DSP-CIS Chapter 8: Modulated Filter Banks Marc Moonen Dept. E.E./ESAT, K.U.Leuven marc.moonen@esat.kuleuven.be www.esat.kuleuven.be/scd/

Part-II : Filter Banks Chapter-6 Chapter-7 Chapter-8 Chapter-9 : 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 : Special Topics Cosine-modulated FBs Non-uniform FBs & Wavelets Frequency domain filtering Chapter-7 Chapter-8 Chapter-9

downsampling/decimation upsampling/expansion Refresh (1) General `subband processing’ set-up (Chapter-6) : PS: subband processing ignored in filter bank design analysis bank synthesis bank subband processing 3 H0(z) H1(z) H2(z) H3(z) IN F0(z) F1(z) F2(z) F3(z) + OUT downsampling/decimation upsampling/expansion

Refresh (2) Two design issues : 4 + - filter specifications, e.g. stopband attenuation, passband ripple, transition band, etc. (for each (analysis) filter!) - perfect reconstruction property (Chapter-6). PS: we are now still considering maximally decimated FB’s, i.e. 4 + u[k-3] u[k] 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)

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 H0 H3 H2 H1 uniform non-uniform

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 (polyphase decomposition) then H0(z) H1(z) H2(z) H3(z) u[k] i.e.

Maximally Decimated DFT-Modulated FBs where F is NxN DFT-matrix (and `*’ is complex conjugate) This means that filtering with the Hi’s can be implemented by first filtering with polyphase components and then DFT 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 Hi(z)’s are automatically `co-designed’ (same passband ripple, etc…) ! i.e. u[k]

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

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)

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

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

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

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

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

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

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

Maximally Decimated DFT-Modulated FBs Will consider only FIR prototype analysis filters, leading to simple polyphase decompositions. However, FIR Ei(z)’s generally still lead to IIR Ri(z)’s, where stability is a concern… Ri(z) ’s are stable only if Ei(z)’s have stable zeros (`minimum-phase filters’). Example: LPC lattice filters with all |ki|<1 (see Chapter-4). The design of such minimum phase FIR filters is (significantly) more difficult.. FIR Ri(z)’s (=guaranteed stability) are only obtained with trivial choices for the Ei(z)’s, i.e. with only 1 non-zero impulse response parameter. E(z) is then unimodular (see Chapter-7). Examples: see next slide.

Maximally Decimated DFT-Modulated FBs Simple example (1) is , which leads to IDFT/DFT bank (Chapter-6) 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 Chapter-9)

Maximally Decimated DFT-Modulated FBs Question (try to answer): Can we have paraunitary FBs here (=desirable property) ? When is maximally decimated DFT-modulated FB at the same time - PR - FIR (both analysis & synthesis) - Paraunitary ? Hint: E(z) is paraunitary only if the Ei(z)’s are all-pass filters. An FIR all-pass filter takes a trivial form, e.g. Ei(z)=1 or Ei(z)=z^{-d}

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-9) Oversampled PR FIR DFT-modulated FB’s (read on)

Oversampled PR Filter Banks So far have considered maximal decimation (M=N), where aliasing makes PR design non-trivial. With downsampling factor (N) smaller than the number of channels (M), aliasing is expected to become a smaller problem, possibly negligible if N<<M. 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 (!)

Oversampled PR Filter Banks Starting point is (see Chapter-7): (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)…?? 4 + u[k-3] u[k]

Oversampled PR Filter Banks ! A more general case is : where E(z) is now MxN (`tall-thin’) and R(z) is NxM (`short-fat’) while still guarantees PR ! 4 + u[k] u[k-3] N=4 decimation M=6 channels

Oversampled PR Filter Banks The PR condition appears to be a `milder’ requirement if M>N for instance for M=2N, we have (where Ei and Ri are NxN 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

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

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

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

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

Oversampled DFT-Modulated FBs Perfect Reconstruction (PR) can now be obtained based on an E(z) that is FIR and paraunitary : If E(z )=F*.B(z) is chosen to be paraunitary, then PR is obtained with R(z)=B~(z).F (=NxM) (=DFT-modulated synthesis bank). E(z) is paraunitary only if B(z) is paraunitary. So how can we make B(z) paraunitary ? 4 + u[k-3] u[k]

Oversampled DFT-Modulated FBs Example 1 (continued) : From the structure of B(z) It follows that B(z) is paraunitary if and only if (for k=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-6 on FIR lossless lattices realizations (!)…

Oversampled DFT-Modulated FBs Lossless 1-in/2-out Design Procedure: Optimize parameters (=angles) of (4) FIR lossless lattices (defining polyphase components of Ho(z) ) such that Ho(z) satisfies specifications. u[k] : 4 p.30 =

Oversampled DFT-Modulated FBs Result = oversampled DFT-modulated FB (M=8, N=4), that is PR/FIR/paraunitary !! All great properties combined in one design !! PS: With 2-fold oversampling (M/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 (for D=integer), paraunitary design will be based on 1-input/D-output lossless systems (see also Chapter-3 on multi-channel FIR lossless lattices). With maximal decimation (D=1), 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) !

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

Oversampled DFT-Modulated FBs DFT modulated (analysis) filter bank can be realized based on an M-point IDFT cascaded with an MxN 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)

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

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

Oversampled DFT-Modulated FBs Perfect Reconstruction by paraunitariness? - E(z) paraunitary iff B(z) 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!

PS: Equivalent PR theory for transmux’s? How does OFDM fit in? 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?