Robust LMS

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TRANSIENT ANALYSIS OF ERROR SATURATION

NONLINEARITY LMS IN PRESENCE OF IMPULSIVE

NOISE

T Panigrahi*1, P M Prdhan1, G Panda1and B Mulgrew2

1Department of ECE, National Institute of Technology, Rourkela

2IDCOM, The University of Edinburgh, UK


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Outline

MotivationIntroduction

LMS with saturation nonlinearity

Model for impulsive noiseTransient analysis

Recursive equations

Results

Conclusion

References


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Motivation

The performance of an adaptive filter is evaluated in terms of its

transient behavior which provides information about how fasta filter

learns, and its steady-state behavior gives how wella filter learns.

In presence of impulsive noise, the classical error square cost

function performs poorly.

Hence different adaptive schemes have to be developed which

should be robust to impulsive noise.

The purpose of this paper is to provide an energy conservation

approach for the performance analysis of adaptive filters with errornonlinearity in presence of Gaussian contaminated impulsive noise.

3


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Introduction

The effect of saturation type of non-linearity on the least-mean

square adaptation for Gaussian inputs and Gaussian noise have been

studied in literature.

The error saturation nonlinearities LMS provides good performance

in presence of impulsive noise. The convergence analysis of any adaptive algorithm is carried out

using weighted-energy relation.

In this paper we perform the transient analysis of saturation

nonlinearity LMS in presence of Gaussian contaminated impulsive

noise.


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Adaptive Algorithm With Saturation Error Nonlinearity

The estimate of an unknown vector by using row regressor oflength Mand output samples that is given as

where is represents the impulsive noise.

The well known LMS algorithm is

Introducing an error nonlinearity into the feedback error signal so

that the weight update equation is modified as

where the nonlinear functionf(e) is defined as

)(=)( ivid i

wu

w iu

)(iv

T

iii ie uww )(= 1

0)]([= 1 iief T

iii uww

s

s

y yerfduuyf

22

=]/2[exp=)( 220


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Transient Analysis

We are interested in studying the time-evolution of the varianceslike and where .

The steady-state values of the variances known as mean-square error

and mean-square deviation have to be studied.

In order to study the time evaluation of above variances, we

introduce the weighted a priori and a posteriori error defined as

where is a symmetric positive definite weighting matrix.

The weighted energy relation is given as

2|)(| ieE 2~|| ||iE w iwww

=~

iipiia ieandie wuwu ~=)(~=)( 1

2

2

2

12

22

||

|)(|~||=|)(|~

i

p

i

i

ai

ieie

uw

uw

||||

||||||||


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Transient Analysis (Cont)

The variance relation can be obtained from the above energy relation

by replacing a posteriori error by its equivalent expression

Taking expectation on both sides we get

Evaluation of 2nd and 3rd term on RHS of above equation is difficult

as it contains the nonlinearity term.

)]([)]([)(2~=~ 2222

1

2iefiefie iaii

|||||||||||| uww

)]]([)([2]~[=]~||[ 212 iefieEEE aii

|||||| ww

)]]([||[ 222

iefE i ||u


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Transient Analysis (Cont)

To evaluate the transient analysis we make the followingassumptions

The noise sequence is iid and independence of input regressor.

For any constant matrix and for all i, and are jointly

Gaussian The adaptive filter is long enough such that the weighted norm of

input regressor and the square of error nonlinearity i.e. are

uncorrelated.

Employing the above assumption and assuming the sequence is

zero-mean iid, and has covariance matrix R, the weighted-energyrecursion relation is given as

)(iv

)(iea )(iea

)]([2

ief

iu

]~[2]~[=]~||[ 2

1

2

1

2

Rwww |||||||||| iGii EhEE Ui hE ]||[

22

||u


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Recursive Equations

The learning curve of the filters refers to the time-evolution of thevariances (excess-mean-square error) and (mean-

square deviation).

In case of white input regressor data, the covariance matrix , so

that

Now recursive equation for MSD in case of white input regressor data

and by taking is given as

Where the time evolution MSD nonlinear parameters are

given as

)]([ 2 ieEa

]~||[ 2||i

E w

IR 2=

u

]~||[=)]([ 21)(

22 ||iua

EieE w

UuuG hMihii 222 1)(21)(=)(

2221)(

)(1=

gsu

srG

i

ph

222 1)(

su

sr

i

p

]1)(

1)(sin)(1= 222

22

12

gsu

ug

srU

i

iph

]1)(

1)(sin 222

22

12

su

usr

i

ip

I=

]~||[=)( 2||iEi w


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Recursive Equations

In the same way we can obtain the recursion equation for EMSE bysimply choosing .

Let the time evolution of EMSE written as

The recursive equation is given as

Where the nonlinearity parameters are

R=

]|~|[=)( 2R

w ||i

Ei

UuuG hMihii 422 1)(21)(=)(

22

1)(

)(1=

gs

srG

i

ph

22

1)(

s

sr

i

p

]1)(

1)(sin)(1= 22

2

12

gs

g

srUi

iph

]1)(

1)(sin 22

2

12

s

sri

ip


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Results and Discussion

It is assumed that the input data is white and the desired data isgenerated according to the defined model.

The unknown vector is set as

The back ground noise is Gaussian with variance and the

impulsive noise is also Gaussian but with high variance .

The theoretical and simulation result of the saturation nonlinearityalgorithm in presence of impulsive noise with different percentage of

impulsive noise is demonstrated.

MT /,1][1,1,

w

2

g

2

w


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Theoretical (Black) and simulated (red) performance of mean-square

deviation (MSD) curve for (no impulsive noise) 0.1, 0.5, and 1.00.0=rp


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Theoretical (Black) and simulated (red) performance of excess-mean-square

deviation (EMSE) curve for (no impulsive noise) 0.1, 0.5, and 1.00.0=rp


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Theoretical (Black) and simulated (red) performance of mean-square

deviation (MSE) curve for (no impulsive noise) 0.1, 0.5, and 1.0

0.0=rp


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Conclusion

We have used energy-weighted conservation arguments to study the

performance of saturation nonlinearity LMS with impulsive noise.

The recursion equations for MSD and EMSE are derived in presence

of impulsive noise.

The simulated results have good agreement with theoretical counter

part.


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References

[1] Abu-Ella, O.A. and El-Jabu, B. Optimal Robust Adaptive LMS Algorithm withoutAdaptation Step-Size, Millimeter Waves, 2008. GSMM 2008. Global Symposium on,

:249-251, 2008.

[2] T. Y. Al-Naffouri and Ali H. Sayed. Adaptive Filters with error Nonlinearities: Mean-

Square analysis and Optimum design, EURASIP Journal on Applied Signal Processing,

192-205, 2001.

[3] T. Y. Al-Naffouri and Ali H. Sayed. Transient Analysis of adaptive filters with errornonlinearities,IEEE_J_SP, 51(3):653-663, 2003.

[4] T. Y. Al-Naffouri and Ali H. Sayed. Transient Analysis of data-Normalized adaptive

filters,IEEE_J_SP, 51(3):639-652, 2003.

[5] N. J. Bershad. On Error saturation nonlinearities for LMS adaptation, IEEE_J_ASSP,

36(4):440-452, 1988.

[6] N. J. Bershad. OnError saturation nonlinearities for LMS adaptation in Impulsive noise,IEEE_J_SP, 56(9):4526-4530, 2008.

[7] N. J. Bershad. On Weight update saturation nonlinearities in LMS adaptation,

IEEE_J_ASSP, 38(2):623-630, 1990.


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References

[8] S. C. Chan and Y. X. Zou, A recursive least M-estimate algorithm for robust adaptive

filtering in impulsive noise: Fast algorithm and convergence performance analysis,

IEEE_J_SP, 52(4):975-991, 2004.

[9] S. Haykin. Adaptive filter theory. Englewood Cliffs,NJ:Prentice-Hall, 2001.

[10] Pawula, R. A modified version of Price's theorem, Information Theory, IEEE

Transactions on, 13(2):285-288, 1967.[11] R. Price. Auseful theorem for non-linear devices having Gaussian inputs, IRE Trans.

Inf. Theory, IT-4:69-72, 1958.

[12] A. H. Sayed, Fundamentals of Adaptive Filtering, John Wiley and Sons. Inc.

Publication, 2003.

[13] X. Wang and H. V. Poor, Jointchannel estimation and symbol detection in Rayleigh flat-

fading channels with impulsive noise,IEEE_J_COML, 1(1):19-21, 1997.[14] B. Widrow and S. D. Strearns, Adaptive Signal Processing, Englewood Cliffs,

NJ:Prentice-Hall, 1985.


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