Digital Signal Processing II Lecture 6: Maximally Decimated Filter Banks

p. 1DSP-II

Digital Signal Processing II

Lecture 6:

Maximally Decimated Filter Banks

Marc Moonen

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

[email protected]

homes.esat.kuleuven.be/~moonen/

DSP-IIVersion 2005-2006 Lecture-6 Maximally Decimated 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


PART-II : Filter Banks

LECTURE-6 : Maximally decimated FBs • Ìnterludium’: Review of multi-rate systems

• Perfect reconstruction (PR) FBs– 2-channel case – M-channel case

• Ìnterludium’: Paraconjugation & paraunitary functions

• Paraunitary PR FBs


Review of Multi-rate Systems (1)

• Decimation : decimator (downsampler)

example : u[k]: 1,2,3,4,5,6,7,8,9,…

2-fold downsampling: 1,3,5,7,9,...

• Interpolation : expander (upsampler)

example : u[k]: 1,2,3,4,5,6,7,8,9,… 2-fold upsampling: 1,0,2,0,3,0,4,0,5,0...

N u[0], u[N], u[2N]...u[0],u[1],u[2]...

N u[0],0,..0,u[1],0,…,0,u[2]...u[0], u[1], u[2],...



• Z-transform (frequency domain) analysis of expander

èxpansion in time domain ~ compression in frequency domain’ expander mostly followed by interpolation filter to remove all images

N u[0],0,..0,u[1],0,…,0,u[2]...u[0], u[1], u[2],...

N)(zU )( NzU

3

ìmages’



• Z-transform (frequency domain) analysis of decimator

decimation introduces ALIASING if input signal occupies frequency band larger than , for hence decimation mostly preceded by anti-aliasing (decimation) filter

N)(zU

1

0

21 ).(.1 N

i

NijN ezUN

3

N u[0], u[N], u[2N]...u[0],u[1],u[2]...

i=0i=2 i=1

N2

3



• Z-transform analysis of decimator (continued)

- Note that is periodic with period while is periodic with period

the summation with i=0…N-1 restores the periodicity with period !

- Example:

)( jeU 2

)( /NjeU N22

0,)(][

1

1...(...).

1)(

1

1)(

0,][

1

1

0

1

kky

zNzY

zzU

kku

kN

N

N

i

k

N)(zU

1

0

21 ).(.1 N

i

NijN ezUN


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)

subband processing 3H1(z)



3

3

3

3 subband processing 3H4(z)

IN

G1(z)

G2(z)

G3(z)

G4(z)

+

OUT



• Interconnection of multi-rate building blocks :

identities also hold if all decimators are replaced by expanders

N x

a

Nx

a=

=

=

N+

u2[k]

Nx

u2[k]

u1[k]

u1[k]

N +

Nu2[k]

u1[k]

N x

Nu2[k]

u1[k]



• `Noble identities’ (I) : (only for rational functions)

Example : N=2 h[0],h[1],0,0,0,…

=N N)( NzH )(zHu[k] u[k]y[k] y[k]

]3[

]2[

]1[

]0[

.0100

0001.

)(

]1[0

]0[]1[

0]0[

...

]3[

]2[

]1[

]0[

.

)2(

]1[000

0]1[00

]0[0]1[0

0]0[0]1[

00]0[0

000]0[

.

010000

000100

000001

]2[

]1[

]0[

ngdownsampli fold-2

ngdownsampli fold-2

u

u

u

u

zH

h

hh

h

u

u

u

u

zH

h

h

hh

hh

h

h

y

y

y



• `Noble identities’ (II) : (only for rational functions)

Example : N=2 h[0],h[1],0,0,0,…

=N N )( NzH)(zHu[k] u[k]y[k] y[k]

]1[

]0[.

00

10

00

01

.

)2(

]1[000

0]1[00

]0[0]1[0

0]0[0]1[

00]0[0

000]0[

...]1[

]0[.

)(

]1[0

]0[]1[

0]0[

.

000

100

000

010

000

001

]5[

]4[

]3[

]2[

]1[

]0[

upsampling fold-2

upsampling fold-2

u

u

zH

h

h

hh

hh

h

h

u

u

zH

h

hh

h

y

y

y

y

y

y



Application of `noble identities : efficient multi-rate filter implementations through…

• Polyphase decomposition: example : (2-fold decomposition)

example : (3-fold decomposition)

general: (N-fold decomposition)

)(

421

)(

642

654321

21

20

)].5[].3[]1[(.)].6[].4[].2[]0[(

].6[].5[].4[].3[].2[].1[]0[)(

zEzE

zhzhhzzhzhzhh

zhzhzhzhzhzhhzH

)(

32

)(

31

)(

63

654321

32

31

30

)].5[]2[(.)].4[]1[(.)].6[].3[]0[(

].6[].5[].4[].3[].2[].1[]0[)(

zEzEzE

zhhzzhhzzhzhh

zhzhzhzhzhzhhzH

k

kl

N

l

Nl

l

k

k zlkNhzEzEzzkhzH ]..[)( , )(.].[)(1

0



• Polyphase decomposition: example : Efficient implementation of a decimation filter

i.e. all filter operations performed at the lowest rate

u[k]

2

)( 20 zE

)( 21 zE

1z+

H(z)

u[k]

2

1z

)(0 zE

)(1 zE +

= 2



• Polyphase decomposition:

example : Efficient implementation of an interpolation filter

i.e. all filter operations performed at the lowest rate

=

u[k]

2

)( 20 zE

)( 21 zE

1z

+

H(z)

u[k]

2

1z

+)(0 zE

)(1 zE

2


Refresh

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




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.

PS: Perfect reconstruction property as such is easily satisfied, if there aren’t any (analysis) filter specs, e.g. (see Lecture-5)

…but this is not very useful/practical. Stringent filter specs. necessary for subband coding, etc.

This lecture : Maximally decimated FB’s :

4444

+1z2z3z

1

u[k-3]444

1z

2z

3z4

1

u[k]

NM


Perfect Reconstruction : 2-Channel Case

It is proved that... (try it!)

• U(-z) represents aliased signals, hence the àlias 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

H0(z)

H1(z)

2

2

u[k] 2

2

F0(z)

F1(z)+

y[k]

)(.

)(

)}()()().(.{2

1)(.

)(

)}()()().(.{2

1)( 11001100 zU

zA

zFzHzFzHzU

zT

zFzHzFzHzY

NM



• Requirement for àlias-free’ filter bank :

• Requirement for `perfect reconstruction’ filter bank (= alias-free + distortion-free): i)

ii)

• PS: 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).

Remarkable !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

H0(z)

H1(z)

2

2

u[k] 2

2

F0(z)

F1(z)+

y[k]

0)( zA

zzT )(

0)( zA



• An initial choice is ….. :

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)

H0(z)

H1(z)

2

2

u[k] 2

2

F0(z)

F1(z)+

y[k]

)()( ),()( ),()( 010001 zHzFzHzFzHzH

)}()({2

1...)( 2

020 zHzHzT 0...)( zA

)(...)()

2(

0

)2

(

1

jjeHeH

2


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.

H0(z)

H1(z)

2

2

u[k] 2

2

F0(z)

F1(z)+

y[k]

2

Ho H1



• A 2nd (better) choice is: [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-called`paraunitary’ perfect reconstruction bank (see below), based on a lossless system Ho,H1 :

H0(z)

H1(z)

22

u[k] 22

F0(z)

F1(z) +y[k]

)()( ),()( 0110 zHzFzHzF

1...)( zT

0...)( zA

1)()(

2)

2(

0

2)

2(

0

jjeHeH

][.)1(][ 01 kLhkh k

1)()(2

1

2

0 jj eHeH

This is already pretty complicated…


Perfect Reconstruction : M-Channel Case

It is proved that... (try it!)

• 2nd term represents aliased signals, hence all àlias 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

).(.

)(

)}()..({.1

)(.

)(

})().(.{1

)(1

1

1

0

1

0

lM

l

M

kk

lk

M

kkk WzU

zlA

zFWzHM

zU

zH

zFzHM

zY

H2(z)

H3(z)

4

4

4

4

F2(z)

F3(z)

y[k]

H0(z)

H1(z)

4

4u[k]

4

4

F0(z)

F1(z)

+

NM

Sigh !!…



• 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)

4444

+u[k-3]

1z

2z

3z

1

1z2z3z

1

u[k] 444

4

)( 4zE )( 4zR

)1(

1|10|1

1|00|0

1

0

:

1

.

)(

)(...)(

::

)(...)(

)(

:

)(

NNNN

NN

NN

N

N z

Nz

zEzE

zEzE

zH

zH E

)(

)(...)(

::

)(...)(

.

1

:

)(

:

)(

1|10|1

1|00|0)1(

1

0

Nz

zRzR

zRzRz

zF

zF

NNN

NN

NN

NTNT

N

R

Do not continue until you understand how formulae correspond to block scheme!



• 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

4444

+u[k-3]

1z

2z

3z

1

1z2z3z

1

u[k] 444

4

)(zE )(zR

)().( zz ER



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->

4444

+u[k-3]

1z

2z

3z

1

1z2z3z

1

u[k] 444

4)(zE )(zR

circulant'-`pseudo)().( zz ER

)()(.)(.)(.

)()()(.)(.

)()()()(.

)()()()(

)().(

031

21

11

1031

21

21031

3210

zpzpzzpzzpz

zpzpzpzzpz

zpzpzpzpz

zpzpzpzp

zz ER



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..

4444

+1z

2z

3z

1

1z2z3z

1

u[k] 444

4)().( 44 zz ER

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

1

...)(.

1

).().()(

43

342

241

140

3

2

1

3

2

144 zUzpzzpzzpzzp

z

z

zzU

z

z

zzz

zT

ER

44444

+1z2z3z

1

T(z)*u[k-3]444

1z

2z

3z

1u[k]

)(zT



Hence necessary & sufficient condition for PR (where T(z)=pure delay):

In is nxn identity matrix, r is arbitrary

(Obvious) example :

4444

+u[k-3]

1z

2z

3z

1

1z2z3z

1

u[k] 444

4)(zE )(zR

10 ,0.

0)().(

1

NrIz

Izzz

r

rN

ER

NIzz )().( ER



For conciseness, will use this from now on :

- Procedure: 1. Design all analysis filters (=DSP-II/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…

4444

444

4

+ u[k-3]1z

2z

3z

1

1z2z3z

1

u[k] )(zE )(zR

NIzz )().( ER

)()( 1 zz ER



Question:

Can we find polynomial (FIR) matrices E(z) that have a FIR inverse?

Answer:

YES, so-called ùnimodular’ matrices (where determinant=constant*z^-d)

Example:

where the Ei’s are constant (=not a function of z) invertible matrices

procedure : optimize Ei’s to obtain analysis filter specs (ripple, etc.)

enough) good( .)().(

.10

0...

10

0.....

10

0..)(

.0

0....

0

0..

0

0.)(

111

11

11

11

11

10

011

111

111

ML

LM

LMM

MML

ML

Izzz

IzIzIzz

z

I

z

I

z

Iz

ER

EEEER

EEEEE



Question:

Can we avoid direct inversion, e.g. through the usage of (again FIR) E(z)

matrices with additional `special properties’ ?

(compare with (real) orthogonal or (complex) unitary matrices)

Answer:

YES, `paraunitary’ matrices (=special class of FIR matrices with FIR inverse)

See next slides….

In the sequel, will focus on paraunitary E(z)

leading to paraunitary PR filter banks


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 = ànalytic extension’ of unit-circle conjugation

)()(~ 1

* zHzH

zazHzazH .1)(~

.1)( *1

*1* })({)()(

~ jjj ezezez

zHzHzH



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 = ànalytic extension’ of unit-circle transpose conjugation

)()(~ 1

* zz THH

zbzazzb

zaz .1.1)(

~

.1

.1)( **

1

1

HH

H

ezez

T

ezjjj

zzz })({)()(~ 1

*

HHH



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 = ànalytic extension’ of unit-circle unitary.

PS: if H1(z) and H2(z) are paraunitary, then H1(z).H2(z) is paraunitary

Izz )().(~

HH

Izz jj ez

H

ez

})(.{})({ HH

1)}({)(~ zz HH



- If E(z) is paraunitary

hence perfect reconstruction is obtained with

If E(z) is FIR, then R(z) is also FIR !! (cfr definition paraconjugation)

4444

+u[k-3]

1z

2z

3z

1

1z2z3z

1

u[k] 444

4

)(zE )(zR

NIzz )().( ER

)(~

)( zz ER

Izz )().(~

EE



• Example: paraunitary FIR E(z) with FIR inverse R(z)

where the Ei’s are constant unitary matrices Procedure : optimize unitary Ei’s to obtain analysis filter specs. ps: 2-channel case with real coefficients, then

hence optimize phi’s... (=lossless lattice, see lecture-2)

M

HL

MHL

MHMH

MML

ML

Izz

IzIzIzz

z

I

z

I

z

Iz

)().(

.10

0...

10

0.....

10

0..)(

.0

0....

0

0..

0

0.)(

11

11

1

11

1

0

011

111

111

ER

EEEER

EEEEE

paraunitary

ii

iii

cossin

sincosE



Properties of paraunitary PR filter banks:

• If polyphase matrix E(z) (and hence E(z^N)) 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)constant )(

1

0

2

N

k

jk eH

)1(

1|10|1

1|00|0

1

0

:

1

.

)(

)(...)(

::

)(...)(

)(

:

)(

)(NN

NNN

N

NN

N

N z

Nz

zEzE

zEzE

zH

zH

z

E

H



Properties of paraunitary PR filter banks (continued):

• Synthesis filter coefficients are obtained by conjugating the analysis filter coefficients + reversing the order :

• Magnitude response of synthesis filter Fk is the same as magnitude response of corresponding analysis filter Hk:

• Analysis filters are power complementary (cfr. supra)• Synthesis filters are power complementary• example: DFT/IDFT bank, Lecture-5 example: 2-channel case, page 21• Great properties/designs.... (proofs omitted)

10 ],[][ * NknLhnf kk

10 ,)()( NkeHeF jk

jk


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 PR FBs

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