Title: Digital Signal Processing II Lecture 6: Maximally Decimated Filter Banks
1Digital Signal Processing IILecture 6
Maximally Decimated Filter Banks
- Marc Moonen
- Dept. E.E./ESAT, K.U.Leuven
- marc.moonen_at_esat.kuleuven.be
- www.esat.kuleuven.be/scd/
2Part-II Filter Banks
- 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
- DFT-modulated FBs
- Cosine-modulated FBs
- Special Topics
- Non-uniform FBs Wavelets
- Oversampled DFT-modulated FBs
- Frequency domain filtering
Lecture-5
Lecture-6
Lecture-7
Lecture-8
3PART-II Filter Banks
- LECTURE-6 Maximally decimated FBs
-
- Perfect reconstruction (PR)
- 2-channel case
- M-channel case
- Interludium Paraconjugation paraunitary
functions - Paraunitary PR FBs
4Refresh (1)
- General subband processing set-up
- - analysis bank synthesis bank
- - multi-rate structure down-sampling after
analysis, up-sampling for synthesis - - aliasing vs. perfect reconstruction
- - applications coding, (adaptive) filtering,
transmultiplexers - - PS subband processing ignored in filter
bank design -
subband processing
3
F0(z)
H0(z)
subband processing
3
IN
F1(z)
H1(z)
subband processing
3
F2(z)
H2(z)
subband processing
3
F3(z)
H3(z)
5Refresh (2)
- Two design issues
- - filter specifications, e.g. stopband
attenuation, passband ripple, transition band,
etc. (for each (analysis) filter!) - - perfect reconstruction property.
- PS Perfect reconstruction property as such is
easily satisfied, if there arent - any (analysis) filter specs, e.g. (see
Lecture-5) -
- but this is not very useful/practical.
Tight filter specs. necessary - for subband coding, etc.
- This lecture Maximally decimated FBs
6PS Filter bank set-up revisited
- - analysis filters Hi(z) are also decimation
(anti-aliasing) filters, to - avoid aliased contributions in subband
signals - - synthesis filters Gi(z) are also
interpolation filters, to remove - images after expanders (upsampling)
7PS Filter bank set-up revisited
- With ideal analysis/synthesis filters, FB
operates as follows (1) -
H1(z)
H2(z)
H3(z)
H4(z)
IN
8PS Filter bank set-up revisited
- With ideal analysis/synthesis filters, FB
operates as follows (2) -
H1(z)
H2(z)
H3(z)
H4(z)
x1
x1
9PS Filter bank set-up revisited
- With ideal analysis/synthesis filters, FB
operates as follows (3) -
H1(z)
H2(z)
H3(z)
H4(z)
x2
x2
10PS Filter bank set-up revisited
- With ideal analysis/synthesis filters, FB
operates as follows (4) -
x1
(ideal subband processing)
x1
x1
11PS Filter bank set-up revisited
- With ideal analysis/synthesis filters, FB
operates as follows (5) -
x2
x2
x2
12PS Filter bank set-up revisited
- With ideal analysis/synthesis filters, FB
operates as follows (6) -
x1
(ideal subband processing)
x1
13PS Filter bank set-up revisited
- With ideal analysis/synthesis filters, FB
operates as follows (7) -
x2
x2
14PS Filter bank set-up revisited
- With ideal analysis/synthesis filters, FB
operates as follows (8) -
G1(z)
G2(z)
G3(z)
G4(z)
x1
(ideal subband processing)
x1
15PS Filter bank set-up revisited
- With ideal analysis/synthesis filters, FB
operates as follows (9) -
G1(z)
G2(z)
G3(z)
G4(z)
x2
x2
16PS Filter bank set-up revisited
- With ideal analysis/synthesis filters, FB
operates as follows (10) -
H1(z)
H2(z)
H3(z)
H4(z)
OUTIN
subband processing
4
G1(z)
H1(z)
OUT
subband processing
4
IN
G2(z)
H2(z)
subband processing
4
G3(z)
H3(z)
subband processing
4
G4(z)
H4(z)
Now try this with non-ideal filters
17Perfect Reconstruction 2-Channel Case
- It is proved that... (try it!)
- U(-z) represents aliased signals, hence the
alias transfer function A(z) should ideally be
zero - T(z) is referred to as distortion function
(amplitude phase distortion). For perfect
reconstruction, T(z) should ideally be a pure
delay
18Perfect Reconstruction 2-Channel Case
- Requirement for alias-free filter bank
- If A(z)0, then Y(z)T(z).U(z), hence the
complete filter bank behaves as a linear time
invariant (LTI) system (despite up-
downsampling) !!!! -
- Requirement for perfect reconstruction filter
bank - ( alias-free distortion-free)
- i)
- ii)
19Perfect Reconstruction 2-Channel Case
- A first attempt is as follows..
-
- so that
-
-
- For the real coefficient case, i.e.
-
- which means the amplitude response of H1 is
the mirror image of the - amplitude response of Ho with respect to the
quadrature frequency - hence the name quadrature mirror filter
(QMF)
?
?
20Perfect Reconstruction 2-Channel Case)
- quadrature mirror filter (QMF)
-
- hence if Ho (Fo) is designed to be a good
lowpass filter, then H1 (-F1) is a good
high-pass filter.
21Perfect Reconstruction 2-Channel Case
- A 2nd (better) attempt is as follows
- Smith Barnwell 1984 Mintzer 1985
- i)
- so that (alias cancellation)
- ii) power symmetric Ho(z) (real
coefficients case) -
-
- iii)
- so that (distortion function)
ignore the details! - This is a so-calledparaunitary perfect
reconstruction bank (see below), - based on a lossless system Ho,H1
?
?
This is already pretty complicated
22Perfect Reconstruction M-Channel Case
- It is proved that... (try it!)
- 2nd term represents aliased signals, hence all
alias transfer functions Al(z) should ideally
be zero (for all l ) - H(z) is referred to as distortion function
(amplitude phase distortion). For perfect
reconstruction, H(z) should ideally be a pure
delay
Sigh !!
23Perfect Reconstruction M-Channel Case
- A simpler analysis results from a polyphase
description -
-
-
i-th row of E(z) has polyphase -
components of Hi(z) -
i-th column of R(z) has polyphase -
components of Fi(z) -
uk
uk-3
Do not continue until you understand how formulae
correspond to block scheme!
24Perfect Reconstruction M-Channel Case
- with the noble identities, this is equivalent
to -
- Necessary sufficient conditions for
- i) alias cancellation
- ii) perfect reconstruction
- are then derived, based on the product
-
-
25Perfect Reconstruction M-Channel Case
- Necessary sufficient condition for alias-free
FB is - a pseudo-circulant matrix is a
circulant matrix with the additional feature - that elements below the main
diagonal are multiplied by 1/z, i.e. - ..and first row of R(z).E(z) are polyphase cmpnts
of distortion function T(z) -
read on-gt
26Perfect Reconstruction M-Channel Case
- PS This can be explained as follows
- first, previous block scheme is equivalent to
(cfr. Noble identities) - then (iff R.E is pseudo-circ.)
- so that finally..
-
4
4
4
4
T(z)uk-3
4
4
uk
4
4
27Perfect Reconstruction M-Channel Case
- Hence necessary sufficient condition for PR
- (where T(z)pure delay)
- I_n is nxn identity matrix, r is arbitrary
-
- (Obvious) example
(r0)
28Perfect Reconstruction M-Channel Case
- For conciseness, will use this from now on
- - Procedure
- 1. Design all analysis filters (see
Part-I). - 2. This determines E(z) (polyphase
matrix). - 3. Assuming E(z) can be inverted (?),
choose synthesis filters - - Example DFT/IDFT Filter bank (Lecture-5)
E(z)F , R(z)F-1 - - FIR E(z) generally leads to IIR R(z), where
stability is a concern
29Perfect Reconstruction M-Channel Case
- PS Inversion of matrix transfer functions ?
- The inverse of a scalar (i.e. 1-by-1 matrix) FIR
transfer function is always IIR (except for
contrived examples) - The inverse of an N-by-N (Ngt1) FIR transfer
function can be FIR -
30Perfect Reconstruction M-Channel Case
- PS Inversion of matrix transfer functions ?
- Compare this to inversion of integers
and integer matrices - The inverse of an integer is always non-integer
(except for E1) - The inverse of an N-by-N (Ngt1) integer matrix
can be integer -
31Perfect Reconstruction M-Channel Case
- Question
- How can we find polynomial (FIR) matrices E(z)
that have a FIR inverse? - Answer
- Unimodular matrices matrices with
determinantconstantz-d - Example
- where the Eis are constant (not a function
of z) invertible matrices - Procedure optimize Eis to obtain analysis
filter specs (ripple, etc.), etc..
32Perfect Reconstruction M-Channel Case
- Question
- Can we avoid direct inversion, e.g. through
the usage FIR - E(z) matrices with additional special
properties, so that - R(z) is trivially obtained and its specs are
better controlled? - (compare with (real) orthogonal or (complex)
unitary matrices, where - inverse is equal to (hermitian) transpose)
- Answer
- YES, paraunitary matrices
- (special class of FIR matrices
with FIR inverse) - See next slides.
- Will focus on paraunitary E(z) leading to
paraunitary PR filter banks
33Paraunitary PR Filter Banks
- Interludium PARACONJUGATION
- For a scalar transfer function H(z),
paraconjugate is - i.e it is obtained from H(z) by
- - replacing z by 1/z
- - replacing each coefficient by its
complex conjugate - Example
-
- On the unit circle, paraconjugation
corresponds to complex conjugation -
- paraconjugation analytic extension of
unit-circle conjugation
34Paraunitary PR Filter Banks
- Interludium PARACONJUGATION
- For a matrix transfer function H(z),
paraconjugate is - i.e it is obtained from H(z) by
- - transposition
- - replacing z by 1/z
- - replacing each coefficient by is
complex conjugate - Example
-
-
- On the unit circle, paraconjugation
corresponds to transpose conjugation -
- paraconjugation analytic extension of
unit-circle transpose conjugation
35Paraunitary PR Filter Banks
- Interludium PARAUNITARY matrix transfer
functions - Matrix transfer function H(z), is paraunitary if
(possibly up to a scalar) - For a square matrix function
-
- A paraunitary matrix is unitary on the unit
circle -
- paraunitary analytic extension of
unit-circle unitary. -
- PS if H1(z) and H2(z) are paraunitary, then
H1(z).H2(z) is paraunitary
36Paraunitary PR Filter Banks
- - If E(z) is paraunitary
- then perfect reconstruction is obtained with
-
(delta to make synthesis causal) - If E(z) is FIR, then R(z) is also FIR !!
(cfr. definition paraconjugation)
37Paraunitary PR Filter Banks
- Example paraunitary FIR E(z) with FIR inverse
R(z) - where the Eis are constant unitary
matrices - Example 1-input/2-output FIR lossless
lattice, see lecture 3, p. 22 - Example 1-input/M-output FIR lossless
lattice, see lecture 3, p. 27 - Procedure optimize unitary Eis (e.g. rotation
angles in lossless lattices) to obtain analysis
filter specs, etc..
38Paraunitary PR Filter Banks
- Properties of paraunitary PR filter banks
(proofs omitted) - If polyphase matrix E(z) (and hence E(zN)) is
paranunitary, and - then vector transfer function H(z) (all
analysis filters ) is paraunitary - If vector transfer function H(z) is paraunitary,
then its components are power complementary
(lossless 1-input/N-output system) -
(see lecture 3 !!)
39Paraunitary PR Filter Banks
- Properties of paraunitary PR filter banks
(continued) - Synthesis filters are obtained from analysis
filters by conjugating the analysis filter
coefficients reversing the order (cfr page 34) - Hence magnitude response of synthesis filter Fk
is the same as magnitude response of
corresponding analysis filter Hk - Hence, as analysis filters are power
complementary (cfr. supra), synthesis filters are
also power complementary - example DFT/IDFT bank (lecture-5), 2-channel
case (page 21) - Great properties/designs....
40Conclusions
- Have derived general conditions for perfect
reconstruction, based on polyphase matrices for
analysis/synthesis bank - Seen example of general PR filter bank design
- Paraunitary FIR PR FBs, e.g. based on FIR
lossless - lattice filters
- Sequel other (better) PR structures
- Lecture 7 Modulated filter banks
- Lecture 8 Oversampled filter banks, etc..
- Reference Multirate Systems Filter Banks ,
P.P. Vaidyanathan Prentice Hall 1993.