# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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The system description and mathematical representation of noise have been discussed. The non-linear noisy system under consideration must be transferred to an equivalent non-linear noise free system with external deterministic and noise generators (sources). This is required to apply the Volterra series technique. The deterministic and noise signals are represented mathematically as Fourier series which are computationally very efficient when Volterra series are used. The mathematical representation of noise sources has been treated in detail. A noise source is given as the response from a non-linear noise free multi-port system with a fundamental (unmodulated) noise source and possibly modulating signals as inputs. In this way it is possible to represent unmodulated as well as modulated (dependent) noise sources. The cross-correlation between Fourier series coefficients of any two (possibly) modulated noise source signals can be described by a cross-correlation matrix for the two fundamental noise sources at various frequencies, and vectors which describe the transfers from fundamental noise sources to modulated noise sources. This way to describe (possibly) modulated noise sources is very flexible, and it is easy to describe even very complicated multi-signal modulations.

Expressions for the noise and deterministic signal response at arbitrary response ports have been determined. The cross-correlation between two Fourier series coefficients at arbitrary response ports has been determined from noise cross-correlation matrices for the fundamental noise sources, and vectors describing the transfer from fundamental to modulated noise sources and from modulated noise sources to the noise responses at the response ports. Also, it has been shown how these rather complicated transfers must be specified for the Volterra series technique.

Expressions have been derived for various average noise powers and noise power densities. The quantities are usually of high interest in the analysis of r.oise. The average noise powers and noise power densities have been determined from noise cross-correlation matrices for the fundamental noise sources, and from the nonlinear transfer of the fundamental noise sources to the given response port.

Three examples have been shown to illustrate the use of the presented method. One example shows the representation and the properties of a type of modulated noise source. Two examples show how to determine noise cross-correlations at a response port, for two type of circuits. All examples have been chosen to make

ii possible I о find an alternative i analytical) solution to the given examples. The results for the presented method and from the alternative (analytical) solutions are in agreement, which indicates the correctness of the presented method. Generally it is not possible to find alternative analytical solutions, and all three examples have been carefully chosen to make comparisons nossible.

9

Multi-port Volterra transfer functions

This final chapter deals with the determination of frequency domain Volterra transfer functions of non-linear multi-port networks containing non-linear multi-port elements (subsystems). A pure frequency domain method is derived which allows commensurate as well as incommensurate frequencies. The method is based on an extension of the probing method to allow multi-port networks and commensurate frequencies. A computer implementation of the method in an algebraic programming language is made which allows determination of Volterra transfer functions in algebraic form up to eighth order on a low end workstation. Examples are presented and the Tesults are compared with existing literature in the special cases where comparison is possible.

9.1 Introduction

In this chapter a method and algorithm are developed to determine multi-port Volterra transfer functions of non-iincar multi-port networks which may contain nonlinear multi-port elements (subsystems) — that is. non-linear elements (subsystems) which are controlled by arbitrary variables. The purpose of this is tw'ofold: (i) multi-port Volterra transfer functions are a fundamental requirement for the iow-and high-level noise analysis, and (ii) it has been pointed out in the literature that the use of Volterra series is limited to one-port non-linear elements which precludes, for example, the analysis of MESFET transistors which ought to contain (at least) two-dimensional non-linear elements [1,2]. The work of the present chapter is also of importance in the analysis of the convergence properties of Voiterra series since relatively high order (> 4) Volterra transfer functions may be determined. In the literature, much material is available on the one-port Volterra series representation, but little on the general multi-port Volterra series representation.

Previously, Bussgang, iihrman and Graham [3], Maas [4,5] and Chua and Mg

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9. Multi-port Volterra transfer functions

[6] have investigated the determination of one-port Volterra transfer functions using the method of non-linear currents by a combination of time and frequency domain analysis. These analysis methods have been limited to one-port networks containing one-port non-linear elements and generators controlled by one variable. The method derived in the present work is based on an extension of the probing method to allow arbitrary (also commensurate) frequencies.1 A recursively based algorithm for the determination of Volterra transfer functions of non-linear multi-port networks is derived. A computer implementation of the method in a symbolic programming language is presented. This makes it possible to determine algebraic expressions of the multi-port Volterra transfer functions. Finally, three examples are considered.

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