Digital Signal Processing II Lecture 7: Modulated Filter Banks

1
Digital Signal Processing IILecture 7
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
Lecture-5

• 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

Lecture-6
Lecture-7
Lecture-8
3
Refresh (1)

• General subband processing set-up (Lecture-5)
  
• PS subband processing ignored in filter bank
  design

synthesis bank
analysis bank
downsampling/decimation
upsampling/interpolation
4
Refresh (2)

• Two design issues
• - filter specifications, e.g. stopband
  attenuation, passband ripple, transition band,
  etc. (for each (analysis) filter!)
• - perfect reconstruction property (Lecture-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 lecture)
• Cosine-modulated FBs (next lecture)

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
  lecture)

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 Lecture-5), 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 Lecture-2).
• 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 Lecture-5).
• 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 Lecture-3).
• 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
  Lecture-6). Examples see next slide.

20
Maximally Decimated DFT-Modulated FBs

• 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 wis
• are constants, which leads to windowed
  IDFT/DFT bank, a.k.a. short-time Fourier
  transform (see Lecture-8)

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...
• i) Perfect
  reconstruction (PR)
• ii) FIR (all
  analysissynthesis filters)
• iii) Paraunitary
• Good news
• Cosine-modulated PR FIR FBs (Lecture-8)
• 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 Lecture-6)
• 
  (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 Lecture-3 on 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 Lecture-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
• Sounded great, but no 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! ?