DSP-CIS Chapter-9: Modulated Filter Banks Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven /

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 2 : 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 Part-II : Filter Banks Chapter-7 Chapter-8 Chapter-9 Chapter-10

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 3 General `subband processing’ set-up (Chapter-7) : PS: subband processing ignored in filter bank design downsampling/decimation Refresh (1) 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 analysis bank synthesis bank upsampling/expansion

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 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 u[k-3] u[k] PS: Equivalent perfect reconstruction condition for transmux’s ? Try it !

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 5 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)

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 6 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) H1(z) H2(z) H3(z) IN H0H3 H2H1 H0H3H2 H1 uniform non-uniform

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 7 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. H0(z) H1(z) H2(z) H3(z) u[k]

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 8 Maximally Decimated DFT-Modulated FBs where F is NxN DFT-matrix (and `*’ is complex conjugate) This means that filtering with the H n ’s can be implemented by first filtering with polyphase components and then DFT i.e.

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 9 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 H n (z)’s are automatically `co-designed’ (same passband ripple, etc…) ! u[k] i.e.

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 10 Maximally Decimated DFT-Modulated FBs Special case: DFT-filter bank, if all E n (z)=1 u[k] Ho(z)H1(z)

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 11 Maximally Decimated DFT-Modulated FBs PS: with F instead of F* (as in Chapter-6), only filter ordering is changed u[k] Ho(z)H1(z)

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 12 Maximally Decimated DFT-Modulated FBs DFT-modulated analysis FB + maximal decimation u[k] = = efficient realization !

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 13 Maximally Decimated DFT-Modulated FBs 2. Synthesis FB y[k] phase shift added for convenience

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 14 Maximally Decimated DFT-Modulated FBs where F is NxN DFT-matrix i.e.

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 15 Maximally Decimated DFT-Modulated FBs i.e. y[k] + + +

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 16 Maximally Decimated DFT-Modulated FBs Expansion + DFT-modulated synthesis FB : y[k] = = efficient realization !

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 17 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 y[k] u[k]

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 18 Maximally Decimated DFT-Modulated FBs Design Procedure : 1. Design prototype analysis filter Ho(z) (see Chapter-3). 2. This determines E n (z) (=polyphase components). 3. Assuming all E n (z) can be inverted (?), choose synthesis filters y[k] u[k]

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 19 Maximally Decimated DFT-Modulated FBs Will consider only FIR prototype analysis filter, leading to simple polyphase decomposition. However, FIR E n (z)’s generally again lead to IIR R n (z)’s, where stability is a concern… FIR unimodular E(Z)?..such that R n (z) are also FIR. Only obtained with trivial choices for the E n (z)’s, with only 1 non-zero impulse response parameter, i.e. E n (z)=α or E n (z)=α.z^{-d}. Examples: next slide all E(z)’s FIR E(z)’s FIR unimodular E(z)’s E(z)=F*.diag{..}

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 20 Maximally Decimated DFT-Modulated FBs Simple example (1) is, which leads to IDFT/DFT bank (Chapter-8) i.e. F n (z) has coefficients of H n (z), but complex conjugated and in reverse order (hence same magnitude response) (remember this?!) Simple example (2) is, where w n’s are constants, which leads to `windowed’ IDFT/DFT bank, a.k.a. `short-time Fourier transform’ (see Chapter-10)

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 21 Maximally Decimated DFT-Modulated FBs FIR paraunitary E(Z)?..such that R n (z) are FIR + power complementary FB’s. Only obtained when the E n (z)’s are all-pass filters (and FIR), i.e. E n (z)=±1 or E n (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 E(z)=F*.diag{..} FIR paraunitary E(z)’s

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 22 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-modulated PR FIR FB’s (Chapter-10) –Oversampled PR FIR DFT-modulated FB’s (read on) SIGH!

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 23 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 (!)

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 24 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-3] u[k]

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 25 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] u[k] N=6 channels D=4 decimation ! PS: Here E(z) has 6 rows (defining 6 analysis filters), with four 4-fold polyphase components in each row

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 26 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 DxDDxN NxD

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 27 Oversampled DFT-Modulated FBs Example-1 : # channels N = 8 Ho(z),H 1 (z),…,H 7 (z) decimation D = 4 prototype analysis filter Ho(z) will consider N’-fold polyphase expansion, with Should not try to understand this…

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 28 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): u[k] Convince yourself that this is indeed correct.. (or see next slide) N=8 channels D=4 decimation

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 29 Oversampled DFT-Modulated FBs Proof is simple: u[k]

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 30 Oversampled DFT-Modulated FBs -With 4-fold decimation, this is… u[k]

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 31 Oversampled DFT-Modulated FBs - Similarly, synthesis FB is… y[k]

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 32 Oversampled DFT-Modulated FBs -Perfect Reconstruction (PR) ? u[k-3] u[k]

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 33 Oversampled DFT-Modulated FBs -Perfect Reconstruction (PR) ? u[k-3] u[k]

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 34 Oversampled DFT-Modulated FBs -FIR Perfect Reconstruction FB (unimodular-like) Design Procedure : 1.Design FIR prototype analysis filter Ho(z). 2.This determines E n (z) (=polyphase components). 3.Compute pairs of FIR Ri(z)’s (Lr+1 coefficients each) from pairs of FIR Ei(z)’s (Le+1 coefficients each) 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-3] u[k] Lr+Le+1 equations in 2(Lr+1) unknowns, can (mostly) be solved if Le-1 ≤ Lr

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 35 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 ? u[k-3] u[k]

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 36 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 (!)…

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 37 Oversampled DFT-Modulated FBs 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 = u[k] : 4 4 : : Lossless 1-in/2-out = not-so-easy but DOABLE !

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 38 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) !

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 39 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 Should not try to understand this…

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 40 Oversampled DFT-Modulated FBs DFT modulated (analysis) filter bank can be realized based on an N-point IDFT cascaded with an NxD polyphase matrix B, which contains the (N’-fold) polyphase components of the prototype Ho(z) u[k] Convince yourself that this is indeed correct.. (or see next slide)

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 41 Oversampled DFT-Modulated FBs Proof is simple: u[k]

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 42 Oversampled DFT-Modulated FBs -With 4-fold decimation, this is -Similar synthesis FB (R(z)=C(z).F), and then PR conditions... u[k]

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 43 Oversampled DFT-Modulated FBs -FIR Perfect Reconstruction FB: try it.. -FIR Paraunitary Perfect Reconstruction FB: E(z) is paraunitary iff 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 !

DSP-CIS / Chapter-9: Modulated Filter Banks / Version p. 44 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?