# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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7.1 Introduction

For non-linear systems the principle of superposition is not valid. This, as a consequence, implies that the (non-Linear) transfer function seen from a given noise source in a system to a given output response port does not only depend on the system itself but also on the applied deterministic signals. This makes non-linear noise analysis much more complicated than noise analysis of linear systems. The purpose of the present chapter is to develop a method, to analyze low level noise in general time invariant non-linear multi-port: non-autonomous systems — general in i.he sense that the equivalent, noise free system may be described by a finite (convergent) multi-port Volterra series. This suggests tliat tile type of system ai* low-- (I) multipl*! input ports which can be exciro.d hy hot ti deterministic л-оцаль

1 In an observation time interval of finite length.

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7. Noise in non-linear systems: Theory

(also dc) and noise, (ii) internal dc sources in the system, (iii) multiple unmodulated (independent) internal noise sources, and (iv) multiple modulated (dependent) internal noise sources which may be modulated by multiple arbitrary deterministic signals. The system may have an arbitrary number of responses (or more generally, arbitrary system variables) which may be, e.g., voltages or currents at any node or branch respectively in the underlying network. The low level noise assumption implies that only systems which are small signal linear may be analyzed — this means that contributions caused by non-linear mixing of noise with noise have to be insignificant. However, most of the known devices are small signal linear. Also dc sources in the system are allowed. Traditionally, dc analysis has not been used in relation to Volterra series analysis. This is because (i) it has not previously been possible to analyze multi-port systems, which is required when dc is applied not only to the signal input port, and (ii) Volterra series have traditionally only been used for weakly non-linear systems in which case the influence of the deterministic signals on the dc due to non-linear phenomena is insignificant.

The objective of the present chapter ts to determine the ensemble cross-correlation (or autocorrelation) between response Fourier series coefficients at two arbitrary frequencies at arbitrary response ports. This makes it possible to determine average noise powers and average noise power densities which are used extensively in noise analysis. Practical applications are expected to be in the analysis and optimization of noise in mixers with moderate local oscillator levels, interconnected mixers and oscillators, some types of frequency multipliers (e.g. FET types), oscillators (once the oscillation frequency and amplitude are determined, using e.g. [1,2]), and communications systems. The theoretical work may also be used in a combined analysis of intermodulation and noise, and to analyze the noise properties when more complicated deterministic excitations (multiple sinusoidal excitations) are used. Another possible application is in the development of non-Linear models of various devices.

The chapter is organized as follows. Section 7.2 presents preliminaries regarding the type of system which is under consideration and discusses the mathematical representation of deterministic signals and noise in the frequency domain. Section 7.3 outlines a method which can be used to represent modulated (dependent) as well as unmodulated (independent) noise sources. Lastly ш Section i .'1 expressions are derived for the responses of a non-linear noisy multi-port system.

7.2 Preliminaries

Two major prerequisites for the analysis of noise in non-linear systems are to consider (i) the system description, and (ii) the mathematical representation of noise and deterministic signals. These two problems are treated in the present section.

7.2. Preliminaries

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7.2.1 System description

The type of system under consideration is shown in Figure 7.1. This is a time invariant non-linear non-autonomons multi-port system with multiple internal unmodulated and modulated noise sources, and the system may also contain dc sources. For the analysis it is required that an equivalent circuit description of the nonlinear noisy system is available. That is, a detailed non-linear network description of the system must be available. The system is excited by ./ input signals ..., xj(f)} applied at ports {(i, 1),.... (x, J)} — these signals may all include deterministic signals (also dc) and noise. The system has L responses { ri (/)>•■• > tl{/)} which are present at ports {( r, 1),.... (r, L)}. The response n(f) at port (r, /) where I £ {1. 2,.... L} may be any open circuit voltage or short circuit current (or, more generally, an arbitrary system variable) at any node or branch respectively in the underlying non-linear network of the system. Using this system formulation it is possible to determine the uoise response, as well as the deterministic signal response (including dc), at any place in the non-linear system. The equivalent noise free non-linear system may contain one- and multi-port non-linear elements and subsystems — e.g. non-linear capacitors, and current generators with a nonlinear dependence of two (or more) controlling variables. The primary objective of the present chapter is to determine expressions for the responses {rj(/),.. ., rj,(/)} and their statistical properties, and to describe the information needed to determine the responses. The statistical properties include the determination of average noise power densities and average noise powers at the response ports as well as the cross-correlation (and autocorrelation) between Fourier series coefficients at arbitrary response ports.

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