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Digital Signal Processing II Chapter 8: Modulated Filter Banks

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Digital Signal Processing II Chapter 8: Modulated Filter Banks Marc Moonen Dept. E.E./ESAT, K.U.Leuven marc.moonen_at_esat.kuleuven.be www.esat.kuleuven.be/scd/ ... – PowerPoint PPT presentation

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Title: Digital Signal Processing II Chapter 8: Modulated Filter Banks


1
Digital Signal Processing IIChapter 8
Modulated Filter Banks
  • Marc Moonen
  • Dept. E.E./ESAT, K.U.Leuven
  • marc.moonen_at_esat.kuleuven.be
  • www.esat.kuleuven.be/scd/

2
Part-II Filter Banks
Chapter-6
  • 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
3
Refresh (1)
  • General subband processing set-up (Chapter-6)
  • PS subband processing ignored in filter bank
    design

synthesis bank
analysis bank
downsampling/decimation
upsampling/expansion
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-6).
  • PS we are now still considering maximally
    decimated FBs, i.e.

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)

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 (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)

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

  • (polyphase decomposition)
  • then

8
Maximally Decimated DFT-Modulated FBs
  • where F is NxN DFT-matrix (and is
    complex conjugate)
  • This means that filtering with the His can
    be implemented by first filtering with polyphase
    components and then DFT

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
    Hi(z)s are
  • automatically co-designed (same
    passband ripple, etc) !

10
Maximally Decimated DFT-Modulated FBs
  • Special case DFT-filter bank, if all Ei(z)1

11
Maximally Decimated DFT-Modulated FBs
  • PS with F instead of F (as in Chapter-6), only
    filter
  • ordering is changed

12
Maximally Decimated DFT-Modulated FBs
  • DFT-modulated analysis FB maximal decimation
    (MN)

efficient realization !
13
Maximally Decimated DFT-Modulated FBs
  • 2. Synthesis FB

phase shift added for convenience
14
Maximally Decimated DFT-Modulated FBs
  • where F is NxN DFT-matrix

15
Maximally Decimated DFT-Modulated FBs

yk
16
Maximally Decimated DFT-Modulated FBs
  • Expansion (MN) DFT-modulated synthesis FB

yk

efficient realization !
yk
17
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

18
Maximally Decimated DFT-Modulated FBs
  • 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

19
Maximally Decimated DFT-Modulated FBs
  • Will consider only FIR prototype analysis
    filters, leading to simple polyphase
    decompositions (see Chapter-6).
  • 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 (i.e. are minimum-phase
  • filters). Example LPC lattice filters
    with all kilt1 (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
  • namely Ei(z)s with only 1 non-zero
    impulse response parameter.
  • E(z) is then unimodular (see
    Chapter-7). Examples see next slide.

20
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 wis
  • are constants, which leads to windowed
    IDFT/DFT bank, a.k.a. short-time Fourier
    transform (see Chapter-9)

21
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

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
    analysissynthesis filters)
  • - Paraunitary
  • Good news
  • Cosine-modulated PR FIR FBs (Chapter-9)
  • Oversampled PR FIR DFT-modulated FBs (read on)

23
Oversampled PR Filter Banks
  • So far have considered maximal decimation (MN),
    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
    NltltM.
  • 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 (!)

24
Oversampled PR Filter Banks
  • Starting point is (see Chapter-7)

  • (delta0 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)??

25
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 !

uk-3
N4 decimation
M6 channels
26
Oversampled PR Filter Banks
  • The PR condition
  • appears to be a milder requirement if MgtN
  • for instance for M2N, 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

27
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

28
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 MN, then NN, and then B is a
    diagonal matrix (cfr. supra)
  • Example-1 (continued)

N4 decimation
M8 channels
Convince yourself that this is indeed correct..
(or see next slide)
29
Oversampled DFT-Modulated FBs
  • Proof is simple

30
Oversampled DFT-Modulated FBs
  • -With (N) 4-fold decimation, this is

31
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 ?

32
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 k0,1,2,3) 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 (!)

33
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.

p.30
34
Oversampled DFT-Modulated FBs
  • Result oversampled DFT-modulated FB (M8, N4),
    that
  • is PR/FIR/paraunitary !!
  • All great properties combined in one design
    !!
  • PS
  • With 2-fold oversampling (M/N2 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
    Dinteger), 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 (D1), 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) !

35
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

36
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)

Convince yourself that this is indeed correct..
(or see next slide)
37
Oversampled DFT-Modulated FBs
  • Proof is simple

38
Oversampled DFT-Modulated FBs
  • -With (N) 4-fold decimation, this is

39
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!

40
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! ?
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