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Digital Signal Processing II Lecture 6: Maximally Decimated Filter Banks

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Title: Digital Signal Processing II Lecture 6: Maximally Decimated Filter Banks


1
Digital 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/

2
Part-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
3
PART-II Filter Banks
  • LECTURE-6 Maximally decimated FBs
  • Perfect reconstruction (PR)
  • 2-channel case
  • M-channel case
  • Interludium Paraconjugation paraunitary
    functions
  • Paraunitary PR FBs

4
Refresh (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)
5
Refresh (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

6
PS 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)

7
PS Filter bank set-up revisited
  • With ideal analysis/synthesis filters, FB
    operates as follows (1)

H1(z)
H2(z)
H3(z)
H4(z)
IN
8
PS 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
9
PS 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
10
PS Filter bank set-up revisited
  • With ideal analysis/synthesis filters, FB
    operates as follows (4)

x1
(ideal subband processing)
x1
x1
11
PS Filter bank set-up revisited
  • With ideal analysis/synthesis filters, FB
    operates as follows (5)

x2
x2
x2
12
PS Filter bank set-up revisited
  • With ideal analysis/synthesis filters, FB
    operates as follows (6)

x1
(ideal subband processing)
x1
13
PS Filter bank set-up revisited
  • With ideal analysis/synthesis filters, FB
    operates as follows (7)

x2
x2
14
PS 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
15
PS 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
16
PS 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
17
Perfect 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

18
Perfect 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)

19
Perfect 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)

?
?
20
Perfect 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.

21
Perfect 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
22
Perfect 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 !!
23
Perfect 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!
24
Perfect 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

25
Perfect 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

26
Perfect 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
27
Perfect 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)

28
Perfect 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

29
Perfect 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

30
Perfect 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

31
Perfect 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..

32
Perfect 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

33
Paraunitary 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

34
Paraunitary 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

35
Paraunitary 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

36
Paraunitary 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)

37
Paraunitary 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..

38
Paraunitary 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 !!)

39
Paraunitary 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....

40
Conclusions
  • 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.
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