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Blind Adaptive Filters & Equalization

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

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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

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

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

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

∑−=

±±==

=⇒=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

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

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

][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

][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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

� Adaptive Filters Theory and

Applications

� B. Farhang-Boroujeny

� wiley