DSP-CIS Chapter-9: Modulated Filter Banks Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven [email protected]

DSP-CIS

Chapter-9: Modulated Filter Banks

Marc MoonenDept. E.E./ESAT-STADIUS, KU Leuven

[email protected]

www.esat.kuleuven.be/stadius/

DSP-CIS / Chapter-9: Modulated Filter Banks / Version 2014-2015 p. 2

: 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

Part-II : Filter Banks

Chapter-7

Chapter-8

Chapter-9

Chapter-10


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

PS: subband processing ignored in filter bank design

downsampling/decimation

Refresh (1)

subband processing 3H0(z)



3

3

3

3 subband processing 3H3(z)

IN

F0(z)

F1(z)

F2(z)

F3(z)

+

OUT

analysis bank synthesis bank

upsampling/expansion


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.

4444

+u[k-3]

1z

2z

3z

1

1z2z3z

1

u[k] 444

4)(zE )(zR

NIzzz )().( ER

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


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)


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)

H0(z)

H1(z)

H2(z)

H3(z)

INH0 H3H2H1

H0 H3H2H1uniform

non-uniform


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

i.e.

H0(z)

H1(z)

H2(z)

H3(z)

u[k]



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

Nj

NN

N

N

N

N

NNN

N

N

N

eW

zU

zEz

zEz

zEz

zE

F

WWWW

WWWW

WWWW

WWWW

zU

zH

zH

zH

zH

/2

11

22

11

0

)1()1(2)1(0

)1(2420

)1(210

0000

1

2

1

0

)(.

)(.

:

)(.

)(.

)(

.

*

...

::::

...

...

...

)(.

)(

:

)(

)(

)(

2

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…) !

*F

u[k]

)( 40 zE

)( 41 zE

)( 42 zE

)( 43 zE

)(0 zH

)(1 zH

)(2 zH

)(3 zH

i.e.



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

*F

u[k]

1 )(0 zH

)(1 zH

)(2 zH

)(3 zH

11

1

Ho(z) H1(z)



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

Fu[k]

1 )(0 zH

)(1 zH

)(2 zH

)(3 zH11

1

Ho(z) H1(z)



• DFT-modulated analysis FB + maximal decimation

*F 4

4

4

4u[k]

)( 40 zE

)( 41 zE

)( 42 zE

)( 43 zE

4

4

4

4u[k]

*F)(0 zE

)(1 zE

)(2 zE

)(3 zE

= = efficient realization !



2. Synthesis FB

+

+

+

)(0 zF][0 ku

)(1 zF][1 ku

)(2 zF][2 ku

)(3 zF][3 kuy[k]

phase shift added

for convenience



where F is NxN DFT-matrix

Nj

NNNN

N

N

NNNNN

N

N

N

eW

zU

zU

zU

zU

F

WWWW

WWWW

WWWW

WWWW

zRzRzzRzzRz

zU

zU

zU

zU

zFzFzFzFzY

/2

1

2

1

0

)1()1(2)1(0

)1(2420

)1(210

0000

011

22

11

1

2

1

0

1210

)(

:

)(

)(

)(

.

...

::::

...

...

...

.)()(.)(....)(.

)(

:

)(

)(

)(

.)(...)()()()(

2

i.e.



i.e.

y[k]

+

+

+)( 40 zR

)( 41 zR

)( 42 zR

)( 43 zR][0 ku

][1 ku

][2 ku

][3 ku

F



• Expansion + DFT-modulated synthesis FB :

y[k]

][0 ku

][1 ku

][2 ku

][3 ku

4

4

4

4

+

+

+)(0 zR

)(1 zR

)(2 zR

)(3 zR

F

y[k]

+

+

+

4

4

4

4 )( 40 zR

)( 41 zR

)( 42 zR

)( 43 zR][0 ku

][1 ku

][2 ku

][3 ku

F

= = efficient realization !



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

y[k]

44

4

4

+

+

+)(0 zR

)(1 zR

)(2 zR

)(3 zR

F4

4

4

4u[k]

*F)(0 zE

)(1 zE

)(2 zE

)(3 zE

NIzzz )().( ER



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

y[k]

4444

+

+

+)(0 zR

)(1 zR

)(2 zR

)(3 zR

F444

4u[k]

*F)(0 zE

)(1 zE

)(2 zE

)(3 zE



• 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

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

FIR paraunitary E(z)’s



• 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-modulated PR FIR FB’s (Chapter-10)

– Oversampled PR FIR DFT-modulated FB’s (read on)

SIG

H!


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 (!)



• 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)…??

4444

+u[k-3]

1z

2z

3z

1

1z2z3z

1

u[k] 444

4

)(zE )(zR



• 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]

4444

+1z

2z

3z

1

1z2z3z

1

u[k] 444

4

)(zE )(zR

N=

6 ch

ann

els

D=

4 d

ecim

atio

n

!

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

)()( 111 zz ER

DxDDxN

NxD


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

Sh

ou

ld n

ot

try

to u

nd

ers

tan

d t

his

…



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):

*88xF

u[k]

)(0 zH

)(6 zH

)(7 zH

484 )( xzB

)(.000

0)(.00

00)(.0

000)(.

)(000

0)(00

00)(0

000)(

)(

87

4

86

4

85

4

84

4

83

82

81

80

4

zEz

zEz

zEz

zEz

zE

zE

zE

zE

zB

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

=8

chan

nel

s

D=

4 d

ecim

atio

n



Proof is simple:*

88xF

u[k]

)(0 zH

)(6 zH

)(7 zH

484 )( xzB

)(.

)(

)(

)(

)(

)(

)(

)(

)(

)(.

)(.

)(.

)(.

)(.

)(.

)(.

)(.

)(

....)(.

1

).(.

7

6

5

4

3

2

1

0

87

7

86

6

85

5

84

4

83

3

82

2

81

1

80

*

3

2

14* zU

zH

zH

zH

zH

zH

zH

zH

zH

zU

zEz

zEz

zEz

zEz

zEz

zEz

zEz

zE

FzU

z

z

zzBF



-With 4-fold decimation, this is…

u[k]

*88xF48)( xzB

4

4

4

4

)(.)( * zBFz E

)(.000

0)(.00

00)(.0

000)(.

)(000

0)(00

00)(0

000)(

)(

27

1

26

1

25

1

24

1

23

22

21

20

zEz

zEz

zEz

zEz

zE

zE

zE

zE

zB



- Similarly, synthesis FB is…

y[k]

4

4

4

4

+

+

+



- Perfect Reconstruction (PR) ?

4

4

4

4

+u[k-3]

1z

2z

3z

1

1z2z3z

1

u[k] 4

4

4

4

)(zE )(zR

)(.)( * zBFz E



- Perfect Reconstruction (PR) ?

4

4

4

4

+u[k-3]

1z

2z

3z

1

1z2z3z

1

u[k] 4

4

4

4

)(zE )(zR

)(.)( * zBFz E



- FIR Perfect Reconstruction FB (unimodular-like)

Design Procedure :1. Design FIR prototype analysis filter Ho(z).

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

3. Compute pairs of FIR Ri(z)’s (Lr+1 coefficients each) from pairs of FIR Ei(z)’s (Le+1 coefficients each)

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 !

4

4

4

4

+u[k-3]

1z

2z

3z

1

1z2z3z

1

u[k] 4

4

4

4

)(zE )(zR

Lr+Le+1 equations in 2(Lr+1) unknowns, can (mostly) be solved if Le-1 ≤ Lr



- 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 ?

)(.)( * zBFz E

4

4

4

4

+u[k-3]

1z

2z

3z

1

1z2z3z

1

u[k] 4

4

4

4

)(zE )(zR



• 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 (!)…



• 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 =

*88F

u[k]

:

4

4 )( 20 zE

)( 24 zE

)( 23 zE

)( 27 zE

:

:

Lossless 1-in/2-out

= 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) !



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

Sh

ou

ld n

ot

try

to u

nd

erst

and

th

is…



DFT modulated (analysis) filter bank can be realized based on an N-point IDFT cascaded with an NxD polyphase matrix B, which contains the (N’-fold) polyphase components of the prototype Ho(z)

)(.0)(.0

0)(.0)(.

)(0)(.0

0)(0)(.

)(.0)(0

0)(.0)(

)(

1211

8125

4

1210

8124

4

123

129

8

122

128

8

127

4121

126

4120

4

zEzzEz

zEzzEz

zEzEz

zEzEz

zEzzE

zEzzE

zB

*66xF

u[k]

)(0 zH

)(1 zH

)(2 zH

)(3 zH

)(4 zH

)(5 zH

464 )( xzB

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



Proof is simple:

)(.

)(

)(

)(

)(

)(

)(

)(.

iondecomposit polyphase fold-6

)(.)(

)(.)(

)(.)(

)(.)(

)(.)(

)(.)(

....)(.

1

).(.

5

4

3

2

1

0

1211

11125

5

1210

10124

4

129

9123

3

128

8122

2

127

7121

1

126

6120

*

3

2

14* ZU

zH

zH

zH

zH

zH

zH

ZU

zEzzEz

zEzzEz

zEzzEz

zEzzEz

zEzzEz

zEzzE

FzU

z

z

zzBF

*66xF

u[k]

)(0 zH

)(1 zH

)(2 zH

)(3 zH

)(4 zH

)(5 zH

464 )( xzB



-With 4-fold decimation, this is

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

)(.)( * zBFz E

u[k]

*66xF46)( xzB4

44

4

)(.0)(.0

0)(.0)(.

)(0)(.0

0)(0)(.

)(.0)(0

0)(.0)(

)(

311

235

1

310

234

1

33

39

2

32

38

2

37

131

36

130

zEzzEz

zEzzEz

zEzEz

zEzEz

zEzzE

zEzzE

zB



- FIR Perfect Reconstruction FB: try it..- FIR Paraunitary Perfect Reconstruction FB:

E(z) is paraunitary iff 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!

)(.)(.

)()(.

)(.)(

and

)(.)(.

)()(.

)(.)(

311

235

1

33

39

2

37

131

310

234

1

32

38

2

36

130

zEzzEz

zEzEz

zEzzE

zEzzEz

zEzEz

zEzzE

= EASY !

= not-so-easy but DOABLE !


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?

DSP-CIS Chapter-9: Modulated Filter Banks Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven [email protected]

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