chapter9-filter bank design-pp40 - ku leuventvanwate/courses... · dsp 2016 / chapter-9: filter...

1

DSP

Chapter-9 : Filter Bank Design

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

[email protected] www.esat.kuleuven.be/stadius/

DSP 2016 / Chapter-9: Filter Bank Design 2 / 32

Filter Bank Preliminaries •  Filter Bank Set-Up •  Filter Bank Applications •  Ideal Filter Bank Operation •  Non-Ideal Filter Banks: Perfect Reconstruction Theory

Filter Bank Design •  Non-Ideal Filter Banks: Perfect Reconstruction Theory

(continued) •  Filter Bank Design Problem Statement •  General Perfect Reconstruction Filter Bank Design •  DFT-Modulated Filter Banks

Chapter-8

Chapter-9

Part-IV : Filter Banks & Subband Systems

2


G0(z) G1(z) G2(z) G3(z)

+ OUT

subband processing 3 H0(z) subband processing 3 H1(z) subband processing 3 H2(z)

3 3 3 3 subband processing 3 H3(z)

IN

N=4 D=3

Filter Bank Set-Up

So this is the picture to keep in mind...

synthesis bank (synthesis & interpolation)

upsampling/expansion

downsampling/decimation

analysis bank (analysis & anti-aliasing)


Perfect Reconstruction Theory (continued)

A simpler analysis results from a polyphase description : n-th row of E(z) has N-fold (=D-fold)

polyphase components of Hn(z) n-th column of R(z) has N-fold polyphase components of Fn(z)

4 4 4 4

+ u[k-3] 1−z

2−z

3−z

1

1−z2−z3−z

1u[k] 4

4 4

4 E(z4 ) )( 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!

D = N

N-by-N N-by-N

D=N=4

3


Perfect Reconstruction Theory

•  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

4 4 4 4

+ u[k-3] 1−z

2−z

3−z

1

1−z2−z3−z

1u[k] 4

4 4

4 )(zE )(zR

)().( zz ER

D=N=4



Necessary & sufficient condition for PR is then (proof omitted)…

In is nxn identity matrix, r is arbitrary Example (r=0) : for conciseness, will use this from now on !

4 4 4 4

+ u[k-3] 1−z

2−z

3−z

1

1−z2−z3−z

1u[k] 4

4 4

4 )(zE )(zR

10 ,0.

0)().( 1 −≤≤⎥

⎦

⎤⎢⎣

⎡=

−−−

−

NrIz

Izzz

r

rNδ

δ

ER

NIzzz δ−=)().( ER

Beautifully simple!!

D=N=4

4



A similar PR condition can be derived for oversampled FBs The polyphase description (compare to p.34) is then…

n-th row of E(z) has D-fold polyphase components of Hn(z)

n-th column of R(z) has D-fold polyphase components of Fn(z)

H0 (z):

HN−1(z)

"

#

$$$$

%

&

''''

=

E0|0 (zD ) ... E0|D−1(z

D ): :

EN−1|0 (zD ) ... EN−1|D−1(z

D )

"

#

$$$$

%

&

''''

E(zD )! "##### $#####

.1:

z−(D−1)

"

#

$$$

%

&

'''

F0 (z):

FN−1(z)

"

#

$$$$

%

&

''''

T

=z−(D−1)

:1

"

#

$$$

%

&

'''

T

.R0|0 (z

D ) ... RN−1|0 (zD )

: :R0|D−1(z

D ) ... RN−1|D−1(zD )

"

#

$$$$

%

&

''''

R(zD )! "##### $#####

Note that E is an N-by-D (‘tall-thin’) matrix, R is a D-by-N (‘short-fat’) matrix !

D < N

1−z2−z3−z

1u[k]

4 4 4 4

4 4 4

4

E(z4 ) + u[k-3]

1−z

2−z

3−z

1

R(z4 )4 4

4 4

N-by-D D-by-N

D=4 N=6



Simplified (cfr. r=0 on p.6) condition for PR is then… In the D=N case (p.6), the PR condition has a product of square matrices. PR-FB design will then involve matrix inversion, which is mostly problematic.

In the D<N case, the PR condition has a product of a ‘short-fat’ matrix and a ‘tall-thin’ matrix. This will lead to additional PR-FB design flexibility.

R(z).E(z) = z−δID

Again beautifully simple!!

DxD DxN

NxD

1−z2−z3−z

1u[k]

4 4 4 4

4 4 4

4

E(z4 ) + u[k-3]

1−z

2−z

3−z

1

R(z4 )4 4

4 4

N-by-D D-by-N

D=4 N=6

5


Filter Bank Design Problem Statement

Two design targets :

✪ Filter specifications, e.g. stopband attenuation, passband ripple, transition band, etc. (for each (analysis) filter!)

✪ Perfect reconstruction (PR) property.

Challenge will be in addressing two design targets at once (e.g. ‘PR only’ (without filter specs) is easy, see ex. Chapter-8)

PS: Can also do ‘Near-Perfect Reconstruction Filter Bank Design’, i.e. optimize filter specifications and at the same time minimize aliasing/distortion (=numerical optimization). Not covered here…


(= N-by-N matrices) •  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, see ex. p.7) •  Will consider only FIR analysis filters, leading to simple polyphase

decompositions (see Chapter-2) •  However, FIR E(z) then generally leads to IIR R(z), where stability is a

concern…

)(.)( 1 zzz −−= ER δ

NIzzz δ−=)().( ER

General PR-FB Design: Maximum Decimation (D=N)

6



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)

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

PS: Compare this to inversion of integers and integer matrices …but…

1))(det(212

2)()(

2221

)( 21

1

1

1

12

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−

−==⇒

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ += −−

−

−

−

−−

z

zzz

zzz

zzz

E

ERE

)2(1)()()2()( 1

11−

−−

−==⇒−=

zzzzz ERE

212 1 ==⇒= −ERE

1)det(5465

5465 1

=

⎥⎦

⎤⎢⎣

⎡

−

−==⇒⎥

⎦

⎤⎢⎣

⎡= −

E

ERE


Question: Can we build FIR E(z)’s (N-by-N)

that have an FIR inverse? Answer: YES, `unimodular’ E(z)’s, i.e. matrices with determinant=constant*zd e.g.

where the El’s are constant (= not a function of z) invertible matrices Design Procedure: Optimize El’s to meet filter specs (ripple, etc.) for all analysis filters (at once)

E(z) = EL.IN−1 0

0 z−1

"

#$$

%

&''.EL−1.

IN−1 0

0 z−1

"

#$$

%

&''...E1.

IN−1 0

0 z−1

"

#$$

%

&''.E0

R(z) = E0−1. z−1.IN−1 0

0 1

"

#$$

%

&''.E1

−1... z−1.IN−1 00 1

"

#$$

%

&''.EL−1

−1 . z−1.IN−1 00 1

"

#$$

%

&''.EL

−1

⇒ R(z).E(z) = z−L.IN

= not-so-easy but DOABLE !


all E(z)’s

FIR E(z)’s

FIR unimodular E(z)’s

7


•  Design Procedure: 1. Design all analysis filters (see Part-II). 2. This determines E(z) (=polyphase matrix). 3. Find R(z) such that PR condition is satisfied (how? read on…)

•  Will consider only FIR analysis filters, leading to simple polyphase decompositions (see Chapter-2)

•  It will turn out that when D<N an FIR R(z) can always be found (except in contrived cases)…


General PR-FB Design: Oversampled FBs (D<N)

DxD DxN

NxD

= easy if step-3 is doable…


•  Given E(z) how can R(z) be computed?

–  Assume every entry in E(z) is LE-th order FIR (i.e. LE +1 coefficients) –  Assume every entry in R(z) is LR-th order FIR (i.e. LR +1 coefficients) –  Hence number of unknown coefficients in R(z) is D.N.(LR +1) –  Every entry in R(z).E(z) is (LE+LR)-th order FIR (i.e. LE+LR+1

coefficients) (cfr. polynomial multiplication / linear convolution) –  Hence PR condition is equivalent to D.D.(LE+LR+1) linear equations

in the unknown coefficients (*) –  Can be solved (except in contrived cases) if è

(*) Try to write down these equations!



DxD DxN

NxD

D.N.(LR +1) ≥ D.D.(LE + LR +1)

LR ≥D

(N −D)LE −1

8


•  Given E(z) how can R(z) be computed?

–  (continued) …

–  Can be solved (except in contrived cases) if è –  If D<N, then LR can be made sufficiently large so that the

(underdetermined) set of equations can be solved, i.e. an R(z) can be found (!).

–  Note that if D=N, then LR in general has to be infinitely large, i.e. R(z) in general has to be IIR



DxD DxN

NxD

D.N.(LR +1) ≥ D.D.(LE + LR +1)

LR ≥D

(N −D)LE −1


DFT-Modulated FBs

- 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 (read on..) •  Cosine-modulated FBs (not covered, but interesting design!)

-  Will consider -  Maximally decimated DFT-modulated FBs -  Oversampled DFT-modulated FBs

9


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)

H0(z) H1(z) H2(z) H3(z)

IN H0 H3 H2 H1

H0 H3 H2 H1 uniform

non-uniform

Hn (z) = H0 (z.e− j2πn/N ) n = 0,...,N −1

hn[k]= h0[k].ej2πk.n/N

Maximally Decimated DFT-Modulated FBs (D=N)

N=4


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

H0 (z),H1(z),...,HN−1(z) with Hn (z) = H0 (z.e− j2πn/N )

Hn (z) = H0 (z.e− j2πn/N ) = z−n .e j2πnn /N .En (zN e− j2πnN /N

1 )

n=0

N−1

∑

= z−n .W −nn .En (zN )n=0

N−1

∑ , with W = e− j2π /N

H0 (z) = z−n .En (zN )n=0

N−1

∑

i.e.

H0(z) H1(z) H2(z) H3(z)

u[k]


N=4

10


where F is NxN DFT-matrix

This means that filtering with the Hn’s can be implemented by first filtering with the polyphase components and then applying an inverse DFT

PS: To simplify formulas the factor N in N.F-1 will be left out from now on (i.e. absorbed in the polyphase components)

H0 (z)H1(z)H2 (z)

:HN−1(z)

"

#

$$$$$$$

%

&

'''''''

.U(z) =

W 0 W 0 W 0 ... W 0

W 0 W −1 W −2 ... W −(N−1)

W 0 W −2 W −4 ... W −2(N−1)

: : : :W 0 W −(N−1) W −2(N−1) ... W −(N−1)2

"

#

$$$$$$$

%

&

'''''''

N.F−1! "####### $#######

.

E0 (zN )

z−1.E1(zN )

z−2.E2 (zN ):

z−N+1.EN−1(zN )

"

#

$$$$$$$$

%

&

''''''''

.U(z)

W = e− j2π /N


i.e.


Conclusion: economy in… –  Implementation complexity (for FIR filters): N filters for the price of 1, plus inverse DFT (=FFT) ! –  Design complexity: Design `prototype’ Ho(z), then other Hn(z)’s are

automatically `co-designed’ (same passband ripple, etc…) !

i.e.

F−1

u[k] Δ

Δ

Δ

)( 40 zE

)( 41 zE

)( 42 zE

)( 43 zE

)(0 zH

)(1 zH

)(2 zH

)(3 zH

E(z4 )Maximally Decimated DFT-Modulated FBs (D=N)

N=4

11


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

F−1

u[k] Δ

Δ

Δ

1 )(0 zH

)(1 zH

)(2 zH

)(3 zH

111

Ho(z) H1(z)


N=4


•  DFT-modulated analysis FB + maximal decimation

F−1 444

4u[k]

Δ

Δ

Δ

)( 40 zE

)( 41 zE

)( 42 zE

)( 43 zE

444

4u[k]

Δ

Δ

Δ

F−1

)(0 zE

)(1 zE

)(2 zE

)(3 zE

= = efficient realization !


N=4

12


2. Synthesis FB

F0 (z),F1(z),...,FN−1(z) with Fn (z) = ej2πn/N .F0 (z.e− j2πn/N )

F0 (z) = z−n .Rn (zN )n=0

N−1

∑ Fn (z) = ... = z−n .Wn(N−1−n ).Rn (zN )n=0

N−1

∑

+

+

+

)(0 zF][0 ku

)(1 zF][1 ku

)(2 zF][2 ku

)(3 zF][3 ku y[k]

phase shift added for convenience


N=4


Similarly simple derivation then leads to…


N=4

y[k]

Δ

+ Δ

+ Δ

+ )( 40 zR

)( 41 zR

)( 42 zR

)( 43 zR][0 ku

][1 ku

][2 ku

][3 ku

F

R(z4 )

13


•  Expansion + DFT-modulated synthesis FB :

y[k]

][0 ku

][1 ku

][2 ku

][3 ku

4444

Δ

+ Δ

+ Δ

+ )(0 zR

)(1 zR

)(2 zR

)(3 zR

Fy[k]

Δ

+ Δ

+ Δ

+

4444 )( 4

0 zR

)( 41 zR

)( 42 zR

)( 43 zR][0 ku

][1 ku

][2 ku

][3 ku

F

= = efficient realization !


N=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

y[k]

4444

Δ

+ Δ

+ Δ

+ )(0 zR

)(1 zR

)(2 zR

)(3 zR

F444

4u[k] Δ

Δ

Δ

F−1)(0 zE

)(1 zE

)(2 zE

)(3 zE

Rn (z) = z−δ.EN−1−n

−1 (z)

E(z) = F−1.diag En (z)[ ] ⇒ R(z) = z−δ.E−1(z) = z−δ.diag En−1(z)#$ %&.FNIzzz δ−=)().( ER


N=4

14


•  Design Procedure: 1. Design prototype analysis filter Ho(z) (see Part-II). 2. This determines En(z) (=polyphase components). 3. Assuming all En(z)’s can be inverted (?), choose synthesis filters •  Will consider only FIR prototype analysis filter, leading to simple

polyphase decomposition (Chapter-2). •  However, FIR En(z)’s generally again lead to IIR Rn(z)’s, where stability

is a concern…

Rn (z) = z−δ.EN−1−n

−1 (z)


y[k]

4444

Δ

+ Δ

+ Δ

+ )(0 zR

)(1 zR

)(2 zR

)(3 zR

F444

4u[k] Δ

Δ

Δ

)(0 zE

)(1 zE

)(2 zE

)(3 zE

F−1

N=4


This does not leave much design freedom… •  FIR E(Z)? ..such that Rn(z) are also FIR

Only obtained when each En(z) is ‘unimodular’, i.e. En(z)=constant.zd

Simple example is , where wn’s are constants, which leads to `windowed’ IDFT/DFT bank,

a.k.a. `short-time Fourier transform’ (see Chapter-14)


En (z) = wn ⇒ RN−1−n (z) = wn−1

all E(z)’s

FIR E(z)’s

FIR unimodular E(z)’s

E(z)=F-1.diag{..}

L

15


•  Bad news: Not much design freedom for maximally decimated DFT-modulated FB’s…

•  Good news: More design freedom with...

–  Cosine-modulated FB’s

–  Oversampled DFT-modulated FB’s


L


•  Procedure: Po(z) = prototype lowpass filter, cutoff at for N filters Then...

etc.. PS: Real-valued filter coefficients here!

•  Details: See literature… •  Maximally decimated & oversampled FB designs •  Design software available (e.g.Matlab)

Cosine Modulated FBs

N2/π±

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

0*0

)5.0(

000N

jN

jezPezPzH

ππ

αα+−

+=

P0

π2

π2

π2

N2/π

N/πH1

Ho ).(.).(.)(

)5.01(

0*1

)5.01(

011N

jN

jezPezPzH

ππ

αα++−

+=

16


•  In maximally decimated DFT-modulated FB, we had (N-by-N matrices)

•  In oversampled DFT-modulated FB, will have with B(z) (tall-thin) and C(z) (short-fat) structured/sparse matrices

constructed with poyphase components of prototype filters

Oversampled DFT-Modulated FBs (D<N)

1−z2−z3−z

1u[k]

4 4 4 4

4 4 4

4

E(z4 ) + u[k-3]

1−z

2−z

3−z

1

R(z4 )4 4

4 4

N-by-D D-by-N

D=4 N=6

E(z) = F−1.diag En (z)[ ]

E(z)N-by-D!

= F−1

N-by-N!.B(z)N-by-D!

R(z) = diag RN−1−n (z)[ ].F

R(z)D-by-N!

=C(z)D-by-N!

.FN-by-N!


•  Details: see literature

•  Design Examples: http://homes.esat.kuleuven.be/~dspuser/DSP-CIS/2016-2017/material.html

•  Design software available (e.g.Matlab)

Oversampled DFT-Modulated FBs (D<N)

1−z2−z3−z

1u[k]

4 4 4 4

4 4 4

4

E(z4 ) + u[k-3]

1−z

2−z

3−z

1

R(z4 )4 4

4 4

N-by-D D-by-N

D=4 N=6

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