Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny...
Transcript of Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny...
![Page 1: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/1.jpg)
Blind Adaptive Filters
&Equalization
![Page 2: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/2.jpg)
Table of contents:
� Introduction about Adaptive Filters� Introduction about Blind Adaptive Filters� Convergence analysis of LMS algorithm� How transforms improve the convergence
rate of LMS� Why Wavelet transform?� Our blind equalization approaches� Simulation results
![Page 3: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/3.jpg)
Equalization
PulseShape
ModulatorChannel equalizer MF
}{ na }ˆ{ na
PulseShape
ModulatorChannel Equalizer
w[n]MF
}{ na }ˆ{ na
Composite Channelh[n]
![Page 4: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/4.jpg)
∑−=
−
±±==
=⇒=N
Nkknk n
nhwnnwnh
...,2,10
01][][*][ δ
∑−=
−=N
Nii inwnw ][][ δ
=
+
−
−
−
−−
−−−
−−−−
0
0
1
0
0
:
2
1
0
1
2
01234
10123
21012
32101
43210
w
w
w
w
w
hhhhh
hhhhh
hhhhh
hhhhh
hhhhh
Example
![Page 5: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/5.jpg)
Adaptive Equalization
]2)([
][ ,22
2
∑∑ −− −+=
=−=
knknknkn
nnnn
xwaxwaE
eEerroryae
InputSignal
Delay
ChannelAdaptiveEqualizer
Random noiseGenerator (2)
xn yn
en
xn
![Page 6: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/6.jpg)
0 5 10 15 2005101520
0
20
40
60
80
100
120
140
160
180
200
![Page 7: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/7.jpg)
][2][2
][2
knnk
nn
k
nn
k
xeEw
yeE
w
eeE
w
error
−−=∂∂−=
∂∂=
∂∂
∑−=
−=N
Nkknkn xwy
][ 2n
nnn
eEerror
yae
=
−=
knkk w
erroranwnw
∂∂−=+ µ
2
1)()1(
![Page 8: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/8.jpg)
][2 knnk
xeEw
error−−=
∂∂
∑−=
−=N
Nkknkn xwy
nnn yae −=
knkk w
erroranwnw
∂∂−=+ µ
2
1)()1(
TNNn
TNnnnnNnn
wwwwwW
xxxxxX
],...,,,...,,[
],...,,,...,,[
101
11
−−
−−++
=
=
nTnn WXY =
nnn yae −=
nnnn xeww µ+=+1
![Page 9: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/9.jpg)
Equalization, Deconvolution,
System compenstation
![Page 10: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/10.jpg)
System Identification
![Page 11: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/11.jpg)
Noise Cancellation
![Page 12: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/12.jpg)
Prediction
![Page 13: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/13.jpg)
Sidelobe cancellation
y(t)=Σ xi(t)
x1
x2
xN
y
![Page 14: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/14.jpg)
y
x1
x2
xN
y(t)=Σai xi(t)
a1
a2
aN
![Page 15: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/15.jpg)
Sound Clip
� Normalized *.wav file (microsoftformat)
� 9,946 bytes� click here
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Number of Samples
Am
plit
ude
of S
igna
l
NORMALIZED INPUT VOICE SIGNAL (*.WAV)
![Page 16: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/16.jpg)
Graphs – w/o and w/ equalization
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Number of Samples
Am
plit
ude
of S
igna
l
WITH NO EQUALIZATION
Voice S ignal, With No EqualizationOriginal Voice Signal
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Number of Samples
Am
plit
ude
of S
igna
l
WITH EQUALIZATION
Voice Signal, With EqualizationOriginal Voice Signal
Simulation under the advisory of Prof. Fontaine (downloaded)
![Page 17: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/17.jpg)
Blind Equalization
InputSignal
Delay
ChannelAdaptive
transversalequalizer
Additive whiteGaussian Noise
![Page 18: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/18.jpg)
Blind equalization approaches
� Stochastic gradient descent approach that minimizes a chosen cost function over all possible choices of the equalizer coefficients in an iterative fashion
� Higher Order Statistics (HOS) method that is using the higher order cumulants spread of the underlying process, and hence to the flatness
� Approaches that exploit statistical cyclostationarityinformation coefficients toward their optimum value, at a given frequency
� Algorithms that are based on the maximum likelihood criterion. depends on the value of the power spectral density of the
![Page 19: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/19.jpg)
Blind Equalization: HOSBlind Equalization: HOS
StatisticsOrder First : )()()}({)1( dxxftxtxEC Xx ∫==
dxtxtxftxtxtxtxEC Xx ∫ ++=+= ))(),(()()()}()({)(
:StatisticsOrder Second)2( ττττ
dxtxtxtxftxtxtx
txtxtxEC
Third
X
x
∫ ++++=
++=
))(),(),(()()()(
)}()()({),(
:StatisticsOrder
2121
2121)3(
ττττ
ττττ
![Page 20: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/20.jpg)
Channel
AWGN
WhiteNoise
x[n] y[n]
)()()()(2)2( fSfSfHfSC ninputyy +=⇒
Blind Equalization: HOSBlind Equalization: HOS
KfHfSC yy
2)2( )()( =⇒
![Page 21: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/21.jpg)
Blind Equalization: HOS
2nfor 0)( >=nxC
� For Gaussian signals:
)4()4()4( nhy CCC +=
Channel
AWGN
WhiteNoise
x[n] y[n]
![Page 22: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/22.jpg)
h1[n]t=nT/P
Y(m)
Fractionally Spaced Equalizer
∑ +−=l
mnlPmhlWmy )()()()(
∑ +−=l
iii mnlmhlWmy )()()()(
)()()( nNnHWnY NLNN h+= +
h2[n]
h1[n]
hi[n]
hN[n]
y1[m]
y2[m]
yi[m]
yN[m]
t=T/P
t=2T/P
t=iT/P
t=NT/P
![Page 23: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/23.jpg)
Fractionally Spaced Equalizer
)()()( nNnHWnY NLNN h+= +
OutputonaryCyclostatiSamplingSpacedlyFractional ⇒
![Page 24: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/24.jpg)
Formula:
� Channel model
� Cost function definition
� Updating equalizer coefficients
� Our proposed Wavelet domain gradient )1(:,)()()(ˆ 2
0 HgnYnYTJ WNN σ−′=∇
![Page 25: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/25.jpg)
Convergence rate of LMS algorithm
� It is well known that the convergence behavior of conventional LMS algorithm depends on the eigenvaluespread of input process
Faster convergencerate of LMS algorithm
Smaller EIG-spread ofinput correlation matrix
![Page 26: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/26.jpg)
EIG-spread of input correlation matrix (R)
vs. flatness of its PSD
� Convergence rate of filter coefficients toward their optimum value, at a given frequency depends on the value of the power spectral density of the underlying process at that frequency relative to all other frequencies.
Smaller EIG-spread ofinput correlation matrix
PSD flatnessof input signal
![Page 27: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/27.jpg)
EIG-spread of R vs. shape of error
surface: Example 1, EIG = 1.22
![Page 28: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/28.jpg)
Example 2, EIG = 3
![Page 29: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/29.jpg)
Example 3, EIG = 100
![Page 30: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/30.jpg)
Summary:
Better convergence rate of LMS algorithm
Shape (circularity) of error surface
Smaller EIG-spread of input correlation
matrix
PSD flatness of input signal
![Page 31: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/31.jpg)
How transforms improve the
convergence rate of LMS?� Band-partitioning property of Wavelet
transform� � Transformed elements are (at least)
approximately uncorrelated with one another� � Correlation matrix is closer to a diagonal
matrix� � An appropriate normalization can convert
the result to a normalized matrix whose EIG spread will be much smaller
![Page 32: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/32.jpg)
Advantages of using Wavelet
transform� Efficient transform algorithms exist (e.g. the
Mallat algorithm)
� Transforms can be implemented as filter banks with FIR filters
� Strong mathematical foundations allow the possibility of custom designing the wavelets e.g. the lifting scheme
![Page 33: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/33.jpg)
Wavelet transform algorithm:
![Page 34: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/34.jpg)
Matrix form implementation of
Wavelet transform
� As mentioned in the previous slide, Wavelet transform consists of two low-pass and high-pass filters
Data Width
Low Pass
High Pass
−−
−−
−−
−−
2301
1032
0123
3210
0123
3210
0123
3210
....
....
cccc
cccc
cccc
cccc
cccc
cccc
cccc
cccc
![Page 35: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/35.jpg)
Why Wavelet transform?
� wavelet analysis filters are ”constant-Q” filters; i.e., the ratio of the bandwidth to the center frequency of the band is constant
![Page 36: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/36.jpg)
band-partitioning property of
Daubechies filters
After transformation, each coefficient shows the
amount of energy passed from one of above filters
![Page 37: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/37.jpg)
The bandwidth of the filters in low frequencies is narrow compared to the bandwidth of the filters in higher frequencies
Most communication signals have a low-pass nature
It’s more probable that the output of filters contain the same amount of energy
more likely to obtain a flat spectrum.
![Page 38: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/38.jpg)
PSD of a typical communication signal
after different transforms
![Page 39: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/39.jpg)
Effect of Wavelet transform on error
surface
![Page 40: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/40.jpg)
MSE of a TD-Godard algorithm vs.
WD-Godard algorithm
![Page 41: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/41.jpg)
Formula:
� Channel model
� Cost function definition
� Updating equalizer coefficients
� Our proposed Wavelet domain gradient
![Page 42: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/42.jpg)
MSE of a TD-FSE algorithm
compared with WDFSE algorithm
![Page 43: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/43.jpg)
References
� Adaptive Filter Theory
� Simon Haykin
� Prentice Hall
![Page 44: Blind Adaptive Filters Equalization...Adaptive Filters Theory and Applications B. Farhang-Boroujeny wiley Title Adaptive Blind FSE Minayi Author Minayi Created Date 12/19/2008 12:00:00](https://reader030.fdocuments.in/reader030/viewer/2022040308/5f05c2a07e708231d41491c7/html5/thumbnails/44.jpg)
� Adaptive Filters Theory and
Applications
� B. Farhang-Boroujeny
� wiley