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

DSP-II p. 2 Version Lecture-7 Modulated 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

DSP-II p. 3 Version Lecture-7 Modulated Filter Banks 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 subband processing 3 H0(z) subband processing 3 H1(z) subband processing 3 H2(z) subband processing 3 H3(z) IN F0(z) F1(z) F2(z) F3(z) + OUT

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

DSP-II p. 5 Version Lecture-7 Modulated Filter Banks 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 p. 6 Version Lecture-7 Modulated Filter Banks DFT-Modulated Filter Banks Uniform versus non-uniform (analysis) filter bank: 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) H0(z) H1(z) H2(z) H3(z) IN H0H3 H2H1 H0H3H2H1 uniform non-uniform

DSP-II p. 7 Version Lecture-7 Modulated Filter Banks DFT-Modulated Filter Banks Uniform filter banks can be implemented cheaply based on polyphase decompositions + DFT(FFT) hence named `DFT modulated FBs’ 1. Analysis FB If then i.e. H0(z) H1(z) H2(z) H3(z) u[k]

DSP-II p. 8 Version Lecture-7 Modulated Filter Banks DFT-Modulated Filter Banks where F is NxN DFT-matrix i.e.

DSP-II p. 9 Version Lecture-7 Modulated Filter Banks DFT-Modulated Filter Banks 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…) !!!!!!!!!!!!!!!!! u[k] i.e.

DSP-II p. 10 Version Lecture-7 Modulated Filter Banks DFT-Modulated Filter Banks Special case: DFT-filter bank, if all Ei(z)=1 u[k] Ho(z)H1(z)

DSP-II p. 11 Version Lecture-7 Modulated Filter Banks DFT-Modulated Filter Banks PS: with F instead of F* (see also Lecture-5), only filter ordering is changed Ho(z)H1(z) u[k]

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

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

DSP-II p. 14 Version Lecture-7 Modulated Filter Banks DFT-Modulated Filter Banks where F is NxN DFT-matrix i.e.

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

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

DSP-II p. 17 Version Lecture-7 Modulated Filter Banks DFT-Modulated Filter Banks Perfect reconstruction (PR) revisited : maximally decimated (M=N) uniform DFT-modulated analysis & synthesis… - 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 y[k] u[k]

DSP-II p. 18 Version Lecture-7 Modulated Filter Banks DFT-Modulated Filter Banks Perfect reconstruction (PR): 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) y[k] u[k]

DSP-II p. 19 Version Lecture-7 Modulated Filter Banks 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. `short- time 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 p. 20 Version Lecture-7 Modulated Filter Banks 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 p. 21 Version Lecture-7 Modulated Filter Banks Cosine-Modulated Filter Banks Uniform DFT-modulated filter banks: Ho(z) is prototype lowpass filter, cutoff at for N filters Cosine-modulated filter banks : Po(z) is prototype lowpass filter, cutoff at for N filters Then... etc... H0H3H2H1 P0 H1 Ho

DSP-II p. 22 Version Lecture-7 Modulated Filter Banks 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 p. 23 Version Lecture-7 Modulated Filter Banks Cosine-Modulated Filter Banks Realization based on polyphase decomposition (analysis): - if Po(z) has 2N-fold polyphase expansion (ps: 2N-fold for N filters!!!) then... u[k] : :

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

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

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

DSP-II p. 27 Version Lecture-7 Modulated Filter Banks Cosine-Modulated Filter Banks Question: How do we obtain Maximal Decimation + FIR + PR + Paraunitariness? Theorem : (proof omitted) -If prototype Po(z) is a real-coefficient (L+1)-taps FIR filter, (L+1)=2N.m for some integer m and po[n]=po[L-n] (linear phase), with polyphase components Ek(z), k=0,1,…2N-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…..this is the hard part…

DSP-II p. 28 Version Lecture-7 Modulated Filter Banks Cosine-Modulated Filter Banks Perfect Reconstruction (continued) Design procedure: 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..this is the hard part…

DSP-II p. 29 Version Lecture-7 Modulated Filter Banks Cosine-Modulated Filter Banks 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. u[k] : N N p.26 = : : lossless

DSP-II p. 30 Version Lecture-7 Modulated Filter Banks 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 DFT- modulated 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. 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. no FIR-design flexibility provides flexibility for FIR-design