Filter Bank Preliminaries and Setup for Efficient Signal Processing

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Dive into the world of filter banks and time-frequency transforms with a comprehensive exploration of analysis filter banks, decimation, and subband processing. Understand the design principles, applications, and implementation of filter banks for optimal signal processing efficiency.

  • Filter Banks
  • Time-Frequency Transforms
  • Signal Processing
  • Subband Processing
  • Decimation
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  1. DSP-CIS Part-IV : Filter Banks & Time-Frequency Transforms Chapter-12 : Filter Bank Preliminaries Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@kuleuven.be www.esat.kuleuven.be/stadius/

  2. Part-IV : Filter Banks & Time-Frequency Transforms Filter Bank Preliminaries Chapter-12 Filter Bank Design Chapter-13 Filter Bank Design (continued) Chapter-14 Time-Frequency Analysis & Scaling Chapter-15 DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 2 / 40

  3. Overview Filter Bank Set-Up Filter Bank Applications Ideal Filter Bank Operation Non-Ideal Filter Banks & Perfect Reconstruction Theory DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 3 / 40

  4. Filter Bank Set-Up What we have in mind is this : subband processing H0(z) OUT IN subband processing H1(z) + subband processing H2(z) subband filters subband processing H3(z) H0 H1 H2 H3 Example with # channels N=4 In practice N can be 1024 or more 2 - Signals split into frequency channels/subbands - Per-channel/per-subband processing - Reconstruction : synthesis of processed signal - Applications : see below (audio coding etc.) - In practice, this is implemented as a multi-rate structure for higher efficiency (see next slides) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 4 / 40

  5. Filter Bank Set-Up Step-1: Analysis filter bank - Collection of N filters (`analysis filters , `decimation filters ) with a common input signal - Ideal (but non-practical) frequency responses = ideal bandpass filters H0 H1 H2 H3 - Typical frequency responses (overlapping, non-overlapping, ) p 2 H1 H2 H3 H0 N=4 H0(z) 2 H0 H1 H2 H3 IN H1(z) 2 H2(z) H0 H1 H2 H3 H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 5 / 40

  6. Filter Bank Set-Up Step-2: Downsamplers/decimators - To increase efficiency, subband sampling rate is reduced by factor D (= Nyquist sampling theorem (for passband signals) ) - Maximally decimated filter banks (=critically downsampled): # subband samples = # fullband samples this sounds like maximum efficiency, but aliasing (see below)! - Oversampled filter banks (=non-critically downsampled): # subband samples > # fullband samples D=N D<N N=4 D=3 3 3 3 3 H0(z) IN H1(z) PS: analysis filters Hn(z) are now also decimation/anti-aliasing filters to avoid aliasing in subband signals after decimation (see Chapter-2) H2(z) H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 6 / 40

  7. Filter Bank Set-Up Step-3: Subband processing - Example : coding (=compression) + (transmission or storage) + decoding - Filter bank design mostly assumes subband processing has `unit transfer function (output signals=input signals), i.e. mostly ignores presence of subband processing N=4 D=3 3 3 3 3 subband processing H0(z) IN subband processing H1(z) subband processing H2(z) subband processing H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 7 / 40

  8. Filter Bank Set-Up Step-4: Upsamplers/Expanders - Restore original fullband sampling rate by D-fold upsampling Step-5: Synthesis filter bank - Upsampling has to be followed by interpolation filtering (to fill the zeroes & remove spectral images, see Chapter-2) - Collection of N filters (`synthesis , `interpolation ) with summed output - Frequency responses : preferably `matched to frequency responses of the analysis filters (see below) F0 F1 F2 F3 2 N=4 D=3 3 3 3 3 3 3 3 3 F0(z) subband processing H0(z) OUT IN F1(z) subband processing H1(z) + F2(z) subband processing H2(z) F3(z) subband processing H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 8 / 40

  9. Filter Bank Set-Up So this is the picture to keep in mind... analysis bank (analysis & anti-aliasing) downsampling/decimation upsampling/expansion synthesis bank (synthesis & interpolation) N=4 D=3 3 3 3 3 3 3 3 3 F0(z) subband processing H0(z) OUT IN F1(z) subband processing H1(z) + F2(z) subband processing H2(z) F3(z) subband processing H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 9 / 40

  10. Filter Bank Set-Up A crucial concept will be Perfect Reconstruction (PR) Assume subband processing does not modify subband signals (e.g. lossless coding/decoding) The overall aim would then be to have PR, i.e. that the output signal is equal to the input signal up to at most a delay: y[k]=u[k-d] But: downsampling introduces aliasing, so achieving PR will be non- trivial N=4 D=3 y[k]=u[k-d]? 3 3 3 3 3 3 3 3 F0(z) output = input H0(z) u[k] F1(z) output = input H1(z) + F2(z) output = input H2(z) F3(z) output = input H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 10 / 40

  11. Overview Filter Bank Set-Up Filter Bank Applications Ideal Filter Bank Operation Non-Ideal Filter Banks & Perfect Reconstruction Theory DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 11 / 40

  12. Filter Bank Applications Noise reduction: (Approach referred to as spectral subtraction ) Assume input signal is a recorded speech signal plus background noise Background noise can be reduced by using a filter bank and applying a gain in each channel (Gi in i-th channel, 0 Gi 1) Gi is a function of the estimated instantaneous signal-to-noise ratio in the i-th channel (noise power measured in speech pauses, etc.) 3 3 3 3 3 3 3 3 F0(z) G0 H0(z) speech + noise speech + noise F1(z) G1 H1(z) + F2(z) G2 H2(z) F3(z) G3 H3(z) Perfect reconstruction if all Gi=1 DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 12 / 40

  13. Filter Bank Applications Subband coding : Coding = Fullband signal split into subbands & downsampled (=analysis filters + decimators) subband signals separately encoded (e.g. subband with smaller energy content encoded with fewer bits) Decoding = reconstruction of subband signals, then fullband signal synthesis (=expanders + synthesis filters) Example : Image coding (e.g. wavelet filter banks) Example : Audio coding e.g. digital compact cassette (DCC), MiniDisc, MPEG, ... Filter bandwidths and bit allocations chosen to further exploit perceptual properties of human hearing (perceptual coding, masking, etc.) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 13 / 40

  14. Filter Bank Applications Subband adaptive filtering : Example : Acoustic echo cancellation Adaptive filter models acoustic echo path, which is challenging (cfr. long & time-varying acoustic impulse responses) Subband filtering = N (simpler) subband modeling problems instead of one (more complicated) fullband modeling problem PR guarantees distortion-free desired near-end speech signal H0(z) 3 ad.filter ad.filter 3 H1(z) H2(z) ad.filter 3 H3(z) 3 ad.filter + F0(z) 3 H0(z) 3 OUT + + F1(z) 3 3 H1(z) + F2(z) 3 H2(z) 3 + F3(z) 3 3 H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 14 / 40

  15. Overview Filter Bank Set-Up Filter Bank Applications Ideal Filter Bank Operation Non-Ideal Filter Banks & Perfect Reconstruction Theory DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 15 / 40

  16. Ideal Filter Bank Operation D= N =4 (*) With ideal analysis/synthesis filters, filter bank operates as follows (1) H0(z) H1(z) H2(z) H3(z) analysis filters p 4 2 IN input signal spectrum p p 4 2 subband processing F0(z) 4 4 H0(z) OUT subband processing F1(z) IN 4 4 H1(z) + subband processing F2(z) 4 4 H2(z) subband processing F3(z) 4 4 H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries (*) Similar figures for other D=N & for oversampled (D<N) case 16 / 40

  17. Ideal Filter Bank Operation With ideal analysis/synthesis filters, filter bank operates as follows (2) H0(z) H1(z) H2(z) H3(z) 2 x1 PS: H0(z) analysis filter lowpass anti-aliasing filter p x1 p 4 2 subband processing F0(z) 4 4 H0(z) OUT subband processing F1(z) IN 4 4 H1(z) + subband processing F2(z) 4 4 H2(z) subband processing F3(z) 4 4 H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 17 / 40

  18. Ideal Filter Bank Operation With ideal analysis/synthesis filters, filter bank operates as follows (3) H0(z) H1(z) H2(z) H3(z) 2 x 1 (ideal subband processing) p p 4 2 x 1 x 1 subband processing F0(z) 4 4 H0(z) OUT subband processing F1(z) IN 4 4 H1(z) + subband processing F2(z) 4 4 H2(z) subband processing F3(z) 4 4 H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 18 / 40

  19. Ideal Filter Bank Operation With ideal analysis/synthesis filters, filter bank operates as follows (4) H0(z) H1(z) H2(z) H3(z) 2 x 1 p p 4 2 x 1 subband processing F0(z) 4 4 H0(z) OUT subband processing F1(z) IN 4 4 H1(z) + subband processing F2(z) 4 4 H2(z) subband processing F3(z) 4 4 H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 19 / 40

  20. Ideal Filter Bank Operation With ideal analysis/synthesis filters, filter bank operates as follows (5) F0(z) F1(z) F2(z) F3(z) PS: F0(z) synthesis filter lowpass interpolation filter 2 x 1 p p 4 2 x 1 (=x1) subband processing F0(z) 4 4 H0(z) OUT subband processing F1(z) IN 4 4 H1(z) + subband processing F2(z) 4 4 H2(z) subband processing F3(z) 4 4 H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 20 / 40

  21. Ideal Filter Bank Operation With ideal analysis/synthesis filters, FB operates as follows (6) H0(z) H1(z) H2(z) H3(z) 2 x2 PS: H1(z) analysis filter bandpass anti-aliasing filter p p 4 2 subband processing F0(z) 4 4 H0(z) x2 OUT subband processing F1(z) IN 4 4 H1(z) + subband processing F2(z) 4 4 H2(z) subband processing F3(z) 4 4 H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 21 / 40

  22. Ideal Filter Bank Operation With ideal analysis/synthesis filters, filter bank operates as follows (7) H0(z) H1(z) H2(z) H3(z) 2 x 2 p p 4 2 3p subband processing F0(z) 4 4 H0(z) x 2 x 2 OUT subband processing F1(z) IN 4 4 H1(z) + subband processing F2(z) 4 4 H2(z) subband processing F3(z) 4 4 H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 22 / 40

  23. Ideal Filter Bank Operation With ideal analysis/synthesis filters, filter bank operates as follows (8) H0(z) H1(z) H2(z) H3(z) 2 IN x 2 p p 4 2 subband processing F0(z) 4 4 H0(z) x 2 OUT subband processing F1(z) IN 4 4 H1(z) + subband processing F2(z) 4 4 H2(z) subband processing F3(z) 4 4 H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 23 / 40

  24. Ideal Filter Bank Operation With ideal analysis/synthesis filters, filter bank operates as follows (9) F0(z) F1(z) F2(z) F3(z) 2 x 2 PS: F1(z) synthesis filter bandpass interpolation filter p p 4 2 subband processing F0(z) 4 4 H0(z) x 2 (=x2) subband processing F1(z) IN 4 4 H1(z) + subband processing F2(z) 4 4 H2(z) subband processing F3(z) 4 4 H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 24 / 40

  25. Ideal Filter Bank Operation With ideal analysis/synthesis filters, filter bank operates as follows (10) H0(z) H1(z) H2(z) H3(z) 2 OUT=IN =Perfect Reconstruction p p 4 2 subband processing F0(z) 4 4 H0(z) OUT subband processing F1(z) IN 4 4 H1(z) + subband processing F2(z) 4 4 H2(z) subband processing F3(z) 4 4 H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries Now try this with non-ideal filters ? 25 / 40

  26. Overview Filter Bank Set-Up Filter Bank Applications Ideal Filter Bank Operation Non-Ideal Filter Banks & Perfect Reconstruction Theory DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 26 / 40

  27. Non-Ideal Filter Bank Operation Question : Can y[k]=u[k-d] be achieved with non-ideal filters i.e. in the presence of aliasing ? Answer : YES !! Perfect Reconstruction Filter Banks (PR-FB) with synthesis bank designed to remove aliasing effects ! D=3 N=4 y[k]=u[k-d]? 3 3 3 3 3 3 3 3 F0(z) output = input H0(z) u[k] F1(z) output = input H1(z) + F2(z) output = input H2(z) F3(z) output = input H3(z) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 27 / 40

  28. Non-Ideal Filter Bank Operation D= N =4 (*) A very simple PR-FB is constructed as follows - Starting point is this 0,0,0,u[0],0,0,0,u[4],0,0,0,... 0,0,u[-1],0,0,0,u[3],0,0,0,0,... Polyphase decomposition ? ? = ?0?4+? 1?1?4+ ? 2?2?4+ ? 3?3?4 ?0?4 4 4 4 4 4 ?0? 1 3 z ? 3(?0?4+? 1?1?4+ ? 2?2?4+ ? 3?3?4) =? 3? ? ? 1?3? 4 4 4 1 2 z z z z + u[k] u[k-3] 2 ? 1?2? 1 z 1 3 ? 1?1? 0,u[-2],0,0,0,u[2],0,0,0,0,0,... u[-3],0,0,0,u[1],0,0,0,0,0,0,... ? 4?1?4 (with lots of aliasing in the subbands!) As y[k]=u[k-d] this can be viewed as a (1st) (maximally decimated)PR-FB All analysis/synthesis filters are seen to be pure delays, hence are not frequency selective (i.e. far from ideal case with ideal bandpass filters, not yet very interesting .) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries (*) Similar figures for other D=N 28 / 40

  29. Non-Ideal Filter Bank Operation - Now insert DFT-matrix (discrete Fourier transform) and its inverse (I-DFT)... 4 4 4 4 4 1 3 z u[k] 4 4 4 F 1 2 z z z z u[k-3] 1 F + 2 1 z 1 3 v FF-1v F.F-1= I as this clearly does not change the input-output relation (hence PR property preserved) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 29 / 40

  30. Non-Ideal Filter Bank Operation - and reverse order of decimators/expanders and DFT- matrices (not done in an efficient implementation!) : 4 4 4 4 4 1 3 z u[k] 4 4 4 F 1 2 1 z z z z u[k-3] F + 2 1 z 1 3 =analysis filter bank =synthesis filter bank This is the `DFT/IDFT filter bank It is a first (or 2nd) example of a (maximally decimated) PR-FB! DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 30 / 40

  31. Non-Ideal Filter Bank Operation What do analysis filters look like? (N-channel case) 3 z H0(z) H1(z) H2(z) : HN-1(z) N=4 W0 W0 W0 : W0 W0 W-1 W-2 : W-(N-1) W0 W-2 W-4 : W-2(N-1) W0 1 ... ... ... W-2(N-1) z-1 z-2 : z-N+1 1 W-(N-1) =1 . u[k] 1 1 z z F N : 2 ... W-(N-1)2 W =e-j2p/N This is seen/known to represent a collection of filters Ho(z),H1(z),..., each of which is a frequency shifted version of Ho(z) : Hn(ejw)= H0(ej(w-n.(2p/N))) H0(z)=1 N.(1+z-1+z-2+...+z-N+1) i.e. the Hn are obtained by uniformly shifting the `prototype Ho over the frequency axis. DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 31 / 40

  32. Non-Ideal Filter Bank Operation H3(z) Ho(z) H2(z) H1(z) The prototype filter Ho(z) is a not-so-great lowpass filter with significant sidelobes. Ho(z) and Hi(z) s are thus far from ideal lowpass/bandpass filters. Synthesis filters are shown to be equal to analysis filters (up to a scaling) N=4 Hence (maximal) decimation introduces significant ALIASING in the decimated subband signals Still, we know this is a PR-FB (see construction previous slides), which means the synthesis filters can apparently restore the aliasing distortion. This is remarkable, it means PR can be achieved even with non-ideal filters! DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 32 / 40

  33. Perfect Reconstruction Theory Now comes the hard part (?) 2-channel case: Simple (maximally decimated, D=N) example to start with N-channel case: Polyphase decomposition based approach DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 33 / 40

  34. Perfect Reconstruction : 2-Channel Case D= N =2 y[k] F0(z) u[k] 2 2 H0(z) + F1(z) 2 2 H1(z) It is proved that...(try it!) 1 1 = + + + ( ) .{ ( ). ( ) ( ) ( )} . ( ) .{ ( ). ( ) ( ) ( )} . ( ) Y z H z F z H z F z U z H z F z H z F z U z 0 0 1 1 0 0 1 1 2 2 ( ) ( ) T z A z U(-z) represents aliased signals (*), hence A(z) is referred to as `alias transfer function T(z) referred to as `distortion function (amplitude & phase distortion) Note that T(z) is also the transfer function obtained after removing the up- and downsampling (up to a scaling) (!) (*) U(-z)z=ejw=U(-ejw)=U(ej(w+p)) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 34 / 40

  35. Perfect Reconstruction : 2-Channel Case y[k] F0(z) u[k] 2 2 H0(z) + F1(z) 2 2 H1(z) Requirement for `alias-free filter bank : = z A ( ) 0 If A(z)=0, then Y(z)=T(z).U(z) hence the complete filter bank behaves as a LTI system (despite/without up- & downsampling)! Requirement for `perfect reconstruction filter bank (= alias-free + distortion-free): T 0 ) ( = z A = z ( ) z DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 35 / 40

  36. Perfect Reconstruction : 2-Channel Case A solution is as follows: (ignore details)[Smith&Barnwell 1984] [Mintzer 1985] i) ) ( ), ( ) ( 1 1 0 z F z H z F = = ( ) H z 0 = = ( ) ... 0 A z so that (alias cancellation) ii) `power symmetric Ho(z) (real coefficients case) 2 2 + ( ) ( ) j j + = ( ) ( ) 1 H e H e 2 2 0 0 = ) 1 k [ ] ( . [ ] h k h L k iii) so that (distortion function) ignore the details! 1 0 = = ( ) ... 1 T z This is a so-called`paraunitary perfect reconstruction bank (see below), based on a lossless system Ho,H1 : 2 2 j j + = ( ) ( ) 1 H e H e 0 1 DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries This is already pretty complicated 36 / 40

  37. Perfect Reconstruction : N-Channel Case D= N D=N=4 F0(z) 4 4 H0(z) u[k] y[k] F1(z) 4 4 H1(z) + F2(z) 4 4 H2(z) F3(z) 4 4 H3(z) It is proved that...(try it!) N-1 N-1 N-1 Y(z)=1 .U(z)+1 Hn(z.Wn).Fn(z)} .U(z.Wn) N.{ Hn(z).Fn(z) } N. { n=0 n=1 n=0 T(z) An(z) 2nd term represents aliased signals, hence all `alias transfer functions An(z) should ideally be zero (for all n=1..N-1) T(z) is referred to as `distortion function (amplitude & phase distortion). For perfect reconstruction, T(z) should be a pure delay Sigh !! Too Complicated!!... DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 37 / 40

  38. Perfect Reconstruction Theory D= N Simpler formulation is obtained with polyphase decompositions: D=N=4 4 4 4 4 4 1 3 z u[k] 4 4 4 1 2 y[k] z z z z 4 R ( ) z E(z4) + 2 1 z 1 n-th row of E(z) has N-fold (=D-fold) polyphase components of Hn(z) (from left to right) 1 | 1 0 | 1 1 ) ( ... ) ( ) ( N N N N z E z E z H N z T N T 3 N E ( ) z N N ( ) ( ) ... ( ) 1 H z E z E z 0 0 | 0 | 0 1 : N = : : . : N ( ) 1 N N z n-th column of R(z) has N-fold polyphase components of Fn(z) (from bottom to top) ) ( ... ) ( 1 ) ( 1 | 1 1 | 0 1 z R z R z F N N N N R ( ) R ( z ) ... R ( z N ) ) 1 ( N ( ) F z z 0 | 1 0 0 | 0 N = : : . : : N N DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 38 / 40

  39. Perfect Reconstruction Theory D= N With the `noble identities , this is equivalent to: 4 4 4 4 4 1 3 z u[k] 4 4 4 1 2 E R y[k] z z z z (z ) (z ) + 2 1 z 1 3 In Chapter 13, it will be demonstrated that leads to a simple/practical perfect reconstruction condition = R E ( ). ( ) z z z N I which will then be used for PR-FB design DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries Beautifully simple!! (compared to p.37) 39 / 40

  40. Perfect Reconstruction Theory D< N Similarly, for oversampled FBs D=4 N=6 4 4 4 4 4 4 4 1 3 z 4 4 4 4 4 y[k] u[k] 2 1 z z z z R(z4) E(z4) + 2 1 z 1 3 + noble identities n-th row of E(z) has D-fold polyphase components of Hn(z) n-th column of R(z) has D-fold polyphase components of Fn(z) R(z).E(z)=z-dID R(z).E(z)=z-dID DxN DxD NxD Here E(z) is an N-by-D ( tall-thin ) matrix, R(z) is a D-by-N ( short-fat ) matrix Again beautifully simple!! (compared to p.37) DSP-CIS 2022-2023 / Chapter 12: Filter Bank Preliminaries 40 / 40

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