The document summarizes adaptive nonlinear filters, including adaptive polynomial filters using truncated Volterra series expansion and adaptive bilinear filters. It discusses the mathematical representations of Volterra series, graphical representations of adaptive Volterra filters, and adaptation algorithms like LMS and RLS. Simulation results demonstrating the learning curves of these algorithms are presented. Finally, applications of adaptive nonlinear filters are discussed in areas like system identification, interference cancellation, radar, and instrumentation.

1. A SURVEY OF
ADAPTIVE NONLINEAR
FILTERS
-By Sandip Joardar
Master of Electrical Engineering
Electrical Measurement and Instrumentation
Dept. of Electrical Engineering
Jadavpur University
SANDIP JOARDAR MEE
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2. CONTENTS
• INTRODUCTION
• ADAPTIVE NONLINEAR FILTERS USING
TRUNCATED VOLTERRA SERIES EXPANSION
• ADAPTIVE BILINEAR FILTERS
• SIMULATION RESULTS
• APPLICATIONS
• CONCLUSION
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3. INTRODUCTION
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4. INTRODUCTION
WHY ARE POLYNOMIAL BASED NONLINEAR
FILTERS REQUIRED ?
WHY DO THESE FILTERS REQUIRE AN
ADAPTATION ALGORITHM ?
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5. TYPES OF ADAPTIVE NONLINEAR
FILTERS
• Adaptive Polynomial Filters using Truncated
Volterra Series Expansion
• Adaptive Lattice Polynomial Filters
• Adaptive Bilinear Filters
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6. ADAPTIVE NONLINEAR FILTER
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7. ADAPTIVE NONLINEAR
FILTERS USING
TRUNCATED
VOLTERRA SERIES
EXPANSION
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8. Volterra Series
Mathematical Representation in the
continuous domain
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9. Volterra Series
Mathematical Representation in the discrete
domain
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10. GRAPHICAL REPRESENTATION
x(t) p1(t)
h(t)
p2(t)
(X)2
p3(t)
y(t)
p4(t)
(X)3
pn-1(t)
SUMMER
(X)4
(X)n-1
pn(t)
(X)n
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11. ADAPTIVE LMS VOLTERRA FILTER
ALGORITHM
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12. ADAPTIVE RLS VOLTERRA FILTER
ALGORITHM
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13. ADAPTIVE
BILINEAR
FILTERS
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14. INTRODUCTION
For a one – dimensional input – output case, its
relationship is given by the Bilinear polynomial
as follows.
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15. GRAPHICAL REPRESENTATION
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16. OPERATIONAL PRINCIPLE
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17. SIMULATION
RESULTS
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18. 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
input signal noise at the system input
3 0.5
0.4
2
0.3
1
0.2
0 0.1
Noise
Input
-1 0
-0.1
-2
-0.2
-3
-0.3
-4 -0.4
0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500
Time Instants Time Instants
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19. LMS ADAPTATION ALGORITHM FOR
SOV FILTER
Learning Curve for LMS Adaptation algorithm
50
40
30
MSE [dB]
20
10
0
-10
0 50 100 150 200 250 300 350 400 450 500
Number of iterations, k
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20. RLS ADAPTATION ALGORITHM FOR
SOV FILTER
Learning Curve for RLS adaptation algorithm
-8
-9
-10
-11
LSE [dB]
-12
-13
-14
-15
-16
0 50 100 150 200 250 300 350 400 450 500
Number of iterations, k
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21. RLS ADAPTATION ALGORITHM FOR
SECOND ORDER BILINEAR FILTER
Learning Curve for LSE
-7
-8
-9
-10
LSE [dB]
-11
-12
-13
-14
-15
0 50 100 150 200 250 300 350 400 450 500
Number of iterations, k
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22. APPLICATION
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23. CLASSES OF APPLICATION
• SYSTEM IDENTIFICATION
• INVERSE MODELLING
• NONLINEAR PREDICTION
• INTERFERENCE CANCELLATION
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24. AREAS OF APPLICATION
• RADAR
• SONAR
• SEISMOLOGY
• SYSTEM MODELLING
• INSTRUMENTATION AND CONTROL
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25. CONCLUSION
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26. CONCLUSION
provides much
more satisfactory result ( )
in when
used for kernel estimation of .
provide much
than
in non-stationary environment.
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27. 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.
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SOMNATH GARAI, CIEM

28. THANK YOU
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