The Volterra Series and Its Application

By Mark Dunn

Modeling of weakly nonlinear systems by means of Volterra series analysis is presented. Necessary conditions for representing nonlinearities by a Volterra series are developed analytically as well as heuristically. A two-condition convergence criterion for Volterra series and a method for determining Volterra transfer functions are established. For systems with multiple nodes, an extension of Volterra series analysis; method of nonlinear currents is developed and applied to a MESFET amplifier. Finally, methods of quantifying nonlinear behavior are discussed.

• Language: English
• Publisher: Lulu.com
• Release date: Feb 1, 2015
• ISBN: 9781312884083

Mark Dunn

Mark Dunn is a lifelong word lover and the author of Ella Minnow Pea (nominee for Book Sense Adult Fiction Book of the Year, and winner of the Borders Original Voices Awards for fiction), Welcome to Higby (short listed by Publishers Weekly as one of the best novels of the year), and Ibid: A Life. He makes his home in Albuquerque, New Mexico.

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The Volterra Series and Its Application

THE VOLTERRA SERIES AND ITS APPLICATION

BY

MARK ROBERT DUNN

B.E.E (Georgia Institute of Technology) 1984

THESIS

Submitted in partial satisfaction of the requirements for the degree of

MASTER OF SCIENCE

in

ELECTRICAL AND COMPUTER ENGINEERING

in the

GRADUATE DIVISION

of the

UNIVERSITY OF CALIFORNIA

DAVIS

Approved:

Committee in Charge

1992

Copyright © 2013 Mark Dunn

All rights reserved.  No part of this book may be reproduced in any form without the permission of the author.

ISBN: 978-1-312-88408-3

ABSTRACT

Modeling of weakly nonlinear systems by means of Volterra series analysis is presented.  Necessary conditions for representing nonlinearities by a Volterra series are developed analytically as well as heuristically.  A two-condition convergence criterion for Volterra series and a method for determining Volterra transfer functions are established.

For systems with multiple nodes, an extension of Volterra series analysis; method of nonlinear currents is developed and applied to a MESFET amplifier.  Finally, methods of quantifying nonlinear behavior are discussed.

ACKNOWLEDGMENT

I would like to thank Hewlett Packard for providing the financial support through its Fellowship program.  I am especially grateful to Kent Stalsberg.  I owe my acceptance in the Fellowship program to his perseverance, tenacity and ultimately to his belief in me.  I also thank Richard Spencer for his flexibility, patience and advice. 

Finally, and most importantly, I thank my wife Sue without whom none of this would have been possible.

Table of Contents

1.0 INTRODUCTION

1.1 DEFINITION OF NONLINEAR SYSTEMS

1.2 STRONGLY vs. WEAKLY NONLINEAR SYSTEMS

1.3 METHODS OF ANALYZING NONLINEAR SYSTEMS

1.3.1 PIECEWISE LINEAR

1.3.2 NONLINEAR DIFFERENTIAL EQUATIONS

1.3.3 HARMONIC BALANCE

1.3.4 VOLTERRA SERIES

2.0 MODELING SYSTEM NONLINEARITIES

2.1 INTRODUCTION

2.2 POLYNOMIAL REPRESENTATION

2.3 CURVE-FITTING TECHNIQUES

3.0 POWER SERIES ANALYSIS

3.1 INTRODUCTION

3.2 GENERAL APPROACH

3.3 SINUSOIDAL RESPONSE OF

MEMORYLESS NONLINEAR SYSTEMS

3.4 LIMITATIONS

4.0 VOLTERRA SERIES ANALYSIS

4.1 CONVERGENCE

4.2 DEVELOPMENT OF VOLTERRA SERIES

4.3 FIRST-ORDER VOLTERRA SYSTEMS

4.3.1 FIRST-ORDER IMPULSE RESPONSE

4.3.2 CAUSALITY

4.3.3 STABILITY

4.3.4 FIRST-ORDER VOLTERRA TRANSFER FUNCTION

4.3.5 SINUSOIDAL RESPONSE OF

FIRST-ORDER VOLTERRA SYSTEMS

4.4 SECOND-ORDER VOLTERRA SYSTEMS

4.4.1 SECOND-ORDER IMPULSE RESPONSE

4.4.2 KERNEL SYMMETRIZATION

4.4.3 CAUSALITY

4.4.4 STABILITY

4.4.5 SECOND-ORDER VOLTERRA

TRANSFER FUNCTION

4.4.6 SINUSOIDAL RESPONSE OF SECOND-ORDER

VOLTERRA SYSTEMS

4.5 nth-ORDER VOLTERRA SYSTEMS

4.5.1 nth-ORDER IMPULSE RESPONSE

4.5.2 KERNEL SYMMETRIZATION

4.5.3 CAUSALITY

4.5.4 STABILITY

4.5.5 nth-ORDER VOLTERRA TRANSFER FUNCTION

4.5.6 SINUSOIDAL RESPONSE OF nth-ORDER

VOLTERRA SYSTEMS

5.0 DETERMINATION OF VOLTERRA TRANSFER FUNCTIONS

5.1 INTRODUCTION

5.2 THEORETICAL DEVELOPMENT

5.3 APPLICATION

6.0 METHOD OF NONLINEAR CURRENTS

6.1 INTRODUCTION

6.2 THEORETICAL DEVELOPMENT

6.3 FREQUENCY-DOMAIN REPRESENTATION OF

NONLINEAR CURRENTS

6.4 APPLICATION TO NONLINEAR SYSTEM

7.0 NETWORK ANALYSIS

7.1 INTRODUCTION

7.2 METHODOLOGY

7.3 APPLICATION

7.4 CONGERGENCE REVISITED

8.0 QUANTIFICATION OF NONLINEAR BEHAVIOR

8.1 INTRODUCTION

8.2 HARMONIC DISTORTION

8.3 INTERMODULATION DISTORTION

8.4 GAIN COMPRESSION

9.0 CONCLUSION

REFERENCES

Table of Figures

Figure 1.1 Block Diagram of Typical Nonlinear System

Figure 2.1 Interpolation vs. Approximation

Figure 3.1 Power Series Analysis Model

Figure 3.2 Complete MESFET Model

Figure 3.3 Modified MESFET Model

Figure 4.1 The basis function

Figure 4.2 Input function & approximation

Figure 4.3 Second-order impulse response

Figure 4.4 Second-order transfer function

Figure 5.1 Nonlinear Circuit

Figure 6.1 First-order Model

Figure 6.2 Second-order Model

Figure 6.3 Third-order Model

Figure 6.4 Fourth-order Model

Figure 6.5 First-order Model

Figure 6.6 Second-order Model

Figure 6.7 Third-order Model

Figure 5.1 Nonlinear Circuit

Figure 7.1a Nonlinear Network

Figure 7.1b Conceptualization of Nonlinear Network with

M nonlinear elements pulled out

Figure 7.1c Associated linear network driven by

M nonlinear current sources

Figure 7.1d First-order Solution

Figure 7.1e Second-order Solution

Figure 7.1f nth-order Solution

Figure 7.2 Complete MESFET Model

Figure 7.3 Simplified MESFET Model

Figure 7.4 MESFET Model with Nonlinear Currents

Figure 7.5 MESFET Model with Second-order

                         Nonlinear Currents

Figure 8.1 Plot of 2nd- and 3rd-order Intercept Points.

Figure 8.2 Plot of Gain Compression.

1.0 INTRODUCTION

Many papers and books have been devoted to the analysis and synthesis of linear systems, in spite of the fact that no system is entirely linear.  All systems, whether electrical, mechanical, biological, or whatever, exhibit varying degrees of nonlinear behavior.

In medicine a patient's condition may improve with a single dose of medication, be cured with a double dose or made critical with a triple.  In electrical engineering one is constantly faced with nonlinearities due to the fact that the constitutive relationships of any electrical device are nonlinear if a large enough range is considered.  Diodes, for example, are nonlinear regardless of the input, whereas resistors will be nonlinear only at the extremes of their operating range, when the power dissipated causes thermal effects to vary their resistance.

Generally when an engineer is faced with a nonlinear problem one of two approaches is taken.  The first approach is to set up a measurement system that quantifies the problem (i.e. measures the power of an undesired harmonic) and then proceed to tweak the design until the undesired response is minimized.  This empirical approach sometimes