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DSP-CIS Chapter-9: Modulated Filter Banks

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DSP-CISChapter-9: Modulated Filter Banks

Marc Moonen

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

marc.moonen@esat.kuleuven.be

www.esat.kuleuven.be/stadius/

: 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

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)

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

PS: subband processing ignored in filter bank design

synthesis bank

analysis bank

downsampling/decimation

upsampling/expansion

u[k-3]

u[k]

4

4

4

4

+

4

4

4

4

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 !

-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)

H0

H1

H2

H3

uniform

H0

H3

non-uniform

H1

H2

H0(z)

IN

H1(z)

H2(z)

H3(z)

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)

H0(z)

u[k]

H1(z)

H2(z)

H3(z)

i.e.

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

i.e.

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

u[k]

i.e.

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…) !

u[k]

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

Ho(z)

H1(z)

u[k]

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

Ho(z)

H1(z)

u[k]

u[k]

4

4

4

4

4

=

4

4

4

- DFT-modulated analysis FB + maximal decimation

= efficient realization !

y[k]

+

+

+

2. Synthesis FB

phase shift added

for convenience

i.e.

where F is NxN DFT-matrix

+

+

+

i.e.

y[k]

4

4

4

4

4

4

4

+

+

+

+

+

+

4

- Expansion + DFT-modulated synthesis FB :

y[k]

=

= efficient realization !

y[k]

4

u[k]

4

4

4

y[k]

+

+

+

4

4

4

4

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

4

u[k]

4

4

4

y[k]

+

+

+

4

4

4

4

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

- 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{..}

- 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)

- 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{..}

- 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!

- 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 (!)

u[k-3]

u[k]

4

4

4

4

+

4

4

4

4

- 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)…??

u[k]

4

4

4

4

+

4

4

4

4

!

- 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

- 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

Should not try

to understand this…

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

u[k]

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)

u[k]

Proof is simple:

4

u[k]

4

4

4

-With 4-fold decimation, this is…

4

4

4

+

+

+

4

- Similarly, synthesis FB is…

y[k]

u[k]

4

4

4

4

+

4

4

4

4

u[k-3]

- Perfect Reconstruction (PR) ?

u[k]

4

4

4

4

+

4

4

4

4

u[k-3]

- Perfect Reconstruction (PR) ?

u[k]

4

4

4

4

+

4

4

4

4

u[k-3]

- 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 !

u[k]

4

4

4

4

+

4

4

4

4

u[k-3]

- 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 ?

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

- 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 (!)…

u[k]

4

:

:

4

:

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 !

- 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) !

Should not try

to understand this…

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

u[k]

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)

u[k]

Proof is simple:

u[k]

4

4

4

4

-With 4-fold decimation, this is

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

- 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 !

- 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?