# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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Using stochastic processes an ensemble of realizations {i A(/q), ;A! Iq), . . ., vA(io)} with similar statistical properties are observed at the same instant of time f(J. The ensemble average of the random noise variable A!f0) is given as

where (■ - •) denotes the ensemble average over realizations with nimiiar otalistical properties, and p(At,,; is the probability density junction tor the random variable Л(iq). Another important quantity is the autocorrelation function defined as

L

A'o ^(-^h ) '^-4,

(A.2)

258

Л. Mathematical concepts

R(h,ti) = (A(i,)A-(/,)) (A.3)

= Jim -J7 У] n'\(ti) п-\'() (A.4!

_i_(

/OO f CO

J a,, x:/p(xh.x;2)d\t, d\;: (a.s)

where ’P(At , Af,) is the joint probability density function for the random variables Л(<j) and A(t-i)-

Sometimes, the time average of a random noise signal is used which is a somewhat different concept from the ensemble average. The time average of the noise signal nA(t) of a given realization n £ {1,2,. .., Лг} in the ensemble is defined as

,, A(() = lim — / ;lA \t)dl (A.(i)

r—*> 2r —

where the bar denotes the time average. An auto-correlation function is defined as

nA(f) „A(< 4- ~‘) = j nMt)n4t + r')dl (A.7)

As the random noise fluctuations are usually observed versus time, it may seem

closest to the physical reality to use time averages instead of ensemble averages.

However, ensemble averages are very useful from a theoretical point of view. This is because the (joint) probability density functions for the random variables in many cases can be deduced from theoretical considerations. Thereby it is possible to evaluate the statistical properties of the random variables without the need for time averaging. Another important factor is that the ensemble average usually equals the time average with regard to electrical networks. In case (A(fo)) = nA(0 the noise process is called ergodic, which means that the ensemble average is equal to the time average for any realization n £ {I, 2,.... Л'}.

The fact that a set of пишет variables are uncorreiated or independent is often used iu the analysis of noise. A set of variables {A.('), AjU), • ■ ■, A.\/(t)} are said to be uncorreiated if

(Ai(t) A2(t)”-AM(i)} = (A,(()) (A2(0) • • • (AmU)) (A.S)

and independent if

n, д \nit\ x,.(/г, ’Pi Ai?'') VP! Л Pi A \/( 0) A.O;

where P( - ■ ■) is the (joint) probabihty density function. This means that il a set of random variables are independent then they are aiso uncorreiated but not necessarily vice versa.

А.2. Fourier series representation

A.2 Fourier series representation

Generally the noise signals considered are extending over all time, -oc < t < со, and have infinite energies. Thus for a real valued random noise signal A(<) where —oc < i < '30 it is given that

Urn j \\{t)\2 dt = оо (АЛО)

This means that the noise signal A(tj is not square iulegrable and thus it does not generally have a Fourier transform.1 Assuming that the noise signal ,\(() has finite energy in the finite time interval -r < t < r where r > 0 then

j |А(0|2Л < 'OO (A.11)

In this case the signal A(t) where — r < t < г ran be represented as a Fourier series given by

-.Xi

A(t) = У] MpZ) exp[_/2-p£;] (A.12)

where

A(p£) = f I \(t) exp[-j'2~p£t} dt (A. 13)

f \ i A. 14)

In Equation (A.1‘2) the quantity A[p£) is a complex valued random variable in the random process describing the statistical properties of the random noise signal Л{(■).

In II] two suggestions for the frequency domain representation of random noise signals are given.

The first suggestion is to represent A(/) in a time interval —r < t < r and to assume A(£) = 0 for U| > r. The frequency domain representation in this case is the (integral) Fourier transform of A(/) given as A{/) — A(/) ехр[-/2тг//] dt.

The second suggestion is to assume that the random noise signal \[t) is periodic with period 2r such that A(t) = Л(i -j- 2nг) 'vhpro n is an in^g^r From this the frequency domain representation is gb'cn as a Fourier scries similar to Копал ions (A. 12), 1ЛЛЗ) and (A.14). However, -he assumption that Л(I) is periodic has tlu* unfortunate consequence ihat. (,1нл autocorrelation function. is also ocriodie surh that H[t\, -г) — i ~ 2 “Г 2 at) where n Is an integer. This also means that the coefficients

lIt .should be noted that it is a sufficient but actually not a necessary condition, for a signal to have a Fourier transform that it is square integrabie. For example the signal cos(2ttft) where

— :o < i < сю has a Fourier transform though ii is not square integrabie.

260

A. Mathematical concepts

of the Fourier series are orthogonal.2 Since R{ii,t2) 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) then it is not a problem that <2) is periodic because there is no need for any evaluation between Fourier series coefficients at different frequencies. However, as some ol the systems considered in the present book are non-linear (multi-response) the periodicity of A(t) can not be assumed since there may very well be a correlation between two Fourier series coefficients at different frequencies.

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