Title: Digital Signal Processing II Chapter 8: Modulated Filter Banks
1Digital 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/
2PartII Filter Banks
Chapter6
 Preliminaries
 Filter bank setup and applications
 Perfect reconstruction problem 1st example
(DFT/IDFT)  Multirate systems review (10 slides)
 Maximally decimated FBs
 Perfect reconstruction filter banks (PR FBs)
 Paraunitary PR FBs
 Modulated FBs
 Maximally decimated DFTmodulated FBs
 Oversampled DFTmodulated FBs
 Special Topics
 Cosinemodulated FBs
 Nonuniform FBs Wavelets
 Frequency domain filtering
Chapter7
Chapter8
Chapter9
3Refresh (1)
 General subband processing setup (Chapter6)

 PS subband processing ignored in filter bank
design
synthesis bank
analysis bank
downsampling/decimation
upsampling/expansion
4Refresh (2)
 Two design issues
  filter specifications, e.g. stopband
attenuation, passband ripple, transition band,
etc. (for each (analysis) filter!)   perfect reconstruction property (Chapter6).
 PS we are now still considering maximally
decimated FBs, i.e.
5Introduction
 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.
 DFTmodulated FBs (this Chapter)
 Cosinemodulated FBs (next Chapter)
6Introduction
 Uniform versus nonuniform (analysis) filter
bank 
 Nchannel 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
 nonuniform everything that is not uniform
 e.g. for speech audio applications
(cfr. human hearing)  example wavelet filter banks (next
Chapter)
7Maximally Decimated DFTModulated FBs
 Uniform filter banks can be realized cheaply
based on  polyphase decompositions DFT(FFT) (hence name
DFTmodulated FB)  1. Analysis FB
 If

(polyphase decomposition)  then
8Maximally Decimated DFTModulated FBs


 where F is NxN DFTmatrix (and is
complex conjugate)  This means that filtering with the His can
be implemented by first filtering with polyphase
components and then DFT
9Maximally Decimated DFTModulated 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 codesigned (same
passband ripple, etc) !
10Maximally Decimated DFTModulated FBs
 Special case DFTfilter bank, if all Ei(z)1

11Maximally Decimated DFTModulated FBs
 PS with F instead of F (as in Chapter6), only
filter  ordering is changed

12Maximally Decimated DFTModulated FBs
 DFTmodulated analysis FB maximal decimation
(MN) 

efficient realization !
13Maximally Decimated DFTModulated FBs
phase shift added for convenience
14Maximally Decimated DFTModulated FBs


 where F is NxN DFTmatrix
15Maximally Decimated DFTModulated FBs
yk
16Maximally Decimated DFTModulated FBs
 Expansion (MN) DFTmodulated synthesis FB



yk
efficient realization !
yk
17Maximally Decimated DFTModulated FBs
 How to achieve Perfect Reconstruction (PR)
 with maximally decimated DFTmodulated FBs?

 i.e. synthesis bank polyphase components are
obtained by inverting analysis bank polyphase
components
18Maximally Decimated DFTModulated FBs
 Design Procedure
 1. Design prototype analysis filter Ho(z)
(see Chapter3).  2. This determines Ei(z) (polyphase
components).  3. Assuming all Ei(z) can be inverted (?),
choose synthesis filters
19Maximally Decimated DFTModulated FBs
 Will consider only FIR prototype analysis
filters, leading to simple polyphase
decompositions (see Chapter6).  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 minimumphase  filters). Example LPC lattice filters
with all kilt1 (see Chapter4).  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 nonzero
impulse response parameter.  E(z) is then unimodular (see
Chapter7). Examples see next slide.
20Maximally Decimated DFTModulated FBs
 Simple example (1) is ,
which leads to  IDFT/DFT bank
(Chapter6)  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. shorttime Fourier
transform (see Chapter9)
21Maximally Decimated DFTModulated FBs
 Question (try to answer)
 Can we have paraunitary FBs here (desirable
property) ?  When is maximally decimated DFTmodulated FB
 at the same time
  PR
  FIR (both analysis synthesis)
  Paraunitary ?
 Hint
 E(z) is paraunitary only if the Ei(z)s
are allpass filters.  An FIR allpass filter takes a trivial
form, e.g. Ei(z)1 or Ei(z)zd
22Maximally Decimated DFTModulated 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
 Cosinemodulated PR FIR FBs (Chapter9)
 Oversampled PR FIR DFTmodulated FBs (read on)
23Oversampled PR Filter Banks
 So far have considered maximal decimation (MN),
where aliasing makes PR design nontrivial.  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 DFTmodulated FBs that are
 PR/FIR/paraunitary (!)
24Oversampled PR Filter Banks
 Starting point is (see Chapter7)

(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)??
25Oversampled PR Filter Banks
!
 A more general case is
 where E(z) is now MxN (tallthin) and R(z) is
NxM (shortfat)  while still
guarantees PR !
uk3
N4 decimation
M6 channels
26Oversampled 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)  DFTmodulated FBs, which can (see below) be
PR/FIR/paraunitary  In the sequel, will give 2 examples of
oversampled DFTmodulated FBs
27Oversampled DFTModulated FBs

 Example1 channels M 8
Ho(z),H1(z),,H7(z)  decimation N 4
 prototype analysis filter
Ho(z)  will consider Nfold
polyphase expansion, with 
28Oversampled DFTModulated FBs
 In general, it is proved that the Mchannel
DFTmodulated (analysis) filter  bank can be realized based on an Mpoint DFT
cascaded with an  MxN polyphase matrix B, which contains the
(Nfold) polyphase  components of the prototype Ho(z)
 ps note that if MN, then NN, and then B is a
diagonal matrix (cfr. supra)  Example1 (continued)
N4 decimation
M8 channels
Convince yourself that this is indeed correct..
(or see next slide)
29Oversampled DFTModulated FBs
30Oversampled DFTModulated FBs
 With (N) 4fold decimation, this is
31Oversampled DFTModulated 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)
(DFTmodulated synthesis bank).  E(z) is paraunitary only if B(z) is paraunitary.
 So how can we make B(z) paraunitary ?
32Oversampled DFTModulated 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 1input/2output system
(explain!)  For 1input/2output power complementary FIR
systems,  see Chapter6 on FIR lossless lattices
realizations (!)
33Oversampled DFTModulated FBs
Lossless 1in/2out



 Design Procedure Optimize parameters (angles)
of (4)  FIR lossless lattices (defining polyphase
components of Ho(z) )  such that Ho(z) satisfies specifications.
p.30
34Oversampled DFTModulated FBs
 Result oversampled DFTmodulated FB (M8, N4),
that  is PR/FIR/paraunitary !!
 All great properties combined in one design
!!  PS
 With 2fold oversampling (M/N2 in
example1), paraunitary design is  based on 1input/2output lossless systems
(see page 3233).  In general, with Dfold oversampling (for
Dinteger), paraunitary design  will be based on 1input/Doutput lossless
systems (see also Chapter3  on multichannel FIR lossless lattices).
 With maximal decimation (D1), paraunitary
design will then be based  on 1input/1output lossless systems, i.e.
allpass (polyphase) filters,  which in the FIR case can only take trivial
forms (page 2122) !
35Oversampled DFTModulated FBs
 Example2 (noninteger oversampling)
 channels M 6
Ho(z),H1(z),,H5(z)  decimation N 4
 prototype analysis filter
Ho(z)  will consider Nfold
polyphase expansion, with 
36Oversampled DFTModulated FBs
 DFT modulated (analysis) filter bank can be
realized based on an  Mpoint IDFT cascaded with an MxN polyphase
matrix B, which contains  the (Nfold) polyphase components of the
prototype Ho(z)
Convince yourself that this is indeed correct..
(or see next slide)
37Oversampled DFTModulated FBs
38Oversampled DFTModulated FBs
 With (N) 4fold decimation, this is
39Oversampled DFTModulated 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)
2input/3output  lossless systems. Such systems can again be
FIR, then  parameterized and optimized. Details
skipped, but doable!
40Conclusions
 Uniform DFTmodulated filter banks are great
 Economy in design and implementation
complexity  Maximally decimated DFTmodulated FBs
 Sounds great, but no PR/FIR design
flexibility ?   Oversampled DFTmodulated FBs
 Oversampling provides additional design
flexibility,  not available in maximally decimated case.
 Hence can have it all at once
PR/FIR/paraunitary! ?