DSP-CIS Chapter-7: Filter Banks -Preliminaries Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven [email protected]

DSP-CIS

Chapter-7: Filter Banks -Preliminaries

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

[email protected]

www.esat.kuleuven.be/stadius/

mailto:[email protected]

DSP-CIS / Chapter 7: Filter Banks – Preliminaries / Version 2014-2015 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


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 p.32)



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 p.30)



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


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

(see also p.32-33)

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

3 output = input 3H4(z)

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 Part IV

- 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)

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) ?



PS : special case is Code Division Multiple Access (CDMA), where filter coefficients=(orthogonal) user codes CDMA basics: (see digital coms courses)…… -Each user (n=1..N) is assigned a length-N pseudo-random code sequence

-Transmission : For each symbol (k-th symbol for user-n), a `chip’ sequence is transmitted

-Mostly binary codes ( ) with BPSK/QPSK symbols -Multiple access based on code-orthogonality (see below)Skip th

is slide


CDMA basics: -Reception : If (x) received signal = transmitted chip sequence (i.e. no

channel effect, no noise), multiply chips with (synchronized) code

sequence + sum.

-Example (user i)

transmitted symbols +1……… -1………. -1……… +1………

code sequence +1,+1,-1,-1

transmitted chips +1,+1,-1,-1 -1,-1,+1,+1 -1,-1,+1,+1 +1,+1,-1,-1

received chips +1,+1,-1,-1 -1,-1,+1,+1 -1,-1,+1,+1 +1,+1,-1,-1

+1,+1,-1,-1 (mult. with code & sum)

received symbols (*1/4) +1………… -1…….…..-1…………+1

(x) PS: real-world CDMA is considerably more complicated (different channels for different users + channel dispersion, asynchronous users, scrambling codes, etc.)


Skip this slid

e



CDMA Transmission/reception block scheme :

-transmitter code-multiplication may be viewed as filtering operation,

with FIR transmit filter

-receiver code-multiplication & summation may be viewed as filtering

operation, with receive filter

-PR for `flat’ channel H(z)=1 and if codes are orthogonal (prove it!)

4444

+444

transmission

channel

)(4 zC

u1[k],u1[k+1]

u2[k],u2[k+1]

u3[k],u3[k+1]

u4[k],u4[k+1]

u2[k+1],u2[k])( 12 zC

)(3 zC

)(2 zC

)(1 zC 4

Bas

e st

atio

n

Use

r-2

term

ina

l

Skip this slid

e


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

3

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 (see p.8 & Chapter-8)

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 (see p.36 & Chapter-8)

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


Example: Ideal Filter Bank

• With ideal analysis/synthesis filters, FB operates as follows (1)




4

4

4


IN

G1(z)

G2(z)

G3(z)

G4(z)

+

OUT

2

H1(z) H2(z) H3(z) H4(z)

IN

2

……







4

4

4


IN

G1(z)

G2(z)

G3(z)

G4(z)

+

OUT

2

H1(z) H2(z) H3(z) H4(z)

x1

x1

2

……PS: H1(z) analysis filter

≈ lowpass anti-alias filter







4

4

4


IN

G1(z)

G2(z)

G3(z)

G4(z)

+

OUT

x’1 x’1

(ideal subband processing)

x’1

2

……







4

4

4


IN

G1(z)

G2(z)

G3(z)

G4(z)

+

OUT

x”1

x”1

2

……







4

4

4


IN

G1(z)

G2(z)

G3(z)

G4(z)

+

OUT

2

G1(z) G2(z) G3(z) G4(z)

x’’’1

x’’’1

2

……

PS: G1(z) synthesis filter

≈ lowpass interpolation filter







4

4

4


IN

G1(z)

G2(z)

G3(z)

G4(z)

+

OUTx2

2

……

2

H1(z) H2(z) H3(z) H4(z)

x2PS: H2(z) analysis filter

≈ bandpass anti-alias filter







4

4

4


IN

G1(z)

G2(z)

G3(z)

G4(z)

+

OUTx’2 x’2

2

……x’2







4

4

4


IN

G1(z)

G2(z)

G3(z)

G4(z)

+

OUTx”2

IN

2

……x”2







4

4

4


IN

G1(z)

G2(z)

G3(z)

G4(z)

+

OUT

2

……PS: G2(z) synthesis filter

≈ bandpass interpolation filter

2

G1(z) G2(z) G3(z) G4(z)

x”’2

x”’2







4

4

4


IN

G1(z)

G2(z)

G3(z)

G4(z)

+

OUT

Now try this with non-ideal filters…

2

……OUT=IN

Documents

DSP-CIS Chapter-7: Filter Banks -Preliminaries Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven [email protected]