paper presentation code no. 081-eei-14

A SURVEY OF

ADAPTIVE NONLINEAR

FILTERS

3/31/2013SANDIP JOARDAR MEE

JADAVPUR UNIVERSITY1

-By Sandip JoardarMaster of Electrical Engineering

Electrical Measurement and Instrumentation Dept. of Electrical Engineering

Jadavpur University

CONTENTS

• INTRODUCTION

• ADAPTIVE NONLINEAR FILTERS USING TRUNCATED VOLTERRA SERIES EXPANSION

• ADAPTIVE BILINEAR FILTERS

• SIMULATION RESULTS

• APPLICATIONS

• CONCLUSION



INTRODUCTION



INTRODUCTION

WHY ARE POLYNOMIAL BASED NONLINEAR FILTERS REQUIRED ?

WHY DO THESE FILTERS REQUIRE AN ADAPTATION ALGORITHM ?



TYPES OF ADAPTIVE NONLINEAR FILTERS

• Adaptive Polynomial Filters using TruncatedVolterra Series Expansion

• Adaptive Lattice Polynomial Filters

• Adaptive Bilinear Filters



ADAPTIVE NONLINEAR FILTER





ADAPTIVE NONLINEAR

FILTERS USING

TRUNCATED

VOLTERRA SERIES

EXPANSION

Volterra Series

Mathematical Representation in the continuous domain



Volterra Series

Mathematical Representation in the discrete domain



GRAPHICAL REPRESENTATION



SUMMER

h(t)

(X)2

(X)3

(X)4

(X)n-1

(X)n

x(t)

y(t)

p1(t)

p2(t)

p3(t)

p4(t)

pn-1

(t)

pn(t)

ADAPTIVE LMS VOLTERRA FILTER

ALGORITHM



ADAPTIVE RLS VOLTERRA FILTER

ALGORITHM





ADAPTIVE

BILINEAR

FILTERS

INTRODUCTION

For a one – dimensional input – output case, itsrelationship is given by the Bilinear polynomialas follows.



GRAPHICAL REPRESENTATION



OPERATIONAL PRINCIPLE





SIMULATION

RESULTS

INPUT AND OTHER PARAMETERS

Parameters:

• Iterations = 500

• Standard deviation of input = 1.01

• Standard deviation of measurement noise = 0.1240

• Length of the adaptive filter = 9



0 50 100 150 200 250 300 350 400 450 500-4

-3

-2

-1

0

1

2

3input signal

Time Instants

Inpu

t

0 50 100 150 200 250 300 350 400 450 500-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5noise at the system input

Time Instants

Nois

e

LMS ADAPTATION ALGORITHM FOR SOV FILTER



0 50 100 150 200 250 300 350 400 450 500-10

0

10

20

30

40

50Learning Curve for LMS Adaptation algorithm

Number of iterations, k

MS

E [d

B]

RLS ADAPTATION ALGORITHM FOR SOV FILTER



0 50 100 150 200 250 300 350 400 450 500-16

-15

-14

-13

-12

-11

-10

-9

-8Learning Curve for RLS adaptation algorithm


LSE

[dB

]

RLS ADAPTATION ALGORITHM FOR SECOND ORDER BILINEAR FILTER



0 50 100 150 200 250 300 350 400 450 500-15

-14

-13

-12

-11

-10

-9

-8

-7Learning Curve for LSE


LSE

[dB

]

APPLICATION



CLASSES OF APPLICATION

• SYSTEM IDENTIFICATION

• INVERSE MODELLING

• NONLINEAR PREDICTION

• INTERFERENCE CANCELLATION



AREAS OF APPLICATION

• RADAR

• SONAR

• SEISMOLOGY

• SYSTEM MODELLING

• INSTRUMENTATION AND CONTROL



CONCLUSION



CONCLUSION

provides muchmore satisfactory result ( )in whenused for kernel estimation of .

provide muchthan

in non-stationary environment.



REFERENCES

[1] Haykin S., “Adaptive Filter Theory”, Fourth Edition.

[2] V.J. Mathews, “Adaptive Polynomial Filters”, IEEE Signal Processing Magazine, July 1991, pp 10-25.

[3] Singh Th. Suka Deba, Chatterjee Amitava, “A comparative study of adaptation algorithms for nonlinear system identification based on second order Volterra and bilinear polynomial filters”, Elsevier Measurement, 2011.

[4] Koh. T. and E.J. Powers. “Second-order Volterra filtering and its application to nonlinear system identification.” IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-33, No. 6, pp 1445-1455, December 1985.

[5] Kenefic R. J., and Weiner D. D., “Application of the Volterra functional expansion in the detection of nonlinear functions of Gaussian processes,” IEEE Transactions on Communications. Vol. COM-31, No.3, pp 407-412, March 1983.

[6] Zhang H., “Volterra Series: Introduction and Application”, ECEN 665(ESS): RF communication Circuits and Systems.

[7] Abrudan T., “Volterra Series and Non – linear Adaptive Filters”, S-88.221 Postgraduate Seminar on Signal Processing 1, Espoo, 30.10.2003 – p. 1/23.

[8] Boyd S., Chua L.O., Desoer C.A., “Analytical Foundation of Volterra Series”, IMA Journal of Mathematical Control & Information (1984) I, 243 – 282.

[9] Niknejad Ali M., “EECS 242: Volterra/Wiener representation of Non-Linear Systems”, Advanced Communication Integrated Circuits, University of California, Berkeley.

[10] Moore J.B., “Global convergence of output error recursions in colored noise”, IEEE Trans, Automatic Control, Vol. AC-27, No. 6, pp. 1189 – 1199, December 1982.

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM27

THANK YOU



paper presentation code no. 081-eei-14

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