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Theory of Linear and nonlinear circuits - Engberg J.

Engberg J. Theory of Linear and nonlinear circuits - Wiley & sons , 1995. - 154 p.
ISBN 0-47-94825
Download (direct link): noisetheoryoflinearandnonlinear1995.pdf
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One way to achieve the goal of a Fourier series representation without the problems of orthogonal Fourier series coefficients due to periodicity of the signals is described in the following. Assuming that the noise signal xj(t) has finite energy in the finite time interval —r < £ < r where г > 0 then fZT [х;(£)|2 dt < oc. In this case Xj(<) can be represented as a Fourier series in the time interval -r < t < r as
CO
xi(t) = *j(.PO exp[7'2;rp£t] (7.1)
p — — CO
where
Xj(pC) - £ I Xj(l) exp{-j2-p£t\dt (7.2)
In this case xAt) is generally non-zero for t > |rj as opposed to the first method in [3] where Xj(t) = 0 for t > |rj. The time interval [-r;r] is referred to as the observation time interval. In Equation (7.1) the quantity Xj(p^) is a complex valued random variable in the random process describing the statistical properties of t he
7.2. Preliminaries
149
noise signal xj{t). The (integral) Fourier transform of the noise signal Xj(t) in Equation (7.1) is given by
cj(/) = (7.4)
roc
= / Xj(t) exp{~j‘2-ft]dt (7.5)
J -'XI
— ^2 xj^pO (Ч/~ .°o (7-6)
p=-cо
oo
= £ *j(W«(/-W. *» = P? (7.7)
p—~oо
where .?"■{•} denotes the (integral) Fourier transform, and <5(-) is the Dirac 6-function
[4]. Note from Equation (7.6) that x}( f) is a two-sided Fourier transform and thus Xj(f) is represented at both positive and negative frequencies. It is very important to maintain both positive and negative frequencies since the mixing of various frequency components is essential to the non-linear noise analysis, and is only correctly analyzed when both positive and negative frequencies are included, [t is also seen from Equation (7.6) that the frequency resolution in the spectrum of x}(f) is given by This frequency resolution can be made arbitrarily small by choosing т sufficiently large.
To prove that Equation (7.1) is fulfilled in the time interval — r < t < r it suffices to show that {|е,(<) — Y.<ZL-ooxJ(p0 exp[j'2"p£i]|2) = 0. This relation may be
proved using standard techniques based on the assumption that Xj(t) is real [5]. It can also be shown that if the autocorrelation function (xj(ti) Xj(t-j)) is periodic with period 2r then the coefficients of the Fourier series expansion are orthogonal. However, in general this property is not valid since the autocorrelation function (ij(ti) x'(t2)) is generally not periodic.
Following the above discussion for both deterministic signals and noise leads to
Л
Ski!) = E ЫФк,1к) b(f - Фк,») (7.8)
jii=i
where the frequencies {Vfc.it ■ ■ •, ’I’k.j-.} are organized such that фкл ф ■ ■ ■ ф Vk.Jk for all к £ {1,2.....K\r and
where q G {l,2,...,Q}.
As an example of a deterministic time domain signal Sk{t) where к 6 {1,2,., A’}, consider a sum of a dc term and smusomai signals as
150
7. Noise in non-linear systems: Theory
■Sjt(i) = Sk.o + ^k•ь^ c,Js(2in3kibkt + <Pk,bk) (7-10)
bk~ 1
where 'Jk.bk £ 72+ for all к 6 {1,2,..., A'} and i/t t {1,2,..., 2?*}, and г^д / • • ■ ^ tffc.jB* f°r all к £ {1,2,..., 7i }. This leads to 5jt(/) = ■^'{■SfcCO} as
•Sfc(/) = i>-t,o6(0)
Rl
+ ^ S ^fe.|6*| «Ptf sgn{b*} *(/ - sgn{6*}
(7Л1)
2 H=-st
кфО
where sgn{-} is the sign function defined by
i 1 for b > 0
sgn{b} = J 0 for b = 0 (7.
( -1 for b < 0
11) can be written according to Equation (7. 8) with Jk = 2 Bk + 1 ;
Фк,2к-1 = $к,ьк , bk <= {1,2,... ,Bk} ("
Фкльк = ~’h.b.K , h e {1,2,. ...Bk} (7.
Фк,2Вк+\ - 0 (7-1й)
and
ЫФкМь-1) = ^ QkJ>k «4>[jV*aJ , bk 6 {1,2, ...,£*} (7.16)
«*(<&*,2»*) = ^ 8кАк «tPhiVM*) > bt 6 {1,2,..., 0*} (7-17)
Sk{fk,2Bk+l) = Sk& (7.IS)
Thus, Equation (7.8) can readily be used.
The notation used in Equations (7.7)-(7.9) implies that sums of pure exponential inputs may be allowed- This type of input may be of significant interest from a theoretical point of view since exponential inputs are frequently used \u Volterra series analysis.
7.3 Noise sources
In the analysis of non-linear noisy networks and systems it is sometimes necessary to take into account modulated (dependent) noise sources as well as unmodulated
7.3. Noise sources
151
(independent or fundamental) noise sources. That is, the noise generated in a given device or network element may depend on some controlling quantity (or quantities) in the system. For example, shot noise in semiconductor devices is dependent on a deterministic signal applied to it (mainly dc, but also the alternating signals may be of importance if they are not much smaller than the dc signals) |6]. This is because the shot noise is usually proportional to an instantaneous current -- this may be viewed as the dc operating point is changing with time due to the excitation, if the deterministic signal is sufficiently large, this causes a modulation of the noise source which m turn leads to correlated noise sidebands — this is even the case if the fundamental noise source (the noise source which is being modulated) generates white noise.
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