Digital Signal Processing II Lecture 7 Modulated Filter

Digital Signal Processing II Lecture 7: Modulated Filter Banks Marc Moonen Dept. E. E. /ESAT, K. U. Leuven marc. moonen@esat. kuleuven. be homes. esat. kuleuven. be/~moonen/ DSP-II p. 1

Part-II : Filter Banks Lecture-5 : Preliminaries • Applications • Intro perfect reconstruction filter banks (PR FBs) Lecture-6 : Maximally decimated FBs • Multi-rate systems review • PR FBs • Paraunitary PR FBs Lecture-7 : Modulated FBs • DFT-modulated FBs • Cosine-modulated FBs Lecture-8 DSP-II : Special Topics • Non-uniform FBs & Wavelets • Oversampled DFT-modulated FBs • Frequency domain filtering Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 2

Refresh General `subband processing’ set-up (Lecture 5) : - 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 IN DSP-II H 0(z) 3 subband processing 3 F 0(z) H 1(z) 3 subband processing 3 F 1(z) H 2(z) 3 subband processing 3 F 2(z) H 3(z) 3 subband processing 3 F 3(z) Version 2005 -2006 Lecture-7 Modulated Filter Banks OUT + p. 3

Refresh Two design issues : - filter specifications, e. g. stopband attenuation, passband ripple, transition band, etc. (for each (analysis) filter!) - perfect reconstruction property (Lecture 6). 4 4 u[k] 4 4 + u[k-3] PS: Lecture 6/7 = maximally decimated FB’s = DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 4

Introduction -All design procedures so far involve monitoring of characteristics (passband ripple, stopband suppression, …) of all (analysis) filters, which may be tedious. -Design complexity may be reduced through usage of `uniform’ and `modulated’ filter banks. • DFT-modulated FBs • Cosine-modulated FBs DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 5

DFT-Modulated Filter Banks Uniform versus non-uniform (analysis) filter bank: H 0(z) IN uniform H 1(z) H 2(z) H 3(z) non-uniform H 0 H 1 H 2 H 3 non-uniform: e. g. for speech & audio applications (cfr. human hearing) example : wavelet filter banks (next lecture) N-Channel uniform filter bank: = frequency responses uniformly shifted over the unit circle Ho(z)= `prototype’ filter (=only filter that has to be designed) DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 6

DFT-Modulated Filter Banks Uniform filter banks can be implemented cheaply based on polyphase decompositions + DFT(FFT) u[k] hence named `DFT modulated FBs’ H 0(z) H 1(z) H 2(z) 1. Analysis FB H 3(z) If then i. e. DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 7

DFT-Modulated Filter Banks i. e. where F is Nx. N DFT-matrix DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 8

DFT-Modulated Filter Banks i. e. u[k] conclusion: economy in… * implementation complexity: N filters for the price of 1, plus DFT (=FFT) * design complexity: design `prototype’ Ho(z), then other Hi(z)’s are automatically `co-designed’ (same passband ripple, etc…) !!!!!!!!! DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 9

DFT-Modulated Filter Banks • Special case: DFT-filter bank, if all Ei(z)=1 u[k] DSP-II Ho(z) Version 2005 -2006 Lecture-7 Modulated Filter Banks H 1(z) p. 10

DFT-Modulated Filter Banks • PS: with F instead of F* (see also Lecture-5), only filter ordering is changed u[k] DSP-II Ho(z) Version 2005 -2006 Lecture-7 Modulated Filter Banks H 1(z) p. 11

DFT-Modulated Filter Banks • Uniform DFT-modulated analysis FB +decimation (M=N) u[k] 4 4 u[k] = 4 4 DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 12

DFT-Modulated Filter Banks 2. Synthesis FB + + phase shift added for convenience + y[k] DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 13

DFT-Modulated Filter Banks i. e. where F is Nx. N DFT-matrix DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 14

DFT-Modulated Filter Banks i. e. + + + y[k] DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 15

DFT-Modulated Filter Banks • Expansion (M=N) + uniform DFT-modulated synthesis FB : 4 4 + 4 + y[k] 4 = DSP-II Version 2005 -2006 4 + 4 + y[k] Lecture-7 Modulated Filter Banks p. 16

DFT-Modulated Filter Banks Perfect reconstruction (PR) revisited : maximally decimated (M=N) uniform DFT-modulated analysis & synthesis… u[k] 4 4 4 4 + + + y[k] - Procedure: 1. Design prototype analysis filter Ho(z) (=DSP-II/Part-I). 2. This determines Ei(z) (=polyphase components). 3. Assuming Ei(z) can be inverted (? ), choose synthesis filters DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 17

DFT-Modulated Filter Banks Perfect reconstruction (PR): u[k] 4 4 4 4 + + + y[k] FIR Ei(z) generally leads to IIR R(z), where stability is a concern… Hence PR with FIR analysis/synthesis bank (=guaranteed stability), only obtained with trivial choices for Ei(z)’s (next slide) DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 18

DFT-Modulated Filter Banks • Simple example (1) is , which leads to IDFT/DFT bank (Lecture-5) i. e. Fl(z) has coefficients of Hl(z), but complex conjugated and in reverse order (hence same magnitude response) (remember this? !) • Simple example (2) is , where wi’s are constants, which leads to `windowed’ IDFT/DFT bank, a. k. a. `shorttime Fourier transform’ (see Lecture-8) • Question (try to answer): when is maximally decimated PR uniform DFT-modulated FB - FIR (both analysis & synthesis) ? - paraunitary ? DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 19

DFT-Modulated Filter Banks • Bad news: From this it is seen that the maximally decimated IDFT/DFT filter bank (or trivial modifications thereof) is the only possible uniform DFT-modulated FB that is at the same time. . . i) maximally decimated ii) perfect reconstruction (PR) iii) FIR (all analysis+synthesis filters) iv) paraunitary • Good news : – Cosine-modulated PR FIR FB’s – Oversampled PR FIR DFT-modulated FB’s (Lecture-8) DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 20

Cosine-Modulated Filter Banks • Uniform DFT-modulated filter banks: Ho(z) is prototype lowpass filter, cutoff at for N filters H 0 H 1 H 2 H 3 • Cosine-modulated filter banks : Po(z) is prototype lowpass filter, cutoff at for N filters P 0 Then. . . Ho H 1 etc. . . DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 21

Cosine-Modulated Filter Banks • Cosine-modulated filter banks : - if Po(z) is prototype lowpass filter designed with real coefficients po[n], n=0, 1, …, L then i. e. `cosine modulation’ (with real coefficients) instead of `exponential modulation’ (for DFT-modulated bank, see page 6) - if Po(z) is `good’ lowpass filter, then Hk(z)’s are `good’ bandpass filters DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 22

Cosine-Modulated Filter Banks Realization based on polyphase decomposition (analysis): - if Po(z) has 2 N-fold polyphase expansion (ps: 2 N-fold for N filters!!!) u[k] then. . . : DSP-II Version 2005 -2006 : Lecture-7 Modulated Filter Banks p. 23

Cosine-Modulated Filter Banks Realization based on polyphase decomposition (continued): ignore all details here !!!!!!!! - if Po(z) has L+1=m. 2 N taps, and m is even (similar formulas for m odd) (m is the number of taps in each polyphase component) then. . . With DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 24

Cosine-Modulated Filter Banks Realization based on polyphase decomposition (continued): - Note that C is Nx. N DCT-matrix (`Type 4’) hence fast implementation (=fast matrix-vector product) based on fast discrete cosine transform procedure, complexity O(N. log. N). - Modulated filter bank gives economy in * design (only prototype Po(z) ) u[k] * implementation (prototype + modulation (DCT)) : : Similar structure for synthesis bank DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 25

Cosine-Modulated Filter Banks Maximally decimated cosine modulated (analysis) bank : u[k] N N : N = u[k] N N : N DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 26

. . th is is Cosine-Modulated Filter t. Banks h eh ard Question: How do we obtain Maximal Decimation + FIR + Paraunitariness? Theorem: par t… (proof omitted) -If prototype Po(z) is a real-coefficient (L+1)-taps FIR filter, (L+1)=2 N. m for some integer m and po[n]=po[L-n] (linear phase), with polyphase components Ek(z), k=0, 1, … 2 N-1, -then the (FIR) cosine-modulated analysis bank is PARAUNITARY if and only if (for all k) are power complementary, i. e. form a lossless 1 input/2 output system Hence FIR synthesis bank (for PR) can be obtained by paraconjugation !!! =Great result… DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 27

. . th is is Cosine-Modulated Filter th Banks eh ard Perfect Reconstruction (continued) Design procedure: par t… Parameterize lossless systems for k=0, 1. . , N-1 Optimize all parameters in this parametrization so that the prototype Po(z) based on these polyphase components is a linear-phase lowpass filter that satisfies the given specifications Example parameterization: Parameterize lossless systems for k=0, 1. . , N-1, -> lattice structure (see Part-I), where parameters are rotation angles DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 28

Cosine-Modulated Filter Banks lossless p. 26 = u[k] N : : N : PS: Linear phase property for po[n] implies that only half of the power complementary pairs have to be designed. The other pairs are then defined by symmetry properties. DSP-II Version 2005 -2006 Lecture-7 Modulated Filter Banks p. 29

Cosine-Modulated Filter Banks PS: Cosine versus DFT modulation In a maximally decimated cosine-modulated (analysis) filter bank 2 polyphase components of the prototype filter, , actually take the place of only 1 polyphase component in the DFTmodulated case. For paraunitariness (hence FIR-PR) in a cosine-modulated bank, each such pair of polyphase filters should form a power complementary pair, i. e. represent a lossless system. provides flexibility for FIR-design In the DFT-modulated case, imposing paraunitariness is equivalent to imposing losslessness for each polyphase component separately, i. e. each polyphase component should be an `allpass’ transfer function. Allpass functions are always IIR, except for trivial cases (pure delays). Hence all FIR paraunitary DFT-modulated banks (with maximal decimation) are trivial modifications of the DFT bank. DSP-II Version 2005 -2006 p. 30 no FIR-design flexibility Lecture-7 Modulated Filter Banks