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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.
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 3
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.
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 5
Sample Rate Alteration
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)
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 6
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)
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 7
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)
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 9
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
=
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 10
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
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 11
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)
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 13
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.
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 14
Decimation (VII)
Xc(j)
1
X(ej)
1/
=
22
YD(ej)
DT
1
22
'T=
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 15
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=
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 17
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)
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 18
Interpolation (II)
=
(a) (b)
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 19
.
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
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 21
Interpolation (V)
Q(ej)
1/
22
'T=
HI(ej)
I
22I/
'T=
I/
YI(ej)
1/T'=I/T
22
'T=
/2/2
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 22
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)
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 23
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
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 25
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 26
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
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 27
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)
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 29
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.
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 30
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 31
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
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 33
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
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 34
.
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.
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 35
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
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 37
...
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.
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 38
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 39
Polyphase Implementation
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)
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 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)
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 42
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.
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 43
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)=
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 45
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.
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 46
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
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 47
)(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
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 49
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.
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 50
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.
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 51
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
N. Tangsangiumvisai Adaptive Signal Processing : Lecture 9 53