DSP-CIS Chapter-7: Introduction to Multi-rate Systems & Filter Banks Marc Moonen Dept. E.E./ESAT, KU Leuven [email protected]

DSP-CIS

Chapter-7: Introduction to Multi-rate Systems & Filter Banks

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

[email protected]

www.esat.kuleuven.be/scd/

mailto:[email protected]

DSP-CIS / Chapter 7: Multi-rate Systems & Filter Banks / Version 2012-2013 p. 2

• Filter banks introduction Introduction Applications Aliasing & perfect reconstruction

• Review of multi-rate systems

• DFT/IDFT filter bank• Maximally decimated DFT-modulated filter banks• Other…

Overview


Filter Banks Introduction

What we have in mind is this… :

- Signals split into frequency channels/subbands - Per-channel/subband processing - Reconstruction : synthesis of processed signal - Applications : see below (audio coding etc.) - In practice, this is implemented as a multi-rate structure for higher efficiency (see next slides)

subband processing

subband processing

subband processing

subband processing

H1(z)

H2(z)

H3(z)

H4(z)

IN+

OUT

H1 H4H3H2

2



Step-1: Analysis filter bank - collection of N filters (ànalysis filters’, `decimation filters’) with a common input signal - ideal (but non-practical) frequency responses = ideal bandpass filters

- typical frequency responses (overlapping, non-overlapping,…)

2

H1 H4H3H2

H1 H4H3H2

H1 H4H3H2

2

2

H1(z)

H2(z)

H3(z)

H4(z)

IN

N=4



Step-2: Decimators (downsamplers) - To increase efficiency, subband sampling rate is reduced by factor D

(= Nyquist (bandpass) sampling theorem)

- Maximally decimated filter banks (=critically downsampled):

# subband samples= # fullband samples this sounds like maximum efficiency, but aliasing (see below)!

- Oversampled filter banks (=non-critically downsampled):

# subband samples> # fullband samples

D=N

D<N

H1(z)

H2(z)

H3(z)

3

3

3

3H4(z)

IN

N=4 D=3

PS: 3-fold downsampling

= ‘remove every 2nd and 3rd sample’ (see below)



Step-3: Subband processing

- Example :

coding (=compression) + (transmission or storage) + decoding

- Filter bank design mostly assumes subband processing has ùnit

transfer function’ (output signals=input signals), i.e. mostly ignores

presence of subband processing

subband processingH1(z)



3

3

3

3 subband processingH4(z)

IN

N=4 D=3



Step-4: Expanders (upsamplers)

- restore original fullband sampling rate by D-fold upsampling

subband processing 3H1(z)



3

3

3

3 subband processing 3H4(z)

IN

N=4 D=3 D=3

PS: 3-fold upsampling

= ‘insert 2 zeros in between every two samples’ (see below)



Step-5: Synthesis filter bank - upsampling has to be followed by (interpolation) filtering (see p.22)

- collection of N `synthesis’ (ìnterpolation’) filters, with a `common’ (summed) output signal - frequency responses : preferably `matched’ to frequency responses of the analysis filters, e.g., to provide perfect reconstruction (see below)

G1(z)

G2(z)

G3(z)

G4(z)

+OUT




3

3

3

3 subband processing 3H4(z)

IN

N=4 D=3 D=3

G1 G4G3G2

2


Aliasing versus Perfect Reconstruction

- 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, especially so in maximally

decimated (but also in non-maximally decimated) filter banks

G1(z)

G2(z)

G3(z)

G4(z)

+

output = input 3H1(z)



3

3

3

3 output = input 3H4(z)

N=4 D=3 D=3

u[k]

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


Aliasing versus Perfect Reconstruction

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 !

G1(z)

G2(z)

G3(z)

G4(z)

+




3

3

3


N=4 D=3 D=3

u[k]

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


Filter Banks Applications

• Subband coding : Coding = Fullband signal split into subbands & downsampled subband signals separately encoded (e.g. subband with smaller energy content encoded with fewer bits)

Decoding = reconstruction of subband signals, then fullband signal synthesis (expanders + synthesis filters) Example : Image coding (e.g. wavelet filter banks) Example : Audio coding e.g. digital compact cassette (DCC), MiniDisc, MPEG, ... Filter bandwidths and bit allocations chosen to further exploit perceptual properties of human hearing (perceptual coding, masking, etc.)



• Subband adaptive filtering :

- See Chapter-8

- Example : Acoustic echo cancellation

Adaptive filter models (time-varying) acoustic echo path and produces

a copy of the echo, which is then subtracted from microphone signal.

= difficult problem !

✪ long acoustic impulse responses

✪ time-varying



- Subband filtering = N (simpler?) subband modeling problems instead of one (more complicated?) fullband modeling problem

- Perfect reconstruction guarantees distortion-free desired near-end speech signal

3H1(z)

3H2(z)

3H3(z)

3H4(z)

3H1(z)

3H2(z)

3H3(z)

3H4(z) ++

++ 3 G1(z)

3 G2(z)

3 G3(z)

3 G4(z)

OUT+

ad.filter

ad.filter

ad.filter

ad.filter



• Transmultiplexers : Frequency Division Multiplexing (FDM) in digital communications - N different source signals multiplexed into 1 transmit signal by expanders & synthesis filters (ps: here interpolation

factor )

- Received signal decomposed into N source signals by analysis filters & decimators

- Again ideal filters = ideal bandpass filters

5 G1(z)

5 G2(z)

5 G3(z)

5 G4(z)

+

5H1(z)

5H2(z)

5H3(z)

5H4(z)

transmission

channel

signal-1

signal-2

signal-3

signal-4

signal-1

signal-2

signal-3

signal-4

D>=N


- Non-ideal synthesis & analysis filters result in aliasing as

well as CROSS-TALK between channels,

i.e. each reconstructed signal contains unwanted

contributions from other signals

- Analysis & synthesis are reversed here, but similar perfect

reconstruction theory (try it!) (where analysis bank removes cross-talk introduced by synthesis bank, if transmission channel = distortion free)




PS: special case is Time Division Multiplexing (TDM), if synthesis and analysis filters are replaced by delay operators (and N=D)

PS: special case is Code Division Multiplexing (CDMA)

4444

+444

transmission

channel

1z2z3z

1

1z

2z

3z

u1[k],u1[k+1]

u2[k],u2[k+1]

u3[k],u3[k+1]

u4[k],u4[k+1]

u1[k],u2[k],u3[k],u4[k],u1[k+1],u2[k+1]...

4 u1[k-1],u1[k]

u2[k-1],u2[k]

u3[k-1],u3[k]

u4[k-1],u4[k]

4z

0,0,0,u4[k],0,0,0,u4[k+1]...

PS: another special case is OFDM… explain! What are Gi(z) & Hi(z) ?


Review of Multi-rate Systems 1/10

• 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...

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

D 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’

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

D)(zU

jez

D

xHz

ìmages’

xHz


Review of Multi-rate Systems 2bis/10

• Z-transform (frequency domain) analysis of expander

expander mostly followed by ìnterpolation filter’ to remove images (and ìnterpolate the zeros’)

interpolation filter can be low-/band-/high-pass

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

3

xHz

ìmages’

xHz

xHz

LP



• Z-transform (frequency domain) analysis of decimator

`compression in time domain ~ expansion in frequency domain’ PS: Note that is periodic with period while is periodic with period

The summation with d=0…D-1 restores the periodicity with period !

D)(zU

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

d=0d=2 d=1

3

xHz

3

xHz

)( jeU 2

2



• Z-transform (frequency domain) analysis of decimator

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

anti-aliasing filter can be low-/band-/high-pass

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

LP

d=0d=2 d=1

3

xHz

3

xHz



• Interconnection of multi-rate building blocks :

identities also hold if all decimators are replaced by expanders

D x

aDx

a=

=

=

D+

u2[k]

Dx

u2[k]

u1[k]

u1[k]

D +

Du2[k]

u1[k]

D x

Du2[k]

u1[k]



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

Example : D=2 h[0],h[1],0,0,0,…

=D D )(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 : D=2 h[0],h[1],0,0,0,…

=D D)(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 realizations of FIR filters through…

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

example : (3-fold decomposition)

general: (D-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



• Polyphase decomposition: Example : efficient implementation of an FIR 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 FIR 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


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)

+




4

4

4


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 p.16

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? (N-channel case)

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 Hn 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

1210 ...1)( NzzzzH

: :



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

lowpass filter with significant sidelobes.

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

1

1F

*(1/N)N=4

::


Maximally Decimated DFT-Modulated FBs

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)

example: wavelet filter banks

H0(z)

H1(z)

H2(z)

H3(z)

INH0 H3H2H1

H0 H3H2H1uniform

non-uniform



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

i.e.

H0(z)

H1(z)

H2(z)

H3(z)

u[k]

NM



where F is NxN DFT-matrix (and `*’ is complex conjugate)

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

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 (for FIR filters):

N filters for the price of 1, plus DFT (=FFT) !– design complexity:

Design `prototype’ Ho(z), then other Hn(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 En(z)=1

*F

u[k]

1 )(0 zH

)(1 zH

)(2 zH

)(3 zH

11

1

Ho(z) H1(z)



• PS: with F instead of F* (see p.32), only filter ordering is changed

Fu[k]

1 )(0 zH

)(1 zH

)(2 zH

)(3 zH11

1

Ho(z) H1(z)



• DFT-modulated analysis FB + maximal decimation

*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

= = efficient realization !



y[k]

+

+

+)( 40 zR

)( 41 zR

)( 42 zR

)( 43 zR][0 ku

][1 ku

][2 ku

][3 ku

F

2. Synthesis FB: similar..



• Expansion + 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

= = efficient realization !



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]

44

4

4

+

+

+)(0 zR

)(1 zR

)(2 zR

)(3 zR

F4

4

4

4u[k]

*F)(0 zE

)(1 zE

)(2 zE

)(3 zE



Design Procedure : 1. Design prototype analysis filter Ho(z) (see Chapter-3).

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

3. Assuming all En(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



• Will consider only FIR prototype analysis filters, leading to simple polyphase decompositions.

• However, FIR En(z)’s generally again lead to IIR Rn(z)’s, where stability is a concern…

• Hence need for other/better design procedures


Other FBs…

• Para-unitary perfect reconstruction filter banks• Cosine-modulated perfect reconstruction filter banks• Non-critically downsampled DFT-modulated perfect

recinstruction filter banks• Non-uniform filter banks• Wavelet filter banks• Etc…

Documents

DSP-CIS Chapter-7: Introduction to Multi-rate Systems & Filter Banks Marc Moonen Dept. E.E./ESAT, KU Leuven [email protected]