adaptive handout 9

8/3/2019 Adaptive Handout 9

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and Filter banksand Filter banks

ContentsContents

Part I : Review of Multi-rate s stems

Sample Rate Alteration

Subband Structure anal sis & s nthesis filter bank

Part II : Filter banks

Aliasing and Perfect Reconstruction

Polyphase Implementation

Summary

N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 2

MultiMulti--rate systemsrate systems

Q : What is a multi-rate system ?

A : It means that multiple sampling rates are used

within such a system.


MultiMulti--rate systemsrate systems

Q : Why do we have to do multi-rate processing?

A : To convert/change the sampling rate for passing

data between two systems, such as

rans err ng a a rom au o pro ess ona s o

Reducing storage space

Reducing the transmission rate of data

Efficient implementation



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MultiMulti--rate Signal Processingrate Signal Processing

How to change the sampling rate of a digital signal?

Convert it back into analog, then re-digitize it at thenew rate

Or process it digitally (until conversion to analog ismandatory)

An efficient technique for sampling frequencya era on o a s gna g a y.



Decimation

- to decrease the sampling rate

- 2 step : filtering

down-sampling

x(n) yD(n)D

Fig.1 A sampling rate compressor (a down-sampler)

(down-sampling : throw away some samples)


Sample Rate Alteration (II)

Interpolation

- to increase the sampling rate

- 2 step : up-sampling

filtering

Ix(n) yI(n)

Fig.2 A sampling rate expander (an up-sampler)

(up-sampling : inserting zero-valued samples betweenoriginal samples)


Sample Rate Alteration (II)

Resampling

- sample rate conversion by a rational (fractional)factor

- combination of decimation and interpolation

e.g. to change the sampling rate by a factor of 1.5=3/2



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Decimation

time-domain :

z-transform :

)()( DnxnyD = ... (1)

=

=n

n

D zDnxzY )()( ... (2)

... (3)=

=n

nzDnx )(int

where )()()(int nxncnx = ... (4)


Decimation (II)

The comb sequenceis defined as

=

= otherwise,0

,2,,0,1

)(

DDn

nc... (5)

which can be represented as

=11

)(D

kn

DWD

nc ... (6)

where is the mth root of unity.

=

D

j

D eW

2

=


Decimation (III)

Therefore

= nznxnczX )()()(int ... (7)=n

= nD

kn

D znxW )(1 1

... (8)

= =n k 0

11 D nkn

= = =

0k n

D znxD

...

1D

1D

= =

=0

)(k n

nk

DzWnxD

... (10))== 0kk

DzWXD


Decimation (IV)

The out ut of the decimator in E . 3 becomes

= DkD zkxzY/

int )()( ... (11)=k

( )DzX /1int= ... (12)

=1

/11D

kDWzXzY

=0kD...

( )

=

=

1

0

/)2(1)(

D

k

DkjjD eX

DeY ... (14)



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Decimation (V)

Fig.3 Illustration of the decimation process by a factor of D=2.

(a) (b)


Decimation (VI)

When decimating with the factor ofD = 2, Eq.(14)becomes

2/2/1

2D eee = ...

Note that

( ) 2/22/ = jj eXeX ... (16)

This can be illustrated in the following page.


Decimation (VII)

Xc(j)

1

X(ej)

1/

=

22

YD(ej)

DT

1

22

'T=


Fig.4 Frequency-domain illustration of the decimation process by a factor of D=2.

Decimation (VIII)

2/jo a as ng s zero or .e

anti-aliasin

hD(n)

DecimatorD

filter

x(n)v(n)

yD(n)

Fig.5 A block diagram of a decimation by a factor D.

The frequency response of the anti-aliasing filter is given

=otherwise0

/,1)(

DHD

...(17)



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Decimation (IX)

HD(ej)

1

=

c 22

Dc / =

V(ej)

1/

= 22 //

YD(ej)

1/D

22

'T=


g. requency- oman us ra on o e ecma on process ( = ), w an -a asng er

Interpolation

time-domain :

=

=,2,,0),/(

)(IInInx

nyI ... (18)

z-transform :

,

=

= nI zInxzY )/()( ... (19)

= mI

zmx)(

... (20)=m

)(I

zX= ... (21)


Interpolation (II)

=

(a) (b)


.

Interpolation (III)

X(ej)

=

Y ejI

1/

'

-

22T=

.



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Interpolation (IV)

To remove all the ima es and to correct the scalin of theamplitude, a post-filter is required.

gain

=otherwise,0

/0,)(

IIHI

... (22)

hI(n)

-

x(n)q(n)

yI

(n)Interpolation

I

Fig.9 A block diagram of an interpolation by a factor ofI


Interpolation (V)

Q(ej)

1/

22

'T=

HI(ej)

I

22I/

'T=

I/

YI(ej)

1/T'=I/T

22

'T=

/2/2


g.1 requency- oma n us ra on o e n erpoa on process (I= ) w a pos - er.

Resampling

The cascade of an interpolator (increase the samplingrate by a factor ofI) with a decimator (decrease thesampling rate by a factor of D) results in a system that

.

anti-aliasing

hD(n)Decimator

D

filterv(n)

yD(n)h

I(n)

post-filter

x(n)q(n) y

I(n)Interpolation

I

see eq.(17) see eq.(22)


Resampling (II)

The equivalent system is given by

Decimatorv(n)y (n)h (n)x(n)

q(n)Interpolation

where hc(n) is a lowpass filter, with gain and cutofffrequency of

=

=

otherwise,0

,,)( IDHC



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Resampling (III)

Example : Change the sampling rate of the signalx(t)

with sampling rate 8 kHz to be 10 kHz.

Solution: To change the sampling rate by a factor of

510==

I

Up-sampling the signal by a factor ofI= 5

48D

Filtering the up-sampled signal with a lowpass filter withgain of 5 and the cutoff frequency of

=c

Down-sampling the signal by a factor ofD = 4


ContentsContents







Summary


Subband Structure

subband

IN

G1(z)

0 0

F1(z)

processing

M

Msubband

processing

OUT

GM-1

(z) FM-1

(z)M Msubband

processing

Fig.11 An example of a subband processing withMsubbands.

analysis filter banks

synthesis filter banks

aliasing V.S. perfect reconstruction polyphase implementation


Analysis & Synthesis filter

The typical frequency response ofthe analysis (or synthesis) filterbanks (AFB & SFB) , forMnumberof subbands, i.e.

G1(z)G

0(z) G

2(z)

GM-1

(z)

marginally overlapping

0 2

(a)

- max ma y ec mate ter an s

- critically down-sampledG1(z)G0(z) G2(z) GM-1(z)

non-overlapping

- oversampled filter banks

0

2

(b)

- non-critically down-sampled

Fig.12 Frequency response of analysis filter banks.



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AFB & SFB (II)

If we consider a symmetric analysis prototype filter, g(n),of length K, which is assumed to be fixed, i.e. g(n) = g, as

T=

By employing the frequency shifting,

,,, ...

( ))( 00 )( jnj eXnxe ... (24)

the ith coefficient of the analysis filter gk(n) in the kth

subband is then obtained as

1,,1,0

1,,1,0,)()(

2

=

==

Ki

Mkeigig M

kij

k

... (25)


AFB & SFB (III)

Normally, anMxMDiscrete Fourier Transform (DFT)matrix is defined as

2

Mklj

kl

,,,,,M ...

Hence, the AFB which employs the DFT matrix WM, iscalled the DFT filter bank.


ContentsContents







Summary


AFB & SFB (IV)AFB & SFB (IV)

By taking thez-transform of the impulse response of theanalysis filter in Eq.(25), we obtain

k=uniform DFT ... 27kfilter bank

...

G1(z)G

0(z) G

2(z) G

M-1(z)

0

2

Fig.13 A uniform filter bank.



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ContentsContents







Summary


AFB & SFB (V)

Let the input signal of the system in Fig.11 bex(n), thesubband input signals can be found as

1 2KM

kij

=0ik ...

which become

)()()( zXWzGzXk

k = ... (29)

One way to cancel the aliasing when Gk(z

)is maximallyoverlapping is by the choice of the synthesis filters.

Consider an example of a 2-band case on the following


.

Quadrature Mirror Filter Bank

A combined system of the AFB and theM-fold decimatorswith the SFB and theM-fold expanders is called aQuadrature-Mirror Filter (QMF) bank.

G0(z) 2 2 F

0(z)

x0(n) u

0(n) c

0(n)

x(n)

G1(z) 2 2 F

1(z)

x(n)

x1(n) u1(n) c1(n)

Analysisbank

Synthesisbank

Fig.14 The two-channel quadrature-mirror filter bank.


QMF Bank (II)

j G ej0

1

/2

0

Fig.15 Frequency response of the analysis filter of a two-channel system.



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Aliasing cancellation

The reconstructed signal is given by

)()()()()( zXzAzXzTzX += ... (30)

where the distortion function is defined as

1FGFGT += ... 31

the transfer function affecting the alias component X(-z) is

2

[ ])()()()(2

1)( 1100 zFzGzFzGzA += ... (32)

It is required that A(z)=0, thus

and GFGF == ... 33


...

Perfect Reconstruction

A perfect reconstruction (PR) system is when

)()(0nnxcnx = ... (34)

or

0),()( 0 = czXczzX n ... (35)

where c is the gain of the subband processing and n0represent the delay of the system.


ContentsContents







Summary



For efficient implementation of decimation andinterpolation filters.

Generally, thez-transform of the analysis prototypefilter, g(n), for anM-subband system is given by

=

=

+++=n

nM

n

nMznMgzznMgzG )1()()(

1

=

++n

nMMzMnMgz )1(

)1( ... (36)

=

=

+=n

nMM

k

k

zknMgz )(

1

0... (37)



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Polyphase Implementation (II)

By defining the polyphase filters of the analysis prototypefilter g(n) as

10),()( += MkknMgnegk

... (38)

which is also represented as

=

=n

ng

k

g

k znezE )()( ... (39)

Thus, the analysis prototype filter becomes1

MgM

k

zEzzG

= Type I polyphase0k= ...

)()()( 1)1(

1

1

0

Mg

M

MMgMg zEzzEzzE +++= ... (41)


Polyphase Implementation (III)

Alternatively,

)()(1

0

)1( Mg

k

M

k

kMzRzzG

=

= Type II polyphase ... (42)

)()()(11

)2(

0

)1( Mg

M

MgMMgMzRzRzzRz

+++= ... (43)

)()()( 02)2(

1

)1( MgMg

M

MMg

M

MzEzEzzEz +++=

... (44)

If the polyphase representation is applied within theuniform DFT filter bank, the analysis filter G

k(z) can be

expressed as

( ) ( )( )MkglM

lk

k zWEzWzG

=

1

)( ... (45)


l =0

Polyphase Implementation (IV)

which is equal to

( )MglM

l

kll

k zEWzzG

=

=1

0

)( ... (46)

for k,l = 0,1,...,M-1 and by using the identity WM= 1. Theexpression of the subband input signals in Eq.(29)ecome

)()()( zXWzGzXk

k =

)()(0

zXzEWzl

Mg

l

kll

=

= ... (47)

which can be illustrated on the following page.


Polyphase Implementation (V)

Eg0(zM) M

x0(n)

x(n)

z-1

u0(n)

Eg1(zM) M

x1

n

u1(n)H

MW

EgM-1

(zM) Mx

M-1(n)

z-1

uM-1

(n)

Fig.16 Polyphase representation of the analysis filter bank.



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Polyphase Implementation (VI)

The Noble Identities

H zM Mx n n H zMx n n=

H(z) Mx(n) y(n) H(zM)Mx(n) y(n)=


Polyphase Implementation (VII)

Eg0(z)M u0(n)x(n)

z-1

Eg1(z)M u1(n)H

MW

EgM-1(z)M uM-1(n)

z-1

Fig.17 Polyphase representation of the AFBafter applying the Noble Identities.


Polyphase Implementation (VIII)

By writing

the polyphase representation of Gk(z) becomes

)()(g

l

g

kl zEWz=E ... (48)

( )MgklM

l

k zzzG E

=1

)( ... (49)

or in matrix form as

=

=

1

1,11110

1,00100

1

0 1

)()()(

)()()(

)(

)(

Mg

M

MgMg

Mg

M

MgMg

zzzz

zzz

zG

zG

EEE

EEE

)1(1,11,10,11 )()()()( MMg MMMgMMgMM zzzzzG EEE


)(Mg

M zEpolyphase component matrix

Polyphase Implementation (IX)

Hence anM-band QMF bank can be illustrated in Fi . 17.

x(n) M M

M M

z-1

g g

z-1

zM zM

M M x(n)

Fig.18 An M-band QMF bank with polyphase representation.



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ContentsContents







Summary


Tree-structured Filter Bank

Another way to obtain the AFB and SFB

a binary tree-structure filter bank technique.

2

2 2

2( ) ( )z10

G

( )z

2

0F

( )z

2

0G

( )2 ( )2

( )( )z10

F

2 2

x(n)

( )1

( )( )z20

G

1

( )( )z20

F

1

( )1

x(n)

2 2

z1

( )( )z21

G( )( )z2

1F

z1

Level 2 Level 2 Level 1Level 1

Fig.19 A two-level tree-structured filter bank.


Application Example : Subband AECApplication Example : Subband AEC

v0(n)d(n)

x n n

M

g0

n

g1(n)

v1(n)

MgM-1(n)v

M-1(n)

e'0(n)

e'1(n)

M

M

v0(n)

v1(n)

-

+-

u0(n)

u1(n)

g0(n)

g1(n)

)( n0h

)(1

nh

e'M-1

(n)M v

M-1(n)

+-uM-1(n)

gM-1

(n) )( 1 nMh

1

Fig.20 A block diagram of an AEC employing subband structure.


Application Example :Application Example : Audio Coding

Coding

- fullband signal is split into subbands

- each subband is separately encoded- subband with fewer energy is encoded with fewer bits

Decoding

- reconstruction of fullband signal



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Summary

Review of Multi-rate systems

Subband Structure

Polyphase Implementations

Tree Structure

Next Lecture : Lecture 10 A lication Exam les


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