Blind Adaptive Filters & Equalization
Blind Adaptive Filters
&Equalization
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
Equalization
PulseShape
ModulatorChannel equalizer MF
}{ na }ˆ{ na
PulseShape
ModulatorChannel Equalizer
w[n]MF
}{ na }ˆ{ na
Composite Channelh[n]
∑−=
−
±±==
=⇒=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
Adaptive Equalization
]2)([
][ ,22
2
∑∑ −− −+=
=−=
knknknkn
nnnn
xwaxwaE
eEerroryae
InputSignal
Delay
ChannelAdaptiveEqualizer
Random noiseGenerator (2)
xn yn
en
xn
0 5 10 15 2005101520
0
20
40
60
80
100
120
140
160
180
200
][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(
][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
Equalization, Deconvolution,
System compenstation
System Identification
Noise Cancellation
Prediction
Sidelobe cancellation
y(t)=Σ xi(t)
x1
x2
xN
y
y
x1
x2
xN
y(t)=Σai xi(t)
a1
a2
aN
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)
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)
Blind Equalization
InputSignal
Delay
ChannelAdaptive
transversalequalizer
Additive whiteGaussian Noise
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
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
Channel
AWGN
WhiteNoise
x[n] y[n]
)()()()(2)2( fSfSfHfSC ninputyy +=⇒
Blind Equalization: HOSBlind Equalization: HOS
KfHfSC yy
2)2( )()( =⇒
Blind Equalization: HOS
2nfor 0)( >=nxC
� For Gaussian signals:
)4()4()4( nhy CCC +=
Channel
AWGN
WhiteNoise
x[n] y[n]
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
Fractionally Spaced Equalizer
)()()( nNnHWnY NLNN h+= +
OutputonaryCyclostatiSamplingSpacedlyFractional ⇒
Formula:
� Channel model
� Cost function definition
� Updating equalizer coefficients
� Our proposed Wavelet domain gradient )1(:,)()()(ˆ 2
0 HgnYnYTJ WNN σ−′=∇
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
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
EIG-spread of R vs. shape of error
surface: Example 1, EIG = 1.22
Example 2, EIG = 3
Example 3, EIG = 100
Summary:
Better convergence rate of LMS algorithm
Shape (circularity) of error surface
Smaller EIG-spread of input correlation
matrix
PSD flatness of input signal
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
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
Wavelet transform algorithm:
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
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
band-partitioning property of
Daubechies filters
After transformation, each coefficient shows the
amount of energy passed from one of above filters
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.
PSD of a typical communication signal
after different transforms
Effect of Wavelet transform on error
surface
MSE of a TD-Godard algorithm vs.
WD-Godard algorithm
Formula:
� Channel model
� Cost function definition
� Updating equalizer coefficients
� Our proposed Wavelet domain gradient
MSE of a TD-FSE algorithm
compared with WDFSE algorithm
References
� Adaptive Filter Theory
� Simon Haykin
� Prentice Hall
� Adaptive Filters Theory and
Applications
� B. Farhang-Boroujeny
� wiley