digital signal processing ii lecture 7: modulated filter banks

p. 1DSP-II

Digital Signal Processing II

Lecture 7: Modulated Filter Banks

Marc Moonen

Dept. E.E./ESAT, K.U.Leuven

[email protected]

homes.esat.kuleuven.be/~moonen/

DSP-IIVersion 2006-2007 Lecture-7 Modulated Filter Banks p. 2

Part-II : Filter Banks

: Preliminaries• Applications

• Intro perfect reconstruction filter banks (PR FBs)

: Maximally decimated FBs• Multi-rate systems review

• 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


Refresh

General `subband processing’ set-up (Lecture-5) :

- 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 3H0(z)



3

3

3

3 subband processing 3H3(z)

IN

F0(z)

F1(z)

F2(z)

F3(z)

+

OUT


Refresh

Two design issues : - filter specifications, e.g. stopband attenuation, passband ripple, transition

band, etc. (for each (analysis) filter!)

- perfect reconstruction property (Lecture-6).

PS: Lecture 6/7 = maximally decimated FB’s = NM

4444

+u[k-3]

1z

2z

3z

1

1z2z3z

1

u[k] 444

4)(zE )(zR

NIzz )().( ER


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 • Cosine-modulated FBs


DFT-Modulated Filter Banks

Uniform versus non-uniform (analysis) filter bank:

non-uniform: e.g. for speech & audio applications (cfr. human hearing) example : wavelet filter banks (next lecture)

N-Channel uniform filter bank:

= frequency responses uniformly shifted over the unit circle

Ho(z)= `prototype’ filter (=only filter that has to be designed)

H0(z)

H1(z)

H2(z)

H3(z)

INH0 H3H2H1

H0 H3H2H1uniform

non-uniform

).()( /20

Nkjk ezHzH

NM

Nnkjk enhnh /.2

0 ].[][



Uniform filter banks can be implemented cheaply based

on polyphase decompositions + DFT(FFT)

hence named `DFT modulated FBs’

1. Analysis FB

If

then

).()( with )(),...,(),( /20110

NkjkN ezHzHzHzHzH

with , )(..

)(..).()(

/21

0

1

0

1

/2/2/20

NjN

l

Nl

kll

N

l

NkNjNl

NkljlNkjk

eWzEWz

ezEezezHzH

)(.)(1

00

N

l

Nl

l zEzzH

i.e.

H0(z)

H1(z)

H2(z)

H3(z)

u[k]



where F is NxN DFT-matrix

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: N filters for the price of 1, plus DFT (=FFT) ! - design complexity: design `prototype’ Ho(z), then other Hi(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 Ei(z)=1

*F

u[k]

1 )(0 zH

)(1 zH

)(2 zH

)(3 zH

11

1

-4 -3 -2 -1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ho(z) H1(z)



• PS: with F instead of F* (see Lecture-5), only filter ordering is changed

-4 -3 -2 -1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ho(z) H1(z)

Fu[k]

1 )(0 zH

)(1 zH

)(2 zH

)(3 zH11

1



• Uniform DFT-modulated analysis FB +decimation (M=N)

*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

=



2. Synthesis FB

).(.)( with )(),...,(),( /20

/2110

NkjNkjkN ezFezFzFzFzF

)(.)(1

00

N

l

Nl

l zRzzF )(..... )(1

0

)1(

N

l

Nl

lNklk zRWzzF

+

+

+

)(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.



y[k]

+

+

+)( 40 zR

)( 41 zR

)( 42 zR

)( 43 zR][0 ku

][1 ku

][2 ku

][3 ku

F

i.e.



• Expansion (M=N) + uniform DFT-modulated synthesis FB :

y[k]

4

4

4

4

+

+

+)(0 zR

)(1 zR

)(2 zR

)(3 zR][0 ku

][1 ku

][2 ku

][3 ku

F

y[k]

+

+

+

4

4

4

4 )( 40 zR

)( 41 zR

)( 42 zR

)( 43 zR][0 ku

][1 ku

][2 ku

][3 ku

F

=



Perfect reconstruction (PR) revisited :maximally decimated (M=N) uniform DFT-modulated analysis & synthesis…

- Procedure:

1. Design prototype analysis filter Ho(z) (=DSP-II/Part-I).

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

3. Assuming Ei(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

)()( 11 zEzR lNl

FERFE .)()()( )(.)( 11* zEdiagzzzEdiagz ii



Perfect reconstruction (PR):

FIR E(z) generally leads to IIR R(z), where stability is a concern…

PR with FIR analysis/synthesis bank (=guaranteed stability), only

obtained with trivial choices for Ei(z)’s (next slide)

y[k]

4444

+

+

+)(0 zR

)(1 zR

)(2 zR

)(3 zR

F444

4u[k]

*F)(0 zE

)(1 zE

)(2 zE

)(3 zE

)()( 11 zEzR lNl



• Simple example (1) is , which leads to IDFT/DFT bank (Lecture-5)

i.e. Fl(z) has coefficients of Hl(z), but complex conjugated and in reverse order (hence same magnitude response) (remember this?!)

• Simple example (2) is , where wi’s are constants, which leads to `windowed’ IDFT/DFT bank, a.k.a. `short- time Fourier transform’ (see Lecture-8)

• Question (try to answer): when is maximally decimated PR uniform DFT-modulated FB - FIR (both analysis & synthesis) ? - paraunitary ?

1)()( zEzR ll

)1()1(33221 .......1)( NlNllll zWzWzWzWzH

)1()2()3(2)4(3)1( ......)( NNlNlNllNl zzWzWzWWzF

11 )()( ilNil wzRwzE



• Bad news: From this it is seen that the maximally

decimated IDFT/DFT filter bank (or trivial modifications

thereof) is the only possible uniform DFT-modulated FB

that is at the same time...

i) maximally decimated

ii) perfect reconstruction (PR)

iii) FIR (all analysis+synthesis filters)

iv) paraunitary• Good news :

– Cosine-modulated PR FIR FB’s – Oversampled PR FIR DFT-modulated FB’s (Lecture-8)


Cosine-Modulated Filter Banks

• Uniform DFT-modulated filter banks: Ho(z) is prototype lowpass filter, cutoff at for N filters

• Cosine-modulated filter banks :

Po(z) is prototype lowpass filter, cutoff at for N filters

Then...

etc...

N/

N2/

H0 H3H2H1

2N/

).(.).(.)()5.0(

0*0

)5.0(

000N

jN

jezPezPzH

P0

2

2

2

N2/

N/H1

Ho

).(.).(.)()5.01(

0*1

)5.01(

011N

jN

jezPezPzH

).()( /20

Nkjk ezHzH



• Cosine-modulated filter banks : - if Po(z) is prototype lowpass filter designed with real coefficients po[n], n=0,1,…,L then

i.e. `cosine modulation’ (with real coefficients) instead of `exponential modulation’ (for DFT-modulated bank, see page 6)

- if Po(z) is `good’ lowpass filter, then Hk(z)’s are `good’ bandpass filters

).(.).(.)()5.0(

0*)5.0(

0N

kj

kN

kj

kk ezPezPzH

}4

.)1()2

)(5.0(cos{].[.2][ 0

kk

Lnk

Nnpnh



Realization based on polyphase decomposition (analysis):

- if Po(z) has 2N-fold polyphase expansion (ps: 2N-fold for N filters!!!)

then...

k

kl

N

l

Nl

lL

k

k zlkNpzEzEzzkpzP ]..2[)( , )(.].[)( 0

12

0

2

000

NNT 2

u[k]

)( 20

NzE

)( 21

NzE

)( 212

NN zE

)(0 zH

)(1 zH

)(1 zH N

: :



Realization based on polyphase decomposition (continued):

- if Po(z) has L+1=m.2N taps, and m is even (similar formulas for m odd) (m is the number of taps in each polyphase component) then...

With

00...1

:::

01...0

10...0

,

1...00

:::

0...10

0...01

)()(... 22

JI

JIJICNT NNNN

})5.0(cos{

})5.01(cos{

})5.0(cos{

...00

:::

0...0

0...0

mN

m

m

)}5.0).(5.0.(cos{2

}{ , qpNN

C qp

ign

ore

all

det

ails

h

ere

!!!!!

!!!!!!

!!!!



Realization based on polyphase decomposition (continued): - Note that C is NxN DCT-matrix (`Type 4’)

hence fast implementation (=fast matrix-vector product) based on fast discrete cosine transform procedure, complexity O(N.logN). - Modulated filter bank gives economy in * design (only prototype Po(z) ) * implementation (prototype + modulation (DCT))

Similar structure for synthesis bank

)}5.0).(5.0.(cos{2

}{ , qpNN

C qp

NNT 2

u[k]

)( 20

NzE

)( 21

NzE

)( 212

NN zE

)(0 zH

)(1 zH

)(1 zH N

: :



Maximally decimated cosine modulated (analysis) bank :

NNT 2

u[k]

)( 20

NzE

)( 21

NzE

)( 212

NN zE

:

N

N

N

NNT 2

u[k]

)( 20 zE

)( 21 zE

)( 212 zE N

:

N

N

N=



Question: How do we obtain Maximal Decimation + FIR + PR + Paraunitariness?

Theorem: (proof omitted)

-If prototype Po(z) is a real-coefficient (L+1)-taps FIR filter, (L+1)=2N.m for some integer m and po[n]=po[L-n] (linear phase), with polyphase components Ek(z), k=0,1,…2N-1, -then the (FIR) cosine-modulated analysis bank is PARAUNITARY if and only if (for all k) are power complementary, i.e. form a lossless 1 input/2 output system

Hence FIR synthesis bank (for PR) can be obtained by paraconjugation !!! =Great result…

)( and )( zEzE Nkk

..this is the hard part…



Perfect Reconstruction (continued)

Design procedure: Parameterize lossless systems for k=0,1..,N-1 Optimize all parameters in this parametrization so that the prototype Po(z) based on these polyphase components is a linear-phase lowpass filter

that satisfies the given specifications

Example parameterization: Parameterize lossless systems for k=0,1..,N-1, -> lattice structure (see Part-I), where parameters are rotation angles

)( and )( zEzE Nkk

)(zEk

)(zE Nk

kl

kl

kl

klk

l

kkkm

km

Nk

k

zzzzE

zE

cossin

sincos

0

1..

0

01....

0

01..

0

01.

)(

)(0111211

E

EEEE

)( and )( zEzE Nkk

..this is the hard part…



PS: Linear phase property for po[n] implies that only half of the power

complementary pairs have to be designed. The other pairs are then

defined by symmetry properties.

NNT 2

u[k]

:

N

Np.26 = )( 20 zE

)( 2zEN

)( 21 zEN

)( 212 zE N

:

:

lossless



PS: Cosine versus DFT modulation In a maximally decimated cosine-modulated (analysis) filter bank 2 polyphase components of the prototype filter, ,

actually take the place of only 1 polyphase component in the DFT- modulated case. For paraunitariness (hence FIR-PR) in a cosine-modulated bank, each such pair of polyphase filters should form a power complementary pair, i.e. represent a lossless system.

In the DFT-modulated case, imposing paraunitariness is equivalent to imposing losslessness for each polyphase component separately, i.e. each polyphase component should be an `allpass’ transfer function. Allpass functions are always IIR, except for trivial cases (pure delays). Hence all FIR paraunitary DFT-modulated banks (with maximal decimation) are trivial modifications of the DFT bank.

)( and )( zEzE Nkk

no FIR-design flexibility

provides flexibility for FIR-design

digital signal processing ii lecture 7: modulated filter banks

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