DSP-CIS Chapter 10: Cosine-Modulated Filter Banks & Special Topics

DSP-CIS

Chapter 10: Cosine-Modulated Filter Banks

& Special Topics

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

[email protected]

www.esat.kuleuven.be/stadius/

DSP-CIS / Chapter-10 : Cosine-Modulated Filter Banks & Special Topics / Version 2013-2014 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


Cosine-Modulated Filter Banks

Motivation :

Cosine-modulated FBs offer an alternative to DFT-modulated FBs…

• Similar to DFT-modulated FBs, cosine-modulated FBs offer economy in design- and implementation complexity

• Unlike DFT-modulated FBs, cosine-modulated FBs can be PR/FIR/paraunitary under maximal decimation (with design flexibility).



• 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



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

i.e. `cosine modulation’ (with real coefficients) instead of èxponential modulation’ (with complex coeffs, see DFT-modulated FBs Chapter-9)

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



Realization based on polyphase decomposition (analysis):

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

then...

NNT 2

u[k]

)( 20

NzE

)( 21

NzE

)( 212

NN zE

)().(0 zUzH

)().(1 zUzH

)().(1 zUzH 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) (i.e. `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

!!!!!

!!!!!!

!!!!

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e



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

hence fast implementation (=fast matrix-vector product) based on fast discrete cosine transform (DCT) procedure, with 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

: :Skip this slid

e



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 + PR/FIR/Paraunitariness?

Theorem: (proof omitted)

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

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

..th

is is

th

e h

ard

par

t…

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e



Perfect Reconstruction (continued)

Design procedure: Parameterize lossless systems for n=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 n=0,1..,N-1, -> lattice structure (see Part-II), where parameters are rotation angles

Skip this slid

e



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.9 = )( 20 zE

)( 2zEN

)( 21 zEN

)( 212 zE N

:

:

lossless 1-in/2-out



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 àllpass’ 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.

no FIR-design flexibility

provides flexibility for FIR-design


Time-Frequency Analysis & Wavelets

Starting point is discrete-time Fourier transform:

= infinitely long sequence u[k] is evaluated at infinitely many frequencies

Inversion/reconstruction/synthesis (=filter bank jargon) is..

= sequence u[k] is represented as weighted sum of basis functions

20 , ].[)(

k

kjj ekueU

2

0

).(2

1][ deeUku kjj



• ùncertainty principle’ says that if u[k] has a narrow support

(i.e. is localized), then U(.) has a wide support (i.e. is non-

localized), and vice versa• Hence notion of `frequency that varies with time’ not

accommodated (e.g. `short lived sine’ will correspond to

non-localized spectrum)

20 , ].[)(

k

kjj ekueU

2

0

).(2

1][ deeUku kjj


Tool to fill this need is `short-time Fourier transform’(STFT)

where w[k] is your favorite window function (typically with `compact support’ (=FIR) )

• Window slides past the data. For each window position k, compute discrete-time Fourier transform

PS: If w[k]=1 (all k) then result is goodold discrete-time FT for all window positions

• In following slides, will provide a filter bank version of STFT, also leading to simple inversion formula




Rewrite STFT formula as…

or similarly…(swap k and k-bar)

• If we forget about the fase factor up front (=modulate window i.o input),

then this corresponds to performing a convolution of u[k] with a filter

(see Chapter-2, p.19)

In practice, will compute this for

a discrete set of (N) frequencies…

leading to a set of filters…

• This is a DFT-modulated analysis bank, prototype filter = window function



• If window length>N, can use efficient implementation

based on polyphase decomposition

of prototype Ho + DFT-modulation

• Often window length=N, hence

1-tap polyphase components u[k]

)(0 zH

)(1 zH

)(2 zH

)(3 zH

*NNF

fre

q.re

solu

tion

N

*NNF

u[k]

)( 40 zE

)( 41 zE

)( 42 zE

)( 43 zE

)(0 zH

)(1 zH

)(2 zH

)(3 zH

window length/N



In practice, will consider ‘window shift’ >1 (decimation D>1)• If maximally decimated (D=N), then analysis FB operation is

xn[k] = decimated subband signals

= `STFT-coefficients’

= infinitely long sequence u[k] is

evaluated at N frequencies,

infinitely many times (i.e. for

infinitely many window positions)

...to be compared to page 14

u[k]

*NNF 4

4

4

4



• Corresponding (PR) synthesis filter bank (also DFT-modulated) is

i.e. synthesis prototype filter fo[k]

= ‘synthesis window’

=

• Reconstruction/synthesis formula

(=inverse STFT) is

..to be compared to page 14

FRFE .)( .)( 1* ii wdiagzwdiagz

4

4

4

4

+

+

+

F



• If oversampled (D=N/d), then analysis FB operation is

Example: d=2 (‘window shift’= N/2)

u[k]

*NNF 2

2

2

2

u[k]

*NNF

2

2

B(z)

=



• Corresponding (PR) synthesis filter bank (also DFT-modulated)

with PR condition for synthesis window v_k

• Can easily generalize this for other oversampling factors d, leading to

PR condition explain…

(and try examples with

Hann windows)

• This is referred to as Weighted OverLap-Add (WOLA)



Now, for some applications (e.g. audio) would like to have

a non-uniform filter bank, hence also with non-uniform

(maximum) decimation, for instance

• non-uniform filters = low frequency resolution at high frequencies, high frequency resolution at low frequencies (as human hearing)

• non-uniform decimation = high time resolution at high frequencies, low time resolution at low frequencies

H2(z)

H3(z)

4

2

H0(z)

H1(z)

8

8u[k] H0 H3H2H1

2

8

4



This can be built as a tree-structure, based on a

2-channel filter bank with

H0 H3H2H1

2

8

4

)(zHLP )(zHHP

u[k]2

2)(zHHP

)(zHLP

2

2)(zHHP

)(zHLP

2

2)(zHHP

)(zHLP

)().().()(

)().().()(

)().()(

)()(

240

241

22

3

zHzHzHzH

zHzHzHzH

zHzHzH

zHzH

LPLPLP

LPLPHP

LPHP

HP

)(),( zHzH HPLP



Note that may be viewed as a prototype filter,

from which a series of filters is derived

The lowpass filters are then needed to turn these

multi-band filters into bandpass filters (i.e. remove images)

)()(1 zHzH HPN

)()( )2( 1

k

zHzH HPkN

2

8

4

)(zHHP

)( 4zHHP

)( 2zHHP

)().().()(

)().().()(

)().()(

)()(

240

241

22

3

zHzHzHzH

zHzHzHzH

zHzHzH

zHzH

LPLPLP

LPLPHP

LPHP

HP



Similar synthesis bank can be constructed with

• If and form a PR FB, then the complete analysis/synthesis structure is PR (why?)

)(),( zFzF HPLP

2

2 +

2

2 + 2

2 + )(zFLP

)(zFHP

)(zFLP

)(zFLP)(zFHP

)(zFHP

)(),( zFzF HPLP)(),( zHzH HPLP



• Analysis bank corresponds to `discrete-time wavelet transform’ (DTWT) (xk[n] = `DTWT-coefficients’)

• With a corresponding (PR) synthesis filter bank, the reconstruction/synthesis formula (inverse DTWT) is

..to be compared to page 14 & 20



• Reconstruction formula may be viewed as an expansion of u[k], using a set of basis functions (infinitely many)

• If the 2-channel filter bank is paraunitary, then this basis is orthonormal (which is a desirable property) :

=òrthonormal wavelet basis’



• Example : `Haar’ wavelet (after Alfred Haar)

• Compare to 2-channel DFT/IDFT bank• Derive formulas for Ho, H1, H2, H3, …

Derive formulas for Fo, F1, F2, F3, …

Paraunitary FB (orthonormal wavelet basis) ?

)1(2

1

)1(2

1

1

1

zH

zHH

LP

HPHaar



Not treated here :• `continuous wavelet transform’ (CWT) of a continuous-time function u(t)

h(t)=prototype p,q are real-valued continuous variables p introduces `dilation’ of prototype, q introduces `shift’ of prototype • `discrete wavelet transform’ (DWT) is CWT with discretized p,q

T is sampling interval p-bar, q-bar are real-valued integer variables mostly a=2

dt

p

tqhtu

pqpxCWT )().(

1),(

Skip this slid

e



Not treated here :• Theory

- multiresolution analysis

- wavelet packets

- 2D transforms

- etc …• Applications :

- audio: de-noising, …

- communications : wavelet modulation

- image : image coding Skip this slid

e


Frequency Domain Filtering

• See DSP-I : cheap FIR realization based on frequency domain processing (`time domain convolution equivalent to component-wise multiplication in the frequency domain’), cfr. òverlap-add’ & òverlap-save’ procedures

• This can be cast in the subband processing setting, as a non-critically downsampled (2-fold oversampled) DFT-modulated filter bank based operation!

• Leads to more general approach to performance/delay trade-off

PS: formulae given for N=4, for conciseness (but without loss of generality)

Chapter 6!



Have to know a theorem from linear algebra here: • A `circulant’ matrix is a matrix where each row is obtained from the previous row using a right-shift (by 1 position), the rightmost element which spills over is circulated back to become the leftmost element • The eigenvalue decomposition of a `circulant’ matrix is trivial…. example (4x4):

with F the NxN DFT-matrix, this means that the eigenvectors are equal to the column-vectors of the IDFT-matrix, and that then eigenvalues are obtained as the DFT of the first column of the circulant matrix (proof by Matlab)

d

c

b

a

F

D

C

B

A

F

D

C

B

A

F

abcd

dabc

cdab

bcda

. with ,.

000

000

000

000

.1

abcd

dabc

cdab

bcda



Starting point is this (see Chapter-8) :

meaning that a filtering with

can be realized in a multirate structure, based on a

pseudo-circulant matrix

T(z)*u[k-3]1z2z3z

1

u[k] 444

4 4444

+1z

2z

3z

1

)(zT

)()(.)(.)(.

)()()(.)(.

)()()()(.

)()()()(

)(

031

21

11

1031

21

21031

3210

zpzpzzpzzpz

zpzpzpzzpz

zpzpzpzpz

zpzpzpzp

zT

)()()()()( 43

342

241

140 zpzzpzzpzzpzT

N*N

filt

ers,

L/N

tap

s ea

ch

at a

n N

-fol

d lo

wer

rat

e (N

=4)



Now some matrix manipulation… :

)()(

)(

44.1

44

7

6

5

4

3

2

1

0

144

44.1

44

3210

3210

3210

3210

0321

1032

2103

3210

44

..

)(0000000

0)(000000

00)(00000

000)(0000

0000)(000

00000)(00

000000)(0

0000000)(

..0

.

0)()()()(000

00)()()()(00

000)()()()(0

0000)()()()(

)(0000)()()(

)()(0000)()(

)()()(0000)(

)()()()(0000

.0

zz

z

x

xx

x

xx

Iz

IF

zP

zP

zP

zP

zP

zP

zP

zP

FI

Iz

I

zpzpzpzp

zpzpzpzp

zpzpzpzp

zpzpzpzp

zpzpzpzp

zpzpzpzp

zpzpzpzp

zpzpzpzp

I

ER

T



• An (8-channel) filter bank representation of this is...

Analysis bank:

Subband processing:

Synthesis bank:

This is a 2N-channel filter bank, with N-fold downsampling. The analysis FB is a 2N-channel uniform DFT filter bank (see Chapter 9, p.30 !). The synthesis FB is matched to the analysis bank, for PR:

..

.)(44

144

x

x

Iz

IFzE

144 .0)( FIz xR

1z2z3z

1

u[k] 444

4 4444

+y[k]

1z

2z

3z

1

)(zR)(zH)(zE

)}(),...(),({)( 710 zPzPzPdiagzH

441)().( xIzzz ER

2N

filte

rs,

L/N

tap

s ea

chat

an

N-f

old

low

er r

ate

||

N/2

-fol

d co

mpl

exity

red

uctio

n



• This is known as an òverlap-save’ realization :– Analysis bank: performs 2N-point DFT (FFT) of a block of (N=4)

samples, together with the previous block of (N) samples (hence òverlap’)

– Synthesis bank: performs 2N-point IDFT (IFFT), throws away the first half of the result, saves the second half

(hence `save’)

– Subband processing corresponds to `frequency domain’ operation

..

.)(44

144

x

x

Iz

IFzE

`block’

`previous block’

144 .0)( FIz xR

`save’`throw away’



Òverlap-add’ can be similarly derived :

)()(

)(

44

44

7

6

5

4

3

2

1

0

14444

.1

44

44

3210

3210

3210

3210

0321

1032

2103

3210

4444.1

0..

)(0000000

0)(000000

00)(00000

000)(0000

0000)(000

00000)(00

000000)(0

0000000)(

..

0.

0)()()()(000

00)()()()(00

000)()()()(0

0000)()()()(

)(0000)()()(

)()(0000)()(

)()()(0000)(

)()()()(0000

.

zz

z

x

xxx

x

xxx

IF

zP

zP

zP

zP

zP

zP

zP

zP

FIIz

I

zpzpzpzp

zpzpzpzp

zpzpzpzp

zpzpzpzp

zpzpzpzp

zpzpzpzp

zpzpzpzp

zpzpzpzp

IIz

ER

T



• This is known as an òverlap-add’ realization :– Analysis bank: performs 2N-point DFT (FFT) of a block of (N=4)

samples, padded with N zero samples

– Synthesis bank: performs 2N-point IDFT (IFFT), adds second half of the result to first half of previous IDFT (hence àdd’)

– Subband processing corresponds to `frequency domain’ operation

.0

.)(44

44

x

xIFzE`block’

`zero padding’

14444

1 ..)( FIIzz xxR

àdd’òverlap’



• Standard Òverlap-add’ and òverlap-save’ realizations are derived when 0th order poly-phase components are used in the above derivation, i.e. each poly-phase component represents 1 tap of an L-tap filter T(z). (N=L)

The corresponding 0th order subband processing (H) then corresponds to what is usually referred to as the `component-wise multiplication’ in the frequency domain.

Note that for an L-tap filter, with large L, this leads to a cheap realization based on FFT/IFFTs instead of DFT/IDFTs.

However, for large L, as 2L-point FFT/IFFTs are needed, this may also lead to an unacceptably large processing delay (latency) between filter input and output.



• In the more general case, with higher-order polyphase components (hence N smaller than the filter length L), a smaller complexity reduction is achieved, but the processing delay is also smaller.

• This provides an interesting trade-off between complexity reduction and latency !!


Conclusions

• Great (=FIR/paraunitary) perfect reconstruction FB designs based on `modulation’:– Oversampled DFT-modulated FBs (Chapter-9)– Maximally decimated (and oversampled (not treated here))

cosine-modulated FBs

• `Perfect reconstruction’ concept provides framework for time-frequency analysis of signals

• Filter bank concept provides framework for frequency domain realization of long FIR filters

DSP-CIS Chapter 10: Cosine-Modulated Filter Banks & Special Topics

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