August 2004Multirate DSP (Part 2/2)1 Multirate DSP Digital Filter Banks Filter Banks and Subband Processing Applications and Advantages Perfect Reconstruction

August 2004 Multirate DSP (Part 2/2) 1

Multirate DSP•Digital Filter Banks

•Filter Banks and Subband Processing

•Applications and Advantages

•Perfect Reconstruction FB

•2-band Quadrature-Mirror Filter Bank

•K-band Filter Bank Case

•Uniform DFT Modulated Filter Banks


Digital Filter Banks

• So far we have mostly concentrated on the design, realisation and applications of single-input, single-output digital filters.

• In certain applications, it is desirable to separate a signal into a set of subband signals occupying, usually nonoverlapping, portions of the original frequency band.

• In other applications, it is necessary to combine many such subband signals into a single composite signal occupying the whole Nyquist range.

• To achieve the above two requirements, we need digital filter banks.


H0(z)

H1(z)

HK-1(z)

. . .

v0[n]x[n]

v1[n]

vK-1[n]

F0(z)

F1(z)

FK-1(z). . .

v0’[n] y[n]

v1’[n]

vK-1’[n]

+

+

Analysis Filter Bank Synthesis Filter Bank

• K-band analysis filter bank with the subfilters Hk(z) known as the analysis filters.

• Decomposes input signal x[n] into a set of K subband signals vk[n] with each subband signal occupying a portion of the original frequency band.

• Synthesis filter bank - a set of subband signals v’k[n] is combined into one signal y[n].

• K-band synthesis bank where each filter Fk(z) is called synthesis filter.


•Decimation: decimator (down-sampler)

example : u[k]: 1,2,3,4,5,6,7,8,9,…

2-fold down-sampling: 1,3,5,7,9,...

•Interpolation: expander (up-sampler)

example : u[k]: 1,2,3,4,5,6,7,8,9,…

2-fold up-sampling: 1,0,2,0,3,0,4,0,5,0...

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

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

Down-sampler and up-sampler (Revisited)


General `subband processing’ set-up/overview:

- signals split into frequency channels/subbands (ànalysis bank’)

- per-channel/subband processing

- reconstruction (`synthesis bank’)

- multi-rate structure: down-sampling / up-sampling

Filter Banks and Subband Processing [1/6]

subband processing 3H1(z)



3

3

3

3 subband processing 3H4(z)

IN

G1(z)

G2(z)

G3(z)

G4(z)

+

OUT


Step-1: Analysis filter bank

- collection of M filters (ànalysis filters’, `decimation filters’) with a

common input signal

- ideal (but non-practical) frequency responses = ideal bandpass filters

- typical frequency responses (overlapping, marginally overlapping,

non-overlapping)

2

H1(z)

H2(z)

H3(z)

H4(z)

IN2

H1 H4H3H2

H1 H4H3H2

H1 H4H3H2

2

K=4

2



Step-2: Decimators (down-samplers)

- subband sampling rate reduction by factor N

- critically decimatedcritically decimated filter banks (= maximally down-sampled filter banks):

N = K (where, K = number filters/subbands)

this sounds like maximum efficiency, but aliasing problem arises!

- over-sampled filter banks (= non-critically down-sampled filter banks):

N < K


H1(z)

H2(z)

H3(z)

H4(z)

IN

3

3

3

3

N=3K=4


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=3K=4



Step-4: Expanders (up-samplers)

- restore original fullband sampling rate by N-fold up-sampling

(= insert N-1 zeros in between every two samples)





3

3

3


IN

K=4 N=3 N=3



Step-5: Synthesis filter bank

- collection of K filters (`synthesis filters’, 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)

2

G1 G4G3G2

2

G1 G4G3G22

G1 G4G3G2G1(z)

G2(z)

G3(z)

G4(z)

+

OUT

K=4

2


Aliasing versus Perfect Reconstruction

Assume subband processing does not modify subband signals

(e.g. lossless coding/decoding)

- The overall aim could 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: down-sampling introduces ALIASING, especially in maximally

decimated (but even so in non-maximally decimated) filter banks

- Question : Can y[k]=u[k-d] be achieved in the presence of aliasing?

- Answer = YES, see below: PERFECT RECONSTRUCTION banks with

synthesis bank designed to remove aliasing effects !

output=input 3H1(z)

3H2(z)

3H3(z)

3333 3H4(z)

u[k]

G1(z)

G2(z)

G3(z)

G4(z)

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

output=input

output=input

output=input










Filter Banks Applications [1/5]

1. Speech coding

2. Image compression

3. Adaptive equalization

4. Echo cancellation

5. Adaptive beamforming

6. Transmultiplexers (TDM)

7. Code division multiple access (CDMA)

coding

adaptive filtering



Subband coding:

Coding = Fullband signal split into subbands & down-sampled

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:

Example 1: Adaptive channel equalization

Adaptive equalizer is employed to mitigate the effect of inter-symbol interference (ISI), that occurs due to limited bandwidth and multipath propagation.

Example 2: 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 adaptive filtering (continued):

Example 1: Adaptive channel equalization

Adaptive equalizer is employed to mitigate the effect of inter-symbol interference (ISI), that occurs due to limited bandwidth and multipath propagation.



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

Subband adaptive filtering (continued):

Example 2: Acoustic echo cancellation

- Subband filtering = K subband modeling problems instead of one fullband

modeling problem

- Perfect reconstruction guarantees distortion-free desired near-end

speech signal



General advantages of subband decomposition techniques

1. Allow parallel processing of signal will slower processor, as the sampling rate is reduces in each subband.

2. Computational complexity reduction. For instance in compression application where subband are compressed at different rates, e.g. speech processing and image compression.

3. Improve/increase convergence speed in adaptive filtering applications, e.g. echo cancellation and adaptive equalization.


- analysis bank + synthesis bank

- multirate structure: down-sampling after analysis, up-sampling for synthesis

- aliasing vs. “perfect reconstruction”

- filter bank implementation (for N = decimation factor, K = number of subband):

1. critically-sampled FB N = K

2. over-sampled FB N < K

- applications: coding, (adaptive) filtering, transmultiplexers

- advantages of subband processingsubband processing 3H1(z)



3

3

3


IN

G1(z)

G2(z)

G3(z)

G4(z)

+

OUT

Conclusion I: General `subband processing’ set-up










Two design issues :

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

2. (nearly) perfect reconstruction property...

Design Issues


Perfect Reconstruction: K = 2 subbands case

H0(z)

H1(z)

2

2

u[k] 2

2

G0(z)

G1(z)+

y[k]

2KN

)(.

)(

)}()()().(.{2

1)(.

)(

)}()()().(.{2

1)( 11001100 zU

zA

zGzHzGzHzU

zT

zGzHzGzHzY

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

• U(-z) represents aliased signals, hence the àlias transfer function’ A(z) should ideally be zero

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


• Alias-free filter bank:

• Perfect reconstruction filter bank (= alias-free + distortion-free):

i)

ii)

• PS: 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- & down-sampling)!


H0(z)

H1(z)

2

2

u[k] 2

2

G0(z)

G1(z)+

y[k]

0)( zA

0)( zA

zzT )(


• An initial choice is ….. :

so that

• For the real coefficient case:

which means the amplitude response of H1 is the mirror image of the

amplitude response of H0 with respect to the quadrature frequency

hence the name `quadrature mirror filter’ (QMF)


H0(z)

H1(z)

2

2

u[k] 2

2

G0(z)

G1(z)+

y[k]

)()( ),()( ),()( 110001 zHzGzHzGzHzH

0...)( zA )}()({2

1...)( 2

020 zHzHzT

)(...)()

2(

0

)2

(

1

jj

eHeH

ignore the details!


• quadrature mirror filter (QMF):

• hence if H0 (=G0) is designed to be a good lowpass filter, then H1 (=-G1) is a good high-pass filter.


H0(z)

H1(z)

2

2

u[k] 2

2

G0(z)

G1(z)+

y[k]

2

H0 H1


Perfect Reconstruction: K number subbands caseKN

H2(z)

H3(z)

4

4

4

4

G2(z)

G3(z)

y[k]

H0(z)

H1(z)

4

4u[k]

4

4

G0(z)

G1(z)

+

• 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 ideally be a pure delay

).(.

)(

)}()..({.1

)(.

)(

})().(.{1

)(1

1

1

0

1

0

lM

l

M

kk

lk

M

kkk WzU

zl

A

zGWzHM

zU

zT

zGzHM

zY

ignore the details!










Uniform versus non-uniform (analysis) filter bank:

• non-uniform: e.g. for speech & audio applications

• uniform filter bank:

= frequency responses uniformly shifted over the unit circle

• H0(z) = `prototype’ filter, is the only filter that has to be designed

Uniform DFT-Modulated Filter Banks

H0(z)

H1(z)

H2(z)

H3(z)

IN

H0 H3H2H1

H0 H3H2H1uniform

non-uniform

).()( /20

Nkjk ezHzH



• Uniform filter banks can be implemented cheaply based on polyphase decompositions + FFT :

1. Uniform DFT-modulated analysis filter banks

If

then

).()( with )(),...,(),( /20110

NkjkN ezHzHzHzHzH

)(.)(1

00

N

l

Nl

l zEzzH

with , )(..

)(..).()(

/21

0

1

0

1

/2/2/20

NjN

l

Nl

kll

N

l

NkNjNl

NkljlNkjk

eWzEWz

ezEezezHzH

H0(z)

H1(z)

H2(z)

H3(z)

u[k]



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

where F is NxN DFT-matrix

*F

u[k]

)( 40 zE

)( 41 zE

)( 42 zE

)( 43 zE

)(0 zH

)(1 zH

)(2 zH

)(3 zH

ignore the details!



• Uniform DFT-modulated analysis FB + decimation (K=N)

*F 4

4

4

4u[k]

)( 40 zE

)( 41 zE

)( 42 zE

)( 43 zE

u[k]

4

4

4

4

*F)(0 zE

)(1 zE

)(2 zE

)(3 zE

=


2. Uniform DFT-modulated synthesis filter banks


+

+

+

)(0 zG][0 ku

)(1 zG][1 ku

)(2 zG][2 ku

)(3 zG][3 kuy[k]

).(.)( with )(),...,(),( /20

/2110

NkjNkjkN ezGezGzGzGzG

)(.)(1

00

N

l

Nl

l zRzzG )(..... )(1

0

)1(

N

l

Nl

lNklk zRWzzG


Nj

NNNN

N

N

NNNNN

N

N

N

eW

zU

zU

zU

zU

F

WWWW

WWWW

WWWW

WWWW

zRzRzzRzzRz

zU

zU

zU

zU

zFzFzFzFzY

/2

1

2

1

0

)1()1(2)1(0

)1(2420

)1(210

0000

011

22

11

1

2

1

0

1210

)(

:

)(

)(

)(

.

...

::::

...

...

...

.)()(.)(....)(.

)(

:

)(

)(

)(

.)(...)()()()(

2

where F is NxN DFT-matrix

y[k]

+

+

+)( 40 zR

)( 41 zR

)( 42 zR

)( 43 zR][0 ku

][1 ku

][2 ku

][3 ku

F

ignore the details!



• Expansion (K=N) + uniform 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

=



• For over-sampled case, with down-sampling factor (N) smaller that the number of subbands (K) [i.e. N < K], aliasing is expected to become a smaller problem, possibly negligible if N<<K.

• Still, PR theory (hence perfect alias cancellation) is not necessarily simpler.


• Design problem = all filter specs + alias cancellation (PR)

• Uniform versus non-uniform filter banks

• Critically-sampled and over-sampled PR filter banks

• Modulated PR filter banks

• DFT-modulated filter banks

• Other methods: GDFT-modulated filter banks and DCT-modulated filter banks (not considered here!)

• Economy in design (only prototype) & implementation

Conclusion II:

Documents

August 2004Multirate DSP (Part 2/2)1 Multirate DSP Digital Filter Banks Filter Banks and Subband Processing Applications and Advantages Perfect Reconstruction