DSP-CIS Chapter-9: Modulated Filter Banks

DSP-CISChapter-9: Modulated Filter Banks Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be www.esat.kuleuven.be/stadius/

Part-II : Filter Banks : Preliminaries • Filter bank set-up and applications • `Perfect reconstruction’ problem + 1st example (DFT/IDFT) • Multi-rate systems review (10 slides) : Maximally decimated FBs • Perfect reconstruction filter banks (PR FBs) • Paraunitary PR FBs : Modulated FBs • Maximally decimated DFT-modulated FBs • Oversampled DFT-modulated FBs : Cosine-modulated FBs & Special topics • Cosine-modulated FBs • Time-frequency analysis & Wavelets • Frequency domain filtering Chapter-7 Chapter-8 Chapter-9 Chapter-10

3 3 F0(z) subband processing H0(z) OUT IN 3 3 F1(z) subband processing H1(z) + 3 3 F2(z) subband processing H2(z) 3 3 F3(z) subband processing H3(z) Refresh (1) General `subband processing’ set-up (Chapter-7) : PS: subband processing ignored in filter bank design synthesis bank analysis bank downsampling/decimation upsampling/expansion

u[k-3] u[k] 4 4 4 4 + 4 4 4 4 Refresh (2) Two design issues : - filter specifications, e.g. stopband attenuation, passband ripple, transition band, etc. (for each (analysis) filter!) - perfect reconstruction property (Chapter-8). PS: still considering maximally decimated FB’s, i.e. PS: Equivalent perfect reconstruction condition for transmux’s ? Try it !

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 (this Chapter) • Cosine-modulated FBs (next Chapter)

H0 H1 H2 H3 uniform H0 H3 non-uniform H1 H2 H0(z) IN H1(z) H2(z) H3(z) Introduction Uniform versus non-uniform (analysis) filter bank: • N-channel uniform FB: i.e. frequency responses are uniformly shifted over the unit circle Ho(z)= `prototype’ filter (=one and only filter that has to be designed) Time domain equivalent is: • non-uniform = everything that is not uniform e.g. for speech & audio applications (cfr. human hearing) example: wavelet filter banks (next Chapter)

H0(z) u[k] H1(z) H2(z) H3(z) i.e. Maximally Decimated DFT-Modulated FBs Uniform filter banks can be realized cheaply based on polyphase decompositions + DFT(FFT) (hence name `DFT-modulated FB) 1. Analysis FB If (N-fold polyphase decomposition) then

i.e. Maximally Decimated DFT-Modulated FBs where F is NxN DFT-matrix(and `*’ is complex conjugate) This means that filtering with the Hn’s can be implemented by first filtering with polyphase components and then DFT

u[k] i.e. Maximally Decimated DFT-Modulated FBs conclusion: economy in… • implementation complexity (for FIR filters): N filters for the price of 1, plus DFT (=FFT) ! • design complexity: Design `prototype’ Ho(z), then other Hn(z)’s are automatically `co-designed’ (same passband ripple, etc…) !

u[k] Maximally Decimated DFT-Modulated FBs • Special case: DFT-filter bank, if all En(z)=1 Ho(z) H1(z)

u[k] Maximally Decimated DFT-Modulated FBs • PS: with F instead of F* (as in Chapter-6), only filter ordering is changed Ho(z) H1(z)

u[k] u[k] 4 4 4 4 4 = 4 4 4 Maximally Decimated DFT-Modulated FBs • DFT-modulated analysis FB + maximal decimation = efficient realization !

y[k] + + + Maximally Decimated DFT-Modulated FBs 2. Synthesis FB phase shift added for convenience

i.e. Maximally Decimated DFT-Modulated FBs where F is NxN DFT-matrix

+ + + i.e. Maximally Decimated DFT-Modulated FBs y[k]

4 4 4 4 4 4 4 + + + + + + 4 Maximally Decimated DFT-Modulated FBs • Expansion + DFT-modulated synthesis FB : y[k] = = efficient realization ! y[k]

4 u[k] 4 4 4 y[k] + + + 4 4 4 4 Maximally Decimated DFT-Modulated FBs How to achieve Perfect Reconstruction (PR) with maximally decimated DFT-modulated FBs? polyphase components of synthesis bank prototype filter are obtained by inverting polyphase components of analysis bank prototype filter

4 u[k] 4 4 4 y[k] + + + 4 4 4 4 Maximally Decimated DFT-Modulated FBs Design Procedure : 1. Design prototype analysis filter Ho(z) (see Chapter-3). 2. This determines En(z) (=polyphase components). 3. Assuming all En(z) can be inverted (?), choose synthesis filters

Maximally Decimated DFT-Modulated FBs • Will consider only FIR prototype analysis filter, leading to simple polyphase decomposition. • However, FIR En(z)’s generally again lead to IIR Rn(z)’s, where stability is a concern… • FIR unimodular E(Z)? ..such that Rn(z) are also FIR. Only obtained with trivial choices for the En(z)’s, with only 1 non-zero impulse response parameter, i.e. En(z)=α or En(z)=α.z^{-d}. Examples: next slide all E(z)’s FIR E(z)’s FIR unimodular E(z)’s E(z)=F*.diag{..}

Maximally Decimated DFT-Modulated FBs • Simple example (1) is , which leads to IDFT/DFT bank (Chapter-8) i.e. Fn(z) has coefficients of Hn(z), but complex conjugated and in reverse order (hence same magnitude response) (remember this?!) • Simple example (2) is , where wn’s are constants, which leads to `windowed’ IDFT/DFT bank, a.k.a. `short-time Fourier transform’(see Chapter-10)

Maximally Decimated DFT-Modulated FBs • FIR paraunitary E(Z)? ..such that Rn(z) are FIR + power complementary FB’s. Only obtained when the En(z)’s are all-pass filters (and FIR), i.e.En(z)=±1 or En(z)=±1.z^{-d}. i.e. only trivial modifications of DFT filter bank ! SIGH ! all E(z)’s FIR E(z)’s FIR unimodular E(z)’s FIR paraunitary E(z)’s E(z)=F*.diag{..}

Maximally Decimated DFT-Modulated FBs • Bad news: It is seen that the maximally decimated IDFT/DFT filter bank (or trivial modifications thereof) is the only possible maximally decimated DFT- modulated FB that is at the same time... - PR - FIR (all analysis+synthesis filters) - Paraunitary • Good news: • Cosine-modulatedPR FIR FB’s (Chapter-10) • OversampledPR FIR DFT-modulated FB’s (read on) SIGH!

Oversampled PR Filter Banks • So far have considered maximal decimation (D=N), where aliasing makes PR design non-trivial. • With downsampling factor (D) smaller than the number of channels (N), aliasing is expected to become a smaller problem, possibly negligible if D<<N. • Still, PR theory (with perfect alias cancellation) is not necessarily simpler ! • Will not consider PR theory as such here, only give some examples of oversampled DFT-modulated FBs that are PR/FIR/paraunitary (!)

u[k-3] u[k] 4 4 4 4 + 4 4 4 4 Oversampled PR Filter Banks • Starting point is(see Chapter-8): delta=0 for conciseness here where E(z) and R(z) are NxN matrices (cfr maximal decimation) • What if we try other dimensions for E(z) and R(z)…??

u[k] 4 4 4 4 + 4 4 4 4 Oversampled PR Filter Banks ! • A more general case is : where E(z) is now NxD(`tall-thin’) and R(z) is DxN(`short-fat’) while still guarantees PR ! u[k-3] D=4 decimation N=6 channels PS: Here E(z) has 6 rows (defining 6 analysis filters), with four 4-fold polyphase components in each row

Oversampled PR Filter Banks • The PR condition appears to be a `milder’ requirement if D<N for instance for D=N/2, we have (where Ei and Ri are DxD matrices) which does not necessarily imply that meaning that inverses may be avoided, creating possibilities for (great) DFT-modulated FBs, which can (see below) be PR/FIR/paraunitary • In the sequel, will give 2 examples of oversampled DFT-modulated FBs DxN DxD NxD

Should not try to understand this… Oversampled DFT-Modulated FBs Example-1 : # channels N = 8 Ho(z),H1(z),…,H7(z) decimation D = 4 prototype analysis filter Ho(z) will consider N’-fold polyphase expansion, with

u[k] Oversampled DFT-Modulated FBs In general, it is proved that the N-channel DFT-modulated (analysis) filter bank can be realized based on an N-point DFT cascaded with an NxD `polyphase matrix’ B, which contains the (N’-fold) polyphase components of the prototype Ho(z) Example-1 (continued): D=4 decimation N=8 channels Convince yourself that this is indeed correct.. (or see next slide)

u[k] Oversampled DFT-Modulated FBs Proof is simple:

4 u[k] 4 4 4 Oversampled DFT-Modulated FBs -With 4-fold decimation, this is…

4 4 4 + + + 4 Oversampled DFT-Modulated FBs - Similarly, synthesis FB is… y[k]

u[k] 4 4 4 4 + 4 4 4 4 u[k-3] Oversampled DFT-Modulated FBs • Perfect Reconstruction (PR) ?

u[k] 4 4 4 4 + 4 4 4 4 u[k-3] Oversampled DFT-Modulated FBs • FIR Unimodular Perfect Reconstruction FB Design Procedure : • Design prototype analysis filter Ho(z). • This determines En(z) (=polyphase components). • Compute pairs of Ri(z)’s from pairs of Ei(z)’s i.e. solve set of linear equations in Ri(z) coefficients : (for sufficiently high synthesis prototype filter order, this set of equations can be solved, except in special cases) = EASY !

u[k] 4 4 4 4 + 4 4 4 4 u[k-3] Oversampled DFT-Modulated FBs • FIR Paraunitary Perfect Reconstruction FB • If E(z)=F*.B(z) is chosen to be paraunitary, then PR is obtained with R(z)=B~(z).F • E(z) is paraunitary only if B(z) is paraunitary So how can we make B(z) paraunitary ?

Oversampled DFT-Modulated FBs • B(z) is paraunitary if and only if i.e. (n=0,1,2,3) are power complementary i.e. form a lossless 1-input/2-output system (explain!) • For 1-input/2-output power complementary FIR systems, see Chapter-5 on FIR lossless lattices realizations (!)…

u[k] 4 : : 4 : Oversampled DFT-Modulated FBs Lossless 1-in/2-out • Design Procedure: Optimize parameters (=angles) of 4 (=D) FIR lossless lattices (defining polyphase components of Ho(z) ) such that Ho(z) satisfies specifications. p.30 = = not-so-easy but DOABLE !

Oversampled DFT-Modulated FBs • Result = oversampled DFT-modulated FB (N=8, D=4), that is PR/FIR/paraunitary !! All great properties combined in one design !! • PS: With 2-fold oversampling(D=N/2 in example-1), paraunitary design is based on 1-input/2-output lossless systems (see page 32-33). In general, with d-fold oversampling(D=N/d), paraunitary design will be based on 1-input/d-output lossless systems (see also Chapter-5 on multi-channel FIR lossless lattices). With maximal decimation(D=N), paraunitary design will then be based on 1-input/1-output lossless systems, i.e. all-pass (polyphase) filters, which in the FIR case can only take trivial forms (=page 21-22) !

Should not try to understand this… Oversampled DFT-Modulated FBs Example-2 (non-integer oversampling) : # channels N = 6 Ho(z),H1(z),…,H5(z) decimation D = 4 prototype analysis filter Ho(z) will consider N’-fold polyphase expansion, with

u[k] Oversampled DFT-Modulated FBs DFT modulated (analysis) filter bank can be realized based on an N-point IDFT cascaded with an NxDpolyphase matrix B, which contains the (N’-fold) polyphase components of the prototype Ho(z) Convince yourself that this is indeed correct.. (or see next slide)

u[k] Oversampled DFT-Modulated FBs Proof is simple:

u[k] 4 4 4 4 Oversampled DFT-Modulated FBs -With 4-fold decimation, this is -Similar synthesis FB (R(z)=C(z).F), and then PR conditions...

Oversampled DFT-Modulated FBs • FIR Unimodular Perfect Reconstruction FB: try it.. • FIR Paraunitary Perfect Reconstruction FB: E(z) is paraunitaryiff B(z) is paraunitary B(z) is paraunitary if and only if submatrices are paraunitary (explain!) Hence paraunitary design based on (two) 2-input/3-output lossless systems. Such systems can again be FIR, then parameterized and optimized. Details skipped, but doable! = EASY ! = not-so-easy but DOABLE !

Conclusions • Uniform DFT-modulated filter banks are great: Economy in design- and implementation complexity • Maximally decimated DFT-modulated FBs: Sounds great, but no PR/FIR design flexibility  - Oversampled DFT-modulated FBs: Oversampling provides additional design flexibility, not available in maximally decimated case. Hence can have it all at once : PR/FIR/paraunitary!  PS: Equivalent PR theory for transmux’s? How does OFDM fit in?

DSP-CIS Chapter-9: Modulated Filter Banks