1. p. 1 DSP-II Digital Signal Processing II Lecture 5: Filter Banks - Preliminaries 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 2005-2006 Lecture-5 Filter Banks - Preliminaries 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 2005-2006 Lecture-5 Filter Banks - Preliminaries PART-II : Filter Banks LECTURE-5 : Filter Banks Preliminaries Filter banks introduction Aliasing vs. perfect reconstruction (PR) Filter banks applications 1st PR FB example : DFT/IDFT filter bank

4. DSP-II p. 4 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Introduction What we have in mind is this… : - Signals split into frequency channels/subbands - Per-channel/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) subband processing H1(z) H2(z) H3(z) H4(z) IN + OUT H1H4H3H2

5. DSP-II p. 5 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Introduction Step-1: Analysis filter bank - collection of M filters (`analysis filters’, `decimation filters’) with a common input signal - ideal (but non-practical) frequency responses = ideal bandpass filters - typical frequency responses (overlapping, marginally overlapping, non-overlapping) H1(z) H2(z) H3(z) H4(z) IN H1H4H3 H2 H1H4H3H2 H1H4H3H2 M=4

6. DSP-II p. 6 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Introduction Step-2: Decimators (downsamplers) - To increase efficiency, subband sampling rate can be reduced by factor N (cfr. Nyquist theorem). maximally decimated - maximally decimated filter banks (=critically downsampled filter banks): -> # subband samples= # fullband samples this sounds like maximum efficiency, but aliasing (see below)! - oversampled filter banks (=non-critically downsampled filter banks): -> # subband samples> # fullband samples H1(z) H2(z) H3(z) H4(z) IN 3 3 3 3 N=3M=4 N=M N<M

7. DSP-II p. 7 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Introduction 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 subband processing H1(z) subband processing H2(z) subband processing H3(z) 3 3 3 3 subband processing H4(z) IN N=3M=4

8. DSP-II p. 8 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Introduction Step-4: Expanders (upsamplers) - restore original fullband sampling rate by N-fold upsampling (=insert N-1 zeros in between every two samples) 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 M=4N=3

9. DSP-II p. 9 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Introduction Step-5: Synthesis filter bank - upsampling has to be followed by (interpolation) filtering (see below p.31) - collection of M `synthesis’ (`interpolation’) filters, with a `common’ (summed) output signal - frequency responses : preferably `matched’ to frequency responses of the analysis filters, e.g., to provide perfect reconstruction (see below) G1(z) G2(z) G3(z) G4(z) + OUT 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 M=4N=3 G1G4G3G2

10. DSP-II p. 10 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Aliasing versus Perfect Reconstruction - Assume subband processing does not modify subband signals (e.g. lossless coding/decoding) -The overall aim would be to have y[k]=u[k-d], i.e. that the output signal is equal to the input signal up to a certain delay -But: downsampling introduces ALIASING, especially so in maximally decimated (but also in non-maximally decimated) filter banks output=input 3 H1(z) 3 H2(z) 3 H3(z) 3 3 3 3 3 H4(z) u[k] G1(z) G2(z) G3(z) G4(z) + y[k]=u[k-d]? output=input

11. DSP-II p. 11 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Aliasing versus Perfect Reconstruction Question : Can y[k]=u[k-d] be achieved in the presence of aliasing ? Answer : YES, see below… PERFECT RECONSTRUCTION banks with synthesis bank designed to remove aliasing effects ! output=input 3 H1(z) 3 H2(z) 3 H3(z) 3 3 3 3 3 H4(z) u[k] G1(z) G2(z) G3(z) G4(z) + y[k]=u[k-d]? output=input

12. DSP-II p. 12 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Applications Subband coding : Coding = Fullband signal split into subbands & downsampled 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.)

13. DSP-II p. 13 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Applications Subband adaptive filtering : - See Lectures 8-11 - Example : Acoustic echo cancellation Adaptive filter models (time-varying) acoustic echo path and produces a copy of the echo, which is then subtracted from microphone signal. = difficult problem ! *long acoustic impulse responses *time-varying

14. DSP-II p. 14 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Applications Subband adaptive filtering (continued): - Subband filtering = M (simpler?) subband modeling problems instead of one (more complicated?) fullband modeling problem - Perfect reconstruction guarantees distortion-free desired near-end speech signal 3 H1(z) 3 H2(z) 3 H3(z) 3 H4(z) 3 H1(z) 3 H2(z) 3 H3(z) 3 H4(z) + + + + 3 G1(z) 3 G2(z) 3 G3(z) 3 G4(z) OUT + ad.filter

15. DSP-II p. 15 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Applications: Transmux’s Transmultiplexers : Frequency Division Multiplexing (FDM) in a digital telephone network: - M different source signals multiplexed into 1 transmit signal by expanders & synthesis filters (ps: here interpolation factor ) - Received signal decomposed into M source signals by analysis filters & decimators - Again ideal filters = ideal bandpass filters 5 G1(z) 5 G2(z) 5 G3(z) 5 G4(z) + 5 H1(z) 5 H2(z) 5 H3(z) 5 H4(z) transmission channel signal-1 signal-2 signal-3 signal-4 signal-1 signal-2 signal-3 signal-4 N>=M

16. DSP-II p. 16 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Applications: Transmux’s Transmultiplexers : (continued) - Non-ideal synthesis & analysis filters result in CROSS-TALK between channels, i.e. each reconstructed signal contains unwanted contributions from other signals - Analysis & synthesis are reversed here, but similar perfect reconstruction theory (try it!) (where analysis bank removes cross-talk introduced by synthesis bank, if transmission channel = distortion free)

17. DSP-II p. 17 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Applications: Transmux’s Transmultiplexers : (continued) PS: special case is Time Division Multiplexing (TDM), if synthesis and analysis filters are replaced by delay operators (and N=M) 4 4 4 4 + 4 4 4 transmission channel u1[k],u1[k+1] u2[k],u2[k+1] u3[k],u3[k+1] u4[k],u4[k+1] u1[k],u2[k],u3[k],u4[k],u1[k+1],u2[k+1]... 4 u1[k-1],u1[k] u2[k-1],u2[k] u3[k-1],u3[k] u4[k-1],u4[k] 0,0,0,u4[k],0,0,0,u4[k+1]...

18. DSP-II p. 18 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Applications: Transmux’s Transmultiplexers : (continued) PS : special case is Code Division Multiple Access (CDMA), where filter coefficients=(orthogonal) user codes CDMA basics : (see digital coms courses)…… -Each user-i is assigned a pseudo-random code sequence -Transmission : For each symbol (k-th symbol for user-i), a `chip’ sequence is transmitted -Mostly binary codes ( ) with BPSK/QPSK symbols -Multiple access based on code-orthogonality (see below)

19. DSP-II p. 19 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Applications: Transmux’s CDMA basics: -Reception : If received signal = transmitted chip sequence (i.e. no channel effect, no noise), multiply chips with (synchronized) code sequence + sum. -Example (user i) transmitted symbols +1……… -1………. -1……… +1……… code sequence +1,+1,-1,-1 transmitted chips +1,+1,-1,-1, -1,-1,+1,+1,-1,-1,+1,+1,+1,+1,-1,-1 received chips +1,+1,-1,-1, -1,-1,+1,+1,-1,-1,+1,+1,+1,+1,-1,-1 +1,+1,-1,-1 (mult. with code & sum) received symbols (*1/4) +1………… -1…….…..-1…………+1

20. DSP-II p. 20 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Banks Applications: Transmux’s CDMA Transmission/reception block scheme : -transmitter code-multiplication may be viewed as filtering operation, with FIR transmit filter -receiver code-multiplication & summation may be viewed as filtering operation, with receive filter -PR for `flat’ channel H(z)=1 and if codes are orthogonal (prove it!) 4 4 4 4 + 4 4 4 transmission channel u1[k],u1[k+1] u2[k],u2[k+1] u3[k],u3[k+1] u4[k],u4[k+1] u2[k+1],u2[k] 4 Base station User-2 terminal

21. DSP-II p. 21 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Filter Bank Applications: Transmux’s CDMA PS: real-world CDMA more complicated (different channels for different users + channel dispersion (instead of H(z)=1 for all users), asynchronous users, scrambling codes, etc.) PS: CDMA application examples : - IS-95 cellular telephony (Qualcomm) - UMTS (`wideband CDMA’) - IEEE 802.11 wireless LANs - GPS - cable modems, power line comms, ….

22. DSP-II p. 22 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries PR-FB Example: DFT/IDFT Filter Bank Fundamental question is..: Downsampling introduces ALIASING, then how can PERFECT RECONSTRUCTION (PR) (i.e. y[k]=u[k-d]) be achieved ? Next slides provide simple PR-FB examples, to demonstrate that PR can indeed (easily) be obtained Discover the magic of aliasing-compensation….

23. DSP-II p. 23 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries DFT/IDFT Filter Bank First attempt to design a perfect reconstruction filter bank: -Starting point is this : as y[k]=u[k-d], this can already be viewed as a perfect reconstruction filter bank... 4 4 4 4 u[k] 4 4 4 4 + + + u[k-3] u[0],0,0,0,u[4],0,0,0,... u[-1],u[0],0,0,u[3],u[4],0,0,... u[-2],u[-1],u[0],0,u[2],u[3],u[4],0,... u[-3],u[-2],u[-1],u[0],u[1],u[2],u[3],u[4],...

24. DSP-II p. 24 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries DFT/IDFT Filter Bank - An equivalent representation is... 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….) ps: transmux version see p.16 4 4 4 4 + u[k-3] 4 4 4 4 u[k]

25. DSP-II p. 25 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries DFT/IDFT Filter Bank - now insert DFT-matrix (discrete Fourier transform) and its inverse (I-DFT)... as this clearly does not change the input-output relation (hence perfect reconstruction property preserved) 4 4 4 4 + u[k-3] 4 4 4 4 u[k]

26. DSP-II p. 26 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries DFT/IDFT Filter Bank - …and reverse order (see p.35) of decimators/expanders and DFT-matrices (not done in an efficient implementation!) : =analysis filter bank =synthesis filter bank This is the `DFT/IDFT filter bank’. It is a first example of a maximally decimated perfect reconstruction filter bank ! 4 4 4 4 u[k] 4 4 4 4 + u[k-3]

27. DSP-II p. 27 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries DFT/IDFT Filter Bank What do analysis filters look like? 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) : i.e. the Hi are obtained by uniformly shifting the `prototype’ Ho over the frequency axis. u[k]

28. DSP-II p. 28 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries DFT/IDFT Filter Bank The prototype filter Ho(z) is a not-so-great lowpass filter with first sidelobe only 13 dB below the main lobe. Ho(z) and Hi(z)’s are thus far from ideal lowpass/ bandpass filters. Hence (maximal) decimation introduces significant ALIASING in the decimated subband signals Still, we know this is a PERFECT RECONSTRUCTION filter bank (see construction p.22-25), which means the synthesis filters can apparently restore the aliasing distortion. This is remarkable! Other perfect reconstruction banks : see Lecture-6 Ho(z)

29. DSP-II p. 29 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries DFT/IDFT Filter Bank What do synthesis filters look like? synthesis filters are (roughly) equal to analysis filters (details omitted, see also Lecture-6) PS: Efficient DFT/IDFT implementation based FFT algorithm (`Fast Fourier Transform’). In the sequel, every time a DFT (IDFT) matrix is shown, FFT (IFFT) -implementation is tacitly assumed. +

30. DSP-II p. 30 Version 2005-2006 Lecture-5 Filter Banks - Preliminaries Conclusions Seen the general subband processing set-up + applications Filter bank system is multi-rate structure, with decimators and expanders, hence ALIASING is a major concern Seen a first (simple & not-so-great) example of a PERFECT RECONSTRUCTION filter bank (DFT/IDFT) Sequel = other (better) PR structures Lecture 6: maximally decimated filter banks Lecture 7: modulated filter banks Lecture 8: oversampled filter banks, etc.. Reference: `Multirate Systems & Filter Banks’, P.P. Vaidyanathan Prentice Hall 1993.