DSP-CIS Chapter-7: Filter Banks -Preliminaries Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 2 : Preliminaries Filter bank (FB) set-up and applications Perfect reconstruction (PR) problem Multi-rate systems review (=10 slides) Example: Ideal filter bank (=10 figures) : Maximally Decimated PR-FBs Example: DFT/IDFT filter bank Perfect reconstruction theory Paraunitary PR-FBs : DFT-Modulated FBs Maximally decimated DFT-modulated FBs Oversampled DFT-modulated FBs : Special Topics Cosine-modulated FBs Non-uniform FBs & Wavelets Frequency domain filtering Chapter-7 Chapter-8 Chapter-9 Chapter-10 Part-III : Filter Banks

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 3 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

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 4 Filter Banks Introduction 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 - typical frequency responses (overlapping, non-overlapping,…) H1H4H3 H2 H1H4H3H2 H1H4H3H2 H1(z) H2(z) H3(z) H4(z) IN N=4

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 5 Filter Banks Introduction Step-2: Decimators (downsamplers) - To increase efficiency, subband sampling rate is reduced by factor D (= Nyquist (bandpass) sampling theorem) - 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 H1(z) H2(z) H3(z) H4(z) IN N=4D=3 PS: 3-fold downsampling = ‘remove every 2 nd and 3 rd sample’ (see p.32)

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 6 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) subband processing H4(z) IN N=4D=3

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 7 Filter Banks Introduction Step-4: Expanders (upsamplers) - restore original fullband sampling rate by D-fold upsampling subband processing 3 H1(z) subband processing 3 H2(z) subband processing 3 H3(z) subband processing 3 H4(z) IN N=4D=3 PS: 3-fold upsampling = ‘insert 2 zeros in between every two samples’ (see p.30)

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 8 Filter Banks Introduction Step-5: Synthesis filter bank - upsampling has to be followed by (interpolation) filtering (see p.22) - collection of N `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) subband processing 3 H4(z) IN N=4D=3 G1G4G3G2

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 9 Aliasing versus Perfect Reconstruction - Assume subband processing does not modify subband signals (e.g. lossless coding/decoding) -The overall aim would then 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 (see also p.32-33) G1(z) G2(z) G3(z) G4(z) + output = input 3 H1(z) output = input 3 H2(z) output = input 3 H3(z) output = input 3 H4(z) N=4D=3 u[k] y[k]=u[k-d]?

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 10 Aliasing versus Perfect Reconstruction Question : Can y[k]=u[k-d] be achieved in the presence of aliasing ? Answer : YES !! PERFECT RECONSTRUCTION banks with synthesis bank designed to remove aliasing effects ! G1(z) G2(z) G3(z) G4(z) + output = input 3 H1(z) output = input 3 H2(z) output = input 3 H3(z) output = input 3 H4(z) N=4D=3 u[k] y[k]=u[k-d]?

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 11 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.)

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 12 Filter Banks Applications Subband adaptive filtering : - See Part IV - 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

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 13 Filter Banks Applications - Subband filtering = N (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) G1(z) 3 G2(z) 3 G3(z) 3 G4(z) OUT + ad.filter

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 14 Filter Banks Applications Transmultiplexers : Frequency Division Multiplexing (FDM) in digital communications - N different source signals multiplexed into 1 transmit signal by expanders & synthesis filters (ps: here interpolation factor ) - Received signal decomposed into N 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 D>=N

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p Non-ideal synthesis & analysis filters result in aliasing as well as 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) Filter Banks Applications

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 16 Filter Banks Applications PS: special case is Time Division Multiplexing (TDM), if synthesis and analysis filters are replaced by delay operators (and N=D) 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]... PS: another special case is OFDM… explain! What are Gi(z) & Hi(z) ?

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 17 Filter Banks Applications PS : special case is Code Division Multiple Access (CDMA), where filter coefficients=(orthogonal) user codes CDMA basics : (see digital coms courses)…… -Each user (n=1..N) is assigned a length-N pseudo-random code sequence -Transmission : For each symbol (k-th symbol for user-n), a `chip’ sequence is transmitted -Mostly binary codes ( ) with BPSK/QPSK symbols -Multiple access based on code-orthogonality (see below) Skip this slide

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 18 CDMA basics: -Reception : If (x) 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 (x) PS: real-world CDMA is considerably more complicated (different channels for different users + channel dispersion, asynchronous users, scrambling codes, etc.) Filter Banks Applications Skip this slide

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 19 Filter Banks Applications 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!) 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 Skip this slide

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 20 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... D u[0], u[N], u[2D]... u[0],u[1],u[2]... D u[0],0,..0,u[1],0,…,0,u[2]...u[0], u[1], u[2],...

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 21 Review of Multi-rate Systems 2/10 Z-transform (frequency domain) analysis of expander `expansion in time domain ~ compression in frequency domain’ D u[0],0,..0,u[1],0,…,0,u[2]...u[0], u[1], u[2],... D D `images’

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 22 Review of Multi-rate Systems 2bis/10 Z-transform (frequency domain) analysis of expander expander mostly followed by `interpolation filter’ to remove images (and `interpolate the zeros’) interpolation filter can be low-/band-/high-pass (see p.8 & Chapter-8) D u[0],0,..0,u[1],0,…,0,u[2]...u[0], u[1], u[2],... 3 `images’ LP

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 23 Review of Multi-rate Systems 3/10 Z-transform (frequency domain) analysis of decimator `compression in time domain ~ expansion in frequency domain’ PS: Note that is periodic with period while is periodic with period The summation with d=0…D-1 restores the periodicity with period ! D D u[0], u[D], u[2D]... u[0],u[1],u[2]... d=0d=2d=1 3

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 24 Review of Multi-rate Systems 4/10 Z-transform (frequency domain) analysis of decimator decimation introduces ALIASING if input signal occupies frequency band larger than, hence mostly preceded by anti-aliasing (decimation) filter anti-aliasing filter can be low-/band-/high-pass (see p.5 & Chapter-8) D u[0], u[N], u[2D]... u[0],u[1],u[2]... LP d=0d=2d=1 3

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 25 Review of Multi-rate Systems 5/10 Interconnection of multi-rate building blocks : identities also hold if all decimators are replaced by expanders D x a D x a = = = D + u2[k] D x u1[k] D + D u2[k] u1[k] D x D u2[k] u1[k]

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 26 Review of Multi-rate Systems 6/10 `Noble identities’ (I) : (only for rational functions) Example : D=2 h[0],h[1],0,0,0,… = DD u[k] y[k]

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 27 Review of Multi-rate Systems 7/10 `Noble identities’ (II) : (only for rational functions) Example : D=2 h[0],h[1],0,0,0,… = DD u[k] y[k]

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 28 Review of Multi-rate Systems 8/10 Application of `noble identities : efficient multi-rate realizations of FIR filters through… Polyphase decomposition: example : (2-fold decomposition) example : (3-fold decomposition) general: (D-fold decomposition)

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 29 Review of Multi-rate Systems 9/10 Polyphase decomposition: Example : efficient implementation of an FIR decimation filter i.e. all filter operations performed at the lowest rate u[k] 2 + H(z) u[k] 2 + = 2

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 30 Review of Multi-rate Systems 10/10 Polyphase decomposition: Example : efficient implementation of an FIR interpolation filter i.e. all filter operations performed at the lowest rate = u[k] 2 + H(z) u[k] 2 + 2

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 31 Example: Ideal Filter Bank With ideal analysis/synthesis filters, FB operates as follows (1) subband processing 4 H1(z) subband processing 4 H2(z) subband processing 4 H3(z) subband processing 4 H4(z) IN G1(z) G2(z) G3(z) G4(z) + OUT H1(z)H2(z)H3(z)H4(z) IN … …

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 32 Example: Ideal Filter Bank With ideal analysis/synthesis filters, FB operates as follows (2) subband processing 4 H1(z) subband processing 4 H2(z) subband processing 4 H3(z) subband processing 4 H4(z) IN G1(z) G2(z) G3(z) G4(z) + OUT H1(z)H2(z)H3(z)H4(z) x1

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 33 Example: Ideal Filter Bank With ideal analysis/synthesis filters, FB operates as follows (3) subband processing 4 H1(z) subband processing 4 H2(z) subband processing 4 H3(z) subband processing 4 H4(z) IN G1(z) G2(z) G3(z) G4(z) + OUT x’1x’1 x’1x’1 x’1x’1 (ideal subband processing)

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 34 Example: Ideal Filter Bank With ideal analysis/synthesis filters, FB operates as follows (4) subband processing 4 H1(z) subband processing 4 H2(z) subband processing 4 H3(z) subband processing 4 H4(z) IN G1(z) G2(z) G3(z) G4(z) + OUT x’’1 (ideal subband processing)

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 35 Example: Ideal Filter Bank With ideal analysis/synthesis filters, FB operates as follows (5) subband processing 4 H1(z) subband processing 4 H2(z) subband processing 4 H3(z) subband processing 4 H4(z) IN G1(z) G2(z) G3(z) G4(z) + OUT G1(z)G2(z)G3(z)G4(z) x’’’1 (ideal subband processing)

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 36 Example: Ideal Filter Bank With ideal analysis/synthesis filters, FB operates as follows (6) subband processing 4 H1(z) subband processing 4 H2(z) subband processing 4 H3(z) subband processing 4 H4(z) IN G1(z) G2(z) G3(z) G4(z) + OUT H1(z)H2(z)H3(z)H4(z) x2

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 37 Example: Ideal Filter Bank With ideal analysis/synthesis filters, FB operates as follows (7) subband processing 4 H1(z) subband processing 4 H2(z) subband processing 4 H3(z) subband processing 4 H4(z) IN G1(z) G2(z) G3(z) G4(z) + OUT x’2x’2 x’2x’2 x’2x’2

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 38 Example: Ideal Filter Bank With ideal analysis/synthesis filters, FB operates as follows (8) subband processing 4 H1(z) subband processing 4 H2(z) subband processing 4 H3(z) subband processing 4 H4(z) IN G1(z) G2(z) G3(z) G4(z) + OUT x’’2

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 39 Example: Ideal Filter Bank With ideal analysis/synthesis filters, FB operates as follows (9) subband processing 4 H1(z) subband processing 4 H2(z) subband processing 4 H3(z) subband processing 4 H4(z) IN G1(z) G2(z) G3(z) G4(z) + OUT G1(z)G2(z)G3(z)G4(z) x’’’2

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version p. 40 Example: Ideal Filter Bank With ideal analysis/synthesis filters, FB operates as follows (10) subband processing 4 H1(z) subband processing 4 H2(z) subband processing 4 H3(z) subband processing 4 H4(z) IN G1(z) G2(z) G3(z) G4(z) + OUT H1(z)H2(z)H3(z)H4(z) Now try this with non-ideal filters… … OUT=IN