# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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The type of system in Figure 7.1 can not readily be analyzed since it contains internal noise and dc sources. The internal sources, as well as the noise sources applied at the input ports, may be applied at separate external ports provided that the internal topology of the underlying non-linear network is not changed. This is shown in Figure 7.2 where the (unmodulated as well as modulated) noise sources {nj(/),..., дд(/)} are applied at ports {(n,l)....,(n,Q)}. These noise sources also describe the noise sources at the input ports. Thus the signals {.s-i(/),.. ., sjy(/)} applied at ports {(5,1),. .., (s, К)} are purely deterministic signals which may also include dc.

The overall system is described up to some maximum order V/. This order ;s the highest order considered, including the possibly non-linear transfer from fundamental (unmodulated) noise source to modulated noise source. For example, ii' the maximum order of a transfer from unmodulated to inouuiaieq nois^ source is M and the system itself is linear, then the order for the analysis must be M to yieid correct results. In this example, of course, the Volterra transfer functions of orders greater than I for the linear system are zero. Only the Volterra transfer functions for the modulated noise source of orders greater than i are non-zero. Suppose that

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7. Noise in non-linear systems: Theory

Figure 7.1: Non-linear noisy time invariant multi-port system containing dc sources. Each input signal ij(f) where j G {1,2,...,./}, which is a voltage or current generator, may contain a deterministic (also dc) signal and noise. Each response r\(f) where I £ {1,2. ...,£} may be either a short circuit current or an open circuit voltage. The noise sources internally in the system may be either unmodulated (independent) or modulated (dependent). The system may also contain dc sources.

the non-linear system in itself is described up to some maximum order \[3, and the maximum order of the transfer from unmodulated to modulated noise source is Mn, then the overall maximum order M must be chosen as M -- тах{Я„ Mn} to yield correct results.

7.2,2 Representation of signals

Fourier series in Volterra series analysis are very attractive since brute force multidimensional integration is replaced by addition of multiplicative terms which in a sense is made in one dimension. This makes a Fourier series representation of the signals computationally very efficient. Also the use of Fourier series has advantages of a simple representation of sinusoidal signals, and in the determination of average noise power densities at specified frequencies. Generally the noise and deterministic signals considered are extending over aii time, -oo < t < oo, and have infinite energies. Thus for a real valued noisy signal x,(t) where -oo < t. < oo it is given

absolutely integra'bie and thus it does not generally have a Fourier transform.

In [3] two standard types of suggestions are given for the frequency domain representation of stationary (random) noise signals. The first suggestion is to represent x.-U"! in a time interval —г < t. < т and to assume = 0 for Ш > r. The

7.2. Preliminaries

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"l (Л nQ(J)

Figure 7.2: Non-linear noise free time invariant multi-port system not containing dc sources with externally applied sources. The signals {si(/), .. ., *‘A'(/)} represent the purely deterministic signals (also dc) applied to the system and internal dc sources. The noise signals {ni(/),...,nQ(/)} represent noise generated internally in the system and noise entering the system through the signal input ports. The noise sources may be either unmodulated or modulated.

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7. Noise in non-linear systems: Theory

frequency domain representation in this case is the (integral) Fourier transform of xj(t) given as Xj(f) = JZ *,(t) exp[-j'2- f t] at. The second suggestion is to assume that the noise signal Xj(t) is periodic with period 2r such that x}(t) = xj(t + 2ir) where i £ Z is an integer. From this, the frequency domain representation is given as a Fourier series. However, the assumption that xj(t) is periodic has the unfortunate consequence that the autocorrelation function is also periodic such that (xj(ti) xj(t2)) = (Xj(ti) x*(t2 + 2ir)) where i 6 2 is an integer. This also implies that the coefficients of the Fourier series are orthogonal — two Fourier series coefficients xnypif) and xj2(p2£) are said to be orthogonal if (Xj1(Pi£) £y2(p2<)} = 0 f°r all integers p1:p2 £ Z except forpi = p2 where (•) denotes the ensemble average over noise processes with identical statistical properties. Since (zj(ti) x*is generally not periodic for the type of signals considered in the present work this suggestion is not useful. If the systems under consideration are linear (single response), which is the case for the work in [3], then it is not a problem that (xj(t.;) jr*(f2)) is periodic, because there is no need for any evaluation between Fourier series coefficients at different frequencies. However, since the systems considered in the present work are non-linear, periodicity of Xj(t) can not be assumed since there may very '.veil be a correlation between Fourier series coefficients at different frequencies. This is, for example, the case for modulated noise sources — even in the case where the fundamental noise source in itself (without modulation) generates white noise (it is very simple to show special cases which illustrate this statement). Since it is most useful to have some kind of a Fourier series representation in the present work, none of the suggestions made in [3] are useful.

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