M-Channel Case For conciseness, will use this from now on : - Procedure: 1. Design all analysis filters (see 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… 4 4 4 4 + u[k-3] u[k] 4 4 4 4

29. DSP-II p. 29 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Perfect Reconstruction : M-Channel Case PS: Inversion of matrix transfer functions ?… –The inverse of a scalar (i.e. 1-by-1 matrix) FIR transfer function is always IIR (except for contrived examples) – The inverse of an N-by-N (N>1) FIR transfer function can be FIR

30. DSP-II p. 30 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Perfect Reconstruction : M-Channel Case PS: Inversion of matrix transfer functions ?… Compare this to inversion of integers and integer matrices: –The inverse of an integer is always non-integer (except for `E=1’) – The inverse of an N-by-N (N>1) integer matrix can be integer

31. DSP-II p. 31 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Perfect Reconstruction : M-Channel Case Question: How can we find polynomial (FIR) matrices E(z) that have a FIR inverse? Answer: `Unimodular’ matrices (=matrices with 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.)

32. DSP-II p. 32 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks Perfect Reconstruction : M-Channel Case Question: Can we avoid direct inversion, e.g. through the usage FIR E(z) matrices with additional `special properties’ ? (compare with (real) orthogonal or (complex) unitary matrices, where inverse is equal to (hermitian) transpose) Answer: YES, `paraunitary’ matrices (=special class of FIR matrices with FIR inverse) See next slides…. Will focus on paraunitary E(z) leading to paraunitary PR filter banks

33. DSP-II p. 33 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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

34. DSP-II p. 34 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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

35. DSP-II p. 35 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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

36. DSP-II p. 36 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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) 4 4 4 4 + u[k-3] u[k] 4 4 4 4

37. DSP-II p. 37 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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-3) paraunitary

38. DSP-II p. 38 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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) (see lecture 3 !!)

39. DSP-II p. 39 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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)

40. DSP-II p. 40 Version 2006-2007 Lecture-6 Maximally Decimated Filter Banks 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 FIR 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.