# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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166

T. Noise in non-linear systems: Theory

(7.52)

Л/-1

provided that m 1 4- ■■■ 4- mi £ {1.2......А/ - 1}. In Equation (7.51) the factor

implies that there are only (possibly) non-zero contributions to ra?(£p) for mi + '-'+mf, 6 {0,1,..., M — 1}. The contribution for mi + - • - + m/4 = 0 is nq(£p) =

(G,f)[,o..о(£p)t2rj(?p) which is the linear transfer from unmodulated to modulated

noise source (actually, this contribution is unmodulated since it does not depend oil the controlling u-signals). This contribution is included in the formulation of frequency sets by defining a frequency set So of order 0 as

which means that Eg = 1. Using the above and collecting terms in Equation (7.51)

where q £ {1,2,.. ., Q}, and (!i0($,it) is a o’th order transfer coefficient from unmodulated noise source wq(£P - Ф0,е) where о £ {0, !,..., А/ - I} - e 6 {1, 2.. .., E0] to modulated noise source Fourier series coefficient n7(fp). It is convenient to rewrite where q £ {1, 2,. .., Q} in Equation | 7.56) into

So = {Фод} = {0}

(7.54)

(7.55)

leads to

M-1 E„

^,о(чр: l^o.e) ^o,e)

(7.56)

o=0 r.= l

(7.57)

whore

Л/-1

{<Pi........®e} = U 5'

,>=o

(7.60)

7.3. Noise sources

167

Note from Equations (7.30), (7.31) and (t .60) that E £ {1, 2,. .., E0 + E\ + • • ■ + E.m-i }■ If the input frequencies {tq.b • ■ ■, Vi.Ji,....., ti’A'.t,..$'K,JK} are incommensurate up to order M - 1 then E = E0 + E\ + • ■ • т Ем-1-

Great care must be taken in determining tq(£v) where q (1,2from E-

qnations (7.56) and (7.57! when frequencies {li'i.b • • •, 5'i j,,......, Фк,\. ■ ■ ■, 'I’KJr }

are commensurate. This is because when the frequencies are commensurate a given tll0(E,,ft) where q £ {1,2,.... 0} and e £ {1,2,.... E} may e.g. consist of the sum ot' Lq.oi (?P, Фе) and tqfi2(£p, 4>e) where ouo2 £ {0,1,..., M - 1} and ox ф o2. This is not the case when the frequencies are incommensurate since in this case there is only one i4,„{£p, ®e) of interest where q £ {1,2,... ,0}. о 6 {0. I, ..., M — 1} and e £ {1, 2,..., j5}.

The cross-correlation between two arbitrary Fourier series coefficients of the two modulated noise sources nqi(f) and nq2(f) where q\,q2 £ {1,2.............0} is

*?,(£?,) (7.6i)

where (•) indicates the ensemble average over noise processes with identical statistical properties, [•]“ indicates the conjugate of a vector (or matrix), and [•]' indicates the conjugate transpose (Hermitian conjugate) of a vector (or matrix). Equation

(7.61) can be written as

<»9,(^)^(?w)) = fej fwH^) Г7.62)

where VF,b92(fPl ,fP2) is a noise cross-correlation matrix for the fundamental (unmodulated) noise sources wqi(f) and xl’52(/) where qi,q2 £ -{1.2.............Q], defined

as

= (*?i(fPr) *'2(?P,))

(и!.п(^л - Ф1) wq2(tP2 - Ф1)) ••• (*qAZP, - $1) *:2{ZPl

. - ф£) “i2(fp2 - 'Pt)> ■ ■ ' («МчР1 “ *c) - Ф e )) _

(7.6-11

iVot.e that the matrix W7. 5. (f„., f32) where £ {1,2, ...,Q} describes the

cross-correlation (and autocorrelation) between Fourier series coetncients in.. It]) and wq2(h) at all frequencies £pi - />. - /> £ {'s4- - - ■.. Ф r,}- Note also from

(7.03) - Фе)> '

^92.71 Up2 ' ) = ^ <7! ,q-2 (^Pl ' ^P2 ) (7.65)

This relation may be used to reduce some computations in the determination of the noise (cross-)correiation matrices for the fundamental noise sources.

168

7. Noise in non-linear systems: Theory

7.3.4 Fundamental noise sources

A typical type of fundamental time domain noise source is a Gaussian white noise process with specified mean and standard deviation. Traditionally, Gaussian white noise sources arc with mean values equal to zero, but in the present work the fundamental noise sources may have non-zero mean values. Due to this the noise is actually not “white” since not all frequencies have the same average power density. However, this type of noise will be referred to as white noise even though it has a non-zero mean. The fundamental time domain Gaussian white noise source with mean p.q and standard deviation <jq has a time domain cross-correlation given by

(wq{tl)wq(t2)) = crq 6(ti - t2) + /iq , ?ь<2е[-г;т] (7.66)

for noise source q £ {1,2,...,0} where S(-) is the Dirac delta-function. It can be shown that the corresponding cross-correlation between two arbitrary frequency domain Fourier series coefficients evaluated in the time interval [~r;rj is given by

sm[sr(pi - л)] , osin[7rpa] sin[^p2]

f ^/(2r) +/4 for Pi = P2 = 0

= | ^/(2r) for Pi = P2 Ф 0 (7.68)

[ 0 otherwise

where p\,pi 6 2. Note that there are only contributions to (Sq(iPl) ™q(ZP2)} when pj - p2. When pi — p2 ф 0 only the standard deviation is of importance, and when p1 — p2 = 0 both the standard deviation and the mean value are of importance.

In some cases it may be useful to make a comparative study by a discrete time

domain simulation. In this case the description in time is discretized such that

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