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DSP-CIS Chapter-9: Modulated Filter Banks. Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven [email protected] www.esat.kuleuven.be / stadius /. Part-II : Filter Banks. : Preliminaries Filter bank set-up and applications

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Dsp cis chapter 9 modulated filter banks

DSP-CISChapter-9: Modulated Filter Banks

Marc Moonen

Dept. E.E./ESAT-STADIUS, KU Leuven

[email protected]

www.esat.kuleuven.be/stadius/


Part ii filter banks

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

  • Maximally decimated DFT-modulated FBs

  • Oversampled DFT-modulated FBs

    : Cosine-modulated FBs & Special topics

  • Cosine-modulated FBs

  • Time-frequency analysis & Wavelets

  • Frequency domain filtering

Chapter-7

Chapter-8

Chapter-9

Chapter-10


Refresh 1

3

3

F0(z)

subband processing

H0(z)

OUT

IN

3

3

F1(z)

subband processing

H1(z)

+

3

3

F2(z)

subband processing

H2(z)

3

3

F3(z)

subband processing

H3(z)

Refresh (1)

General `subband processing’ set-up (Chapter-7) :

PS: subband processing ignored in filter bank design

synthesis bank

analysis bank

downsampling/decimation

upsampling/expansion


Refresh 2

u[k-3]

u[k]

4

4

4

4

+

4

4

4

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-8).

PS: still considering maximally decimated FB’s, i.e.

PS: Equivalent perfect reconstruction condition for transmux’s ? Try it !


Introduction

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)


Introduction1

H0

H1

H2

H3

uniform

H0

H3

non-uniform

H1

H2

H0(z)

IN

H1(z)

H2(z)

H3(z)

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 (=one and 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)


Maximally decimated dft modulated fbs

H0(z)

u[k]

H1(z)

H2(z)

H3(z)

i.e.

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

(N-fold polyphase decomposition)

then


Maximally decimated dft modulated fbs1

i.e.

Maximally Decimated DFT-Modulated FBs

where F is NxN DFT-matrix(and `*’ is complex conjugate)

This means that filtering with the Hn’s can be implemented by first filtering with polyphase components and then DFT


Maximally decimated dft modulated fbs2

u[k]

i.e.

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 Hn(z)’s are

    automatically `co-designed’ (same passband ripple, etc…) !


Maximally decimated dft modulated fbs3

u[k]

Maximally Decimated DFT-Modulated FBs

  • Special case: DFT-filter bank, if all En(z)=1

Ho(z)

H1(z)


Maximally decimated dft modulated fbs4

u[k]

Maximally Decimated DFT-Modulated FBs

  • PS: with F instead of F* (as in Chapter-6), only filter

    ordering is changed

Ho(z)

H1(z)


Maximally decimated dft modulated fbs5

u[k]

u[k]

4

4

4

4

4

=

4

4

4

Maximally Decimated DFT-Modulated FBs

  • DFT-modulated analysis FB + maximal decimation

= efficient realization !


Maximally decimated dft modulated fbs6

y[k]

+

+

+

Maximally Decimated DFT-Modulated FBs

2. Synthesis FB

phase shift added

for convenience


Maximally decimated dft modulated fbs7

i.e.

Maximally Decimated DFT-Modulated FBs

where F is NxN DFT-matrix


Maximally decimated dft modulated fbs8

+

+

+

i.e.

Maximally Decimated DFT-Modulated FBs

y[k]


Maximally decimated dft modulated fbs9

4

4

4

4

4

4

4

+

+

+

+

+

+

4

Maximally Decimated DFT-Modulated FBs

  • Expansion + DFT-modulated synthesis FB :

y[k]

=

= efficient realization !

y[k]


Maximally decimated dft modulated fbs10

4

u[k]

4

4

4

y[k]

+

+

+

4

4

4

4

Maximally Decimated DFT-Modulated FBs

How to achieve Perfect Reconstruction (PR)

with maximally decimated DFT-modulated FBs?

polyphase components of synthesis bank prototype filter are obtained by inverting polyphase components of analysis bank prototype filter


Maximally decimated dft modulated fbs11

4

u[k]

4

4

4

y[k]

+

+

+

4

4

4

4

Maximally Decimated DFT-Modulated FBs

Design Procedure :

1. Design prototype analysis filter Ho(z) (see Chapter-3).

2. This determines En(z) (=polyphase components).

3. Assuming all En(z) can be inverted (?), choose synthesis filters


Maximally decimated dft modulated fbs12

Maximally Decimated DFT-Modulated FBs

  • Will consider only FIR prototype analysis filter, leading to simple polyphase decomposition.

  • However, FIR En(z)’s generally again lead to IIR Rn(z)’s, where stability is a concern…

  • FIR unimodular E(Z)? ..such that Rn(z) are also FIR.

    Only obtained with trivial choices for the En(z)’s, with

    only 1 non-zero impulse response parameter,

    i.e. En(z)=α or En(z)=α.z^{-d}.

    Examples: next slide

all E(z)’s

FIR E(z)’s

FIR unimodular E(z)’s

E(z)=F*.diag{..}


Maximally decimated dft modulated fbs13

Maximally Decimated DFT-Modulated FBs

  • Simple example (1) is , which leads to

    IDFT/DFT bank (Chapter-8)

    i.e. Fn(z) has coefficients of Hn(z), but complex conjugated and in

    reverse order (hence same magnitude response) (remember this?!)

  • Simple example (2) is , where wn’s

    are constants, which leads to `windowed’ IDFT/DFT bank, a.k.a. `short-time Fourier transform’(see Chapter-10)


Maximally decimated dft modulated fbs14

Maximally Decimated DFT-Modulated FBs

  • FIR paraunitary E(Z)?

    ..such that Rn(z) are FIR + power complementary FB’s.

    Only obtained when the En(z)’s are all-pass filters (and

    FIR), i.e.En(z)=±1 or En(z)=±1.z^{-d}.

    i.e. only trivial modifications

    of DFT filter bank !

    SIGH !

all E(z)’s

FIR E(z)’s

FIR unimodular E(z)’s

FIR paraunitary E(z)’s

E(z)=F*.diag{..}


Maximally decimated dft modulated fbs15

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 analysis+synthesis filters)

    - Paraunitary

  • Good news:

    • Cosine-modulatedPR FIR FB’s (Chapter-10)

    • OversampledPR FIR DFT-modulated FB’s (read on)

SIGH!


Oversampled pr filter banks

Oversampled PR Filter Banks

  • So far have considered maximal decimation (D=N), where aliasing makes PR design non-trivial.

  • With downsampling factor (D) smaller than the number of channels (N), aliasing is expected to become a smaller problem, possibly negligible if D<<N.

  • 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 (!)


Oversampled pr filter banks1

u[k-3]

u[k]

4

4

4

4

+

4

4

4

4

Oversampled PR Filter Banks

  • Starting point is(see Chapter-8):

    delta=0 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)…??


Oversampled pr filter banks2

u[k]

4

4

4

4

+

4

4

4

4

Oversampled PR Filter Banks

!

  • A more general case is :

    where E(z) is now NxD(`tall-thin’) and R(z) is DxN(`short-fat’)

    while still guarantees PR !

u[k-3]

D=4 decimation

N=6 channels

PS: Here E(z) has 6 rows (defining 6 analysis filters),

with four 4-fold polyphase components in each row


Oversampled pr filter banks3

Oversampled PR Filter Banks

  • The PR condition

    appears to be a `milder’ requirement if D<N

    for instance for D=N/2, we have (where Ei and Ri are DxD 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

DxN

DxD

NxD


Oversampled dft modulated fbs

Should not try

to understand this…

Oversampled DFT-Modulated FBs

Example-1 : # channels N = 8 Ho(z),H1(z),…,H7(z)

decimation D = 4

prototype analysis filter Ho(z)

will consider N’-fold polyphase expansion, with


Oversampled dft modulated fbs1

u[k]

Oversampled DFT-Modulated FBs

In general, it is proved that the N-channel DFT-modulated (analysis) filter

bank can be realized based on an N-point DFT cascaded with an

NxD `polyphase matrix’ B, which contains the (N’-fold) polyphase

components of the prototype Ho(z)

Example-1 (continued):

D=4 decimation

N=8 channels

Convince yourself that this is indeed correct.. (or see next slide)


Oversampled dft modulated fbs2

u[k]

Oversampled DFT-Modulated FBs

Proof is simple:


Oversampled dft modulated fbs3

4

u[k]

4

4

4

Oversampled DFT-Modulated FBs

-With 4-fold decimation, this is…


Oversampled dft modulated fbs4

4

4

4

+

+

+

4

Oversampled DFT-Modulated FBs

- Similarly, synthesis FB is…

y[k]


Oversampled dft modulated fbs5

u[k]

4

4

4

4

+

4

4

4

4

u[k-3]

Oversampled DFT-Modulated FBs

  • Perfect Reconstruction (PR) ?


Oversampled dft modulated fbs6

u[k]

4

4

4

4

+

4

4

4

4

u[k-3]

Oversampled DFT-Modulated FBs

  • Perfect Reconstruction (PR) ?


Oversampled dft modulated fbs7

u[k]

4

4

4

4

+

4

4

4

4

u[k-3]

Oversampled DFT-Modulated FBs

  • FIR Unimodular Perfect Reconstruction FB

    Design Procedure :

    • Design prototype analysis filter Ho(z).

    • This determines En(z) (=polyphase components).

    • Compute pairs of Ri(z)’s from pairs of Ei(z)’s

      i.e. solve set of linear equations in Ri(z) coefficients :

      (for sufficiently high synthesis prototype filter order,

      this set of equations can be solved, except in special cases)

= EASY !


Oversampled dft modulated fbs8

u[k]

4

4

4

4

+

4

4

4

4

u[k-3]

Oversampled DFT-Modulated FBs

  • FIR Paraunitary Perfect Reconstruction FB

    • If E(z)=F*.B(z) is chosen to be paraunitary,

      then PR is obtained with R(z)=B~(z).F

    • E(z) is paraunitary only if B(z) is paraunitary

      So how can we make B(z) paraunitary ?


Oversampled dft modulated fbs9

Oversampled DFT-Modulated FBs

  • B(z) is paraunitary if and only if

    i.e. (n=0,1,2,3) 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-5 on FIR lossless lattices realizations (!)…


Oversampled dft modulated fbs10

u[k]

4

:

:

4

:

Oversampled DFT-Modulated FBs

Lossless 1-in/2-out

  • Design Procedure: Optimize parameters (=angles) of 4 (=D)

    FIR lossless lattices (defining polyphase components of Ho(z) )

    such that Ho(z) satisfies specifications.

p.30 =

= not-so-easy but DOABLE !


Oversampled dft modulated fbs11

Oversampled DFT-Modulated FBs

  • Result = oversampled DFT-modulated FB (N=8, D=4), that

    is PR/FIR/paraunitary !!

    All great properties combined in one design !!

  • PS:

    With 2-fold oversampling(D=N/2 in example-1), paraunitary design is

    based on 1-input/2-output lossless systems (see page 32-33).

    In general, with d-fold oversampling(D=N/d), paraunitary design

    will be based on 1-input/d-output lossless systems (see also Chapter-5

    on multi-channel FIR lossless lattices).

    With maximal decimation(D=N), 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) !


Oversampled dft modulated fbs12

Should not try

to understand this…

Oversampled DFT-Modulated FBs

Example-2 (non-integer oversampling) :

# channels N = 6 Ho(z),H1(z),…,H5(z)

decimation D = 4

prototype analysis filter Ho(z)

will consider N’-fold polyphase expansion, with


Oversampled dft modulated fbs13

u[k]

Oversampled DFT-Modulated FBs

DFT modulated (analysis) filter bank can be realized based on an

N-point IDFT cascaded with an NxDpolyphase 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)


Oversampled dft modulated fbs14

u[k]

Oversampled DFT-Modulated FBs

Proof is simple:


Oversampled dft modulated fbs15

u[k]

4

4

4

4

Oversampled DFT-Modulated FBs

-With 4-fold decimation, this is

-Similar synthesis FB (R(z)=C(z).F), and then PR conditions...


Oversampled dft modulated fbs16

Oversampled DFT-Modulated FBs

  • FIR Unimodular Perfect Reconstruction FB: try it..

  • FIR Paraunitary Perfect Reconstruction FB:

    E(z) is paraunitaryiff B(z) is 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!

= EASY !

= not-so-easy but DOABLE !


Conclusions

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! 

PS: Equivalent PR theory for transmux’s? How does OFDM fit in?


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