DSP-CIS Chapter-8: Maximally Decimated PR-FBs

DSP-CIS

Chapter-8: Maximally Decimated PR-FBs

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

[email protected]

www.esat.kuleuven.be/scd/

DSP-CIS / Chapter-8: Maximally Decimated PR-FBs / Version 2012-2013 p. 2

: Preliminaries• Filter bank (FB) set-up and applications • Perfect reconstruction (PR) problem • Multi-rate systems review (=10 slides)• Example: Ideal filter bank (=10 figures)

: Maximally Decimated PR-FBs• Example: DFT/IDFT filter bank• Perfect reconstruction theory• Paraunitary PR-FBs

: DFT-Modulated FBs• Maximally decimated DFT-modulated FBs• Oversampled DFT-modulated FBs

: Special Topics• Cosine-modulated FBs• Non-uniform FBs & Wavelets• Frequency domain filtering

Chapter-7

Chapter-8

Chapter-9

Chapter-10

Part-III : Filter Banks


General `subband processing’ set-up (Chapter-7) : PS: subband processing ignored in filter bank design

Refresh (1)

subband processing 3H0(z)



3

3

3

3 subband processing 3H3(z)

IN

F0(z)

F1(z)

F2(z)

F3(z)

+

OUT

downsampling/decimation

analysis bank synthesis bank

upsampling/expansion


PS: analysis filters Hi(z) are also decimation/anti-aliasing filters, to avoid aliasing in subband signals after decimation (downsampling) PS: synthesis filters Gi(z) are also interpolation filters, to remove images after expanders (upsampling)

downsampling/decimation upsampling/expansion

Refresh (2)




3

3

3


IN

F0(z)

F1(z)

F2(z)

F3(z)

+

OUT


Refresh (3)

Assume subband processing does not modify subband signals

(e.g. lossless coding/decoding)

The overall aim would then be to have y[k]=u[k-d], i.e. that the output signal is equal to the input signal up to a certain delay

But: downsampling introduces ALIASING…




3

3

3


IN

F0(z)

F1(z)

F2(z)

F3(z)

+

OUT

y[k]=u[k-d]?


Refresh (4)

Question : Can y[k]=u[k-d] be achieved in the presence of aliasing ?Answer : Yes !! PERFECT RECONSTRUCTION banks with synthesis bank designed to remove aliasing effects !




3

3

3


IN

F0(z)

F1(z)

F2(z)

F3(z)

+

OUT

y[k]=u[k-d]?


Refresh (5)

Two design issues :

✪ Filter specifications, e.g. stopband attenuation,

passband ripple, transition band, etc.

(for each (analysis) filter!)

✪ Perfect reconstruction (PR) property.

Challenge will be in addressing these two design issues at once (‘PR only’ (without filter specs) is easy, see next slides)

This chapter : Maximally decimated FB’s : NM


Example: DFT/IDFT Filter Bank

• Basic question is..: Downsampling introduces ALIASING, then how can PERFECT RECONSTRUCTION (PR) (i.e. y[k]=u[k-d]) be achieved ?

• Next slides provide simple PR-FB example, to

demonstrate that PR can indeed (easily) be obtained• Discover the magic of aliasing-compensation….

G1(z)

G2(z)

G3(z)

G4(z)

+

output = input 4H1(z)



4

4

4

4 output = input 4H4(z)

u[k]

y[k]=u[k-d]?



First attempt to design a perfect reconstruction filter bank

- Starting point is this :

convince yourself that y[k]=u[k-3] …

4

4

4

4

u[k]

4

4

4

4

+

+

+u[k-3]

u[0],0,0,0,u[4],0,0,0,...

u[-1],u[0],0,0,u[3],u[4],0,0,...

u[-2],u[-1],u[0],0,u[2],u[3],u[4],0,...

u[-3],u[-2],u[-1],u[0],u[1],u[2],u[3],u[4],...



- An equivalent representation is ...

As y[k]=u[k-d], this can already be viewed as a (1st) perfect reconstruction filter bank (with lots of aliasing in the subbands!) All analysis/synthesis filters are seen to be pure delays, hence are not frequency selective (i.e. far from ideal case with ideal bandpass filters….)

PS: Transmux version (TDM) see Chapter-7

4444

+1z2z3z

1

u[k-3]444

1z

2z

3z4

1

u[k]



-now insert DFT-matrix (discrete Fourier transform)

and its inverse (I-DFT)...

as this clearly does not change the input-output relation (hence perfect reconstruction property preserved)

4444

+u[k-3]

1z

2z

3z

1

1z2z3z

1

444

4u[k]

F 1F

IFF .1



- …and reverse order of decimators/expanders and DFT-matrices (not done in an efficient implementation!) :

=analysis filter bank =synthesis filter bank

This is the `DFT/IDFT filter bank’. It is a first (or 2nd) example of a maximally decimated perfect reconstruction filter bank !

4444

1z2z3z

1

u[k] 444

4

F +u[k-3]

1z

2z

3z

1

1F



What do analysis filters look like?

This is seen (known) to represent a collection of filters Ho(z),H1(z),..., each of which is a frequency shifted version of Ho(z) :

i.e. the Hi are obtained by uniformly shifting the `prototype’ Ho over the frequency axis.

F

u[k]

Nj

N

F

NNN

N

N

N

eW

z

z

z

WWWW

WWWW

WWWW

WWWW

zH

zH

zH

zH

/2

1

2

1

)1()1(210

)1(2420

1210

0000

1

2

1

0

:

1

.

...

::::

...

...

...

)(

:

)(

)(

)(

2

)()( ))/2.((0

Nijji eHeH 121

0 ...1)( NzzzzH



The prototype filter Ho(z) is a not-so-great

lowpass filter with first sidelobe only

13 dB below the main lobe. Ho(z) and

Hi(z)’s are thus far from ideal lowpass/

bandpass filters.

Hence (maximal) decimation introduces

significant ALIASING in the decimated subband signals

Still, we know this is a PERFECT RECONSTRUCTION filter bank (see construction previous slides), which means the synthesis filters can apparently restore the aliasing distortion. This is remarkable!

Other perfect reconstruction banks : read on..

Ho(z)H1(z)

N=4

H3(z)H2(z)



Synthesis filters ?

synthesis filters are (roughly) equal to analysis filters…

PS: Efficient DFT/IDFT implementation based on FFT algorithm

(`Fast Fourier Transform’).

...

)(

:

)(

)(

)(

1

2

1

0

zF

zF

zF

zF

N

+1z

2z

3z

1

1F

*(1/N)N=4


Perfect Reconstruction Theory

Now comes the hard part…(?)

2-channel case:

Examples…

✪ M-channel case:

Polyphase decomposition based approach


Perfect Reconstruction : 2-Channel Case

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

• U(-z) represents aliased signals, hence A(z) referred to as àlias transfer function’’.

• T(z) referred to as `distortion function’ (amplitude & phase distortion).

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 :

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) !!!!

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

ii)

H0(z)

H1(z)

2

2

u[k] 2

2

F0(z)

F1(z)+

y[k]

0)( zA

zzT )(

0)( zA



• A first attempt is as follows….. :

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]

)}()({2

1...)( 2

020 zHzHzT 0...)( zA

)(...)()

2(

0

)2

(

1

jjeHeH

2



`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) attempt is as follows: [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 :

)()( ),()( 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 )

• T(z) is referred to as `distortion function’ (amplitude & phase distortion). For perfect reconstruction, T(z) should be a pure delay

).(.

)(

)}()..({.1

)(.

)(

})().(.{1

)(1

1

1

0

1

0

lM

l

M

kk

lk

M

kkk WzU

zlA

zFWzHM

zU

zT

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|11|0

0|10|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



Necessary & sufficient condition for PR is…: (i.e. where T(z)=pure delay, hence p_r(z)=pure delay, and all other p_i(z)=0)

I_n is nxn identity matrix, r is arbitrary

Example (r=0) : for conciseness, will use this from now on !

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

NIzzz )().( ER



• Example : DFT/IDFT Filter bank : E(z)=F , R(z)=F^-1

• Design Procedure:

1. Design all analysis filters (see Part-II).

2. This determines E(z) (=polyphase matrix).

3. Assuming E(z) can be inverted (?), synthesis filters are

(delta to make synthesis causal)

• Will consider only FIR analysis filters, leading to simple polyphase decompositions (see Chapter-7). However, FIR E(z) generally leads to IIR R(z), where stability is a concern…

)(.)( 1 zzz ER

NIzzz )().( ER



PS: Inversion of matrix transfer functions ?…

– The inverse of a scalar (i.e. 1-by-1 matrix) FIR transfer function is always IIR (except for contrived examples)

– The inverse of an N-by-N (N>1) FIR transfer function can be FIR

1))(det(

2

12

2)()(

222

1)( 21

1

1

1

12

z

zz

zzz

z

zzz

E

ERE

)2(

1)()()2()(

111

zzzzz ERE



PS: Inversion of matrix transfer functions ?…

Compare this to inversion of integers and integer matrices:

– The inverse of an integer is always non-integer (except for È=1’)

– The inverse of an N-by-N (N>1) integer matrix can be integer

1)det(

54

65

54

65 1

E

ERE

2

12 1 ERE



Question:

Can we build FIR matrices E(z) that have an FIR inverse?

Answer:

YES, Ùnimodular’ matrices = matrices with {determinant=constant*z^-d}

e.g.

where the Ei’s are constant (≠ a function of z) invertible matrices

Example: E(z) = FIR LPC lattice , see Chapter-5 (not a good filter bank though… explain!)

Design Procedure: optimize Ei’s to obtain filter specs (ripple, etc.), etc..

.)().(

.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. build FIR E(z) matrices with additional `special properties’, so that R(z) is trivially obtained and its specs are better controlled? (compare with (real) orthogonal or (complex) unitary matrices, where inverse is equal to (Hermitian) transpose)

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

See next slides….

Will focus on paraunitary E(z) leading to PR/FIR/paraunitary filter banks


Paraunitary PR Filter Banks

Review : `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, paraconjugate corresponds to complex conjugate

paraconjugate = ànalytic extension’ of unit-circle complex conjugate

)()(~ 1

* zHzH

zazHzazH .1)(~

.1)( *1

*1* })({)()(

~ jjj ezezez

zHzHzH



Review : `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, paraconjugate corresponds to conjugate transpose(*)

paraconjugate = ànalytic extension’ of unit-circle conjugate transpose(*)

)()(~ 1

* zz THH

zbzazzb

zaz .1.1)(

~

.1

.1)( **

1

1

HH

H

ezez

T

ezjjj

zzz })({)()(~ 1

*

HHH

(*)

=H

erm

itia

n t

ran

spo

se



Review : `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

then perfect reconstruction is obtained with

(delta to make synthesis causal)

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

)(~

.)( zzz ER

Izz )().(~

EE

NIzzz )().( ER



• Question: Can we build paraunitary FIR E(z) (with FIR inverse R(z))? • Answer:

Yes!

where the Ei’s are constant unitary matrices

• Example: 1-input/2-output FIR lossless lattice, see Chapter-5

Example: 1-input/M-output FIR lossless lattice, see Chapter-5

• Design Procedure: optimize unitary Ei’s (e.g. rotation angles in lossless lattices) to obtain analysis filter specs, etc..

ML

HL

MHL

MHMH

M

MML

ML

Izzz

IzIzIzz

Izz

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

EE

EEEEE



Properties of paraunitary PR filter banks: (proofs omitted)

• 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)

(see Chapter 5)constant)other (or

1

0

2 1 )(

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 filters are obtained from analysis filters by conjugating the

analysis filter coefficients + reversing the order (cfr page 34):

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

• Hence, as analysis filters are power complementary (cfr. supra),

synthesis filters are also power complementary• Examples: DFT/IDFT bank, 2-channel case (p. 21)• Great properties/designs....

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 :

PR/FIR/Paraunitary FBs, e.g. based on FIR lossless

lattice filters• Sequel = other (better) PR structures

Chapter 9: Modulated filter banks

Chapter 10: Oversampled filter banks, etc..

• Reference: `Multirate Systems & Filter Banks’ , P.P. Vaidyanathan Prentice Hall 1993.

Documents

DSP-CIS Chapter-8: Maximally Decimated PR-FBs