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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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(iteration number) 7 at frequencies £•,, and <fp,. Equation (S.l) can be used to illustrate the convergence of the result as more and more ilei.iiions are processed. If the result of a simulation is to be given as a single number it is most convenient to take the average over a number of iterations. This can be do.ie as
181
182
S. Noise in non-linear systems: Examples and Conclusion
) n\iip2 ))ri ~ n p , i 22 (ni(£pi) ”i(£p2))r (S-2)
i2 - Lt+ 1Г=Г]
where the average is made over the simulated results for the iteration number Г 6 {14,14 + 1,..., Г2}.
8.1 Jiixample i
PROBLEM: First an example of a modulated noise source is considered. The nonlinear noisy system under consideration is shown in Figure 8.1. For this system the number of signal input ports К = 1 and the number of controlling variables Q = 1 (and the number of output ports is L = 1 though this is not of interest in this example, since only the noise source is investigated). The modulated noise source current iii depends on the controlling current и 1д. The objective of the example is to determine (ni(^?1) KJ(fP2)) where pi,p-> 6 Z are integers, and to compare the theoretical results with numerical experiments. The maximum order is chosen to be M = 3.
Theory: The system in Figure 8.1 is excited by the deterministic signal s-i(f) given by Equation (7.8) as
4
M/) = (8-3)
ji=i
where ^1,1 = A.i > 0, ^1,2 = — ^1.1, Фг,з = $1,2 > 0, and Ф 1.4 = -^1,2 with *4,1 Ф $1,2 and
5i(^i,i) = I «1,1 exp[i v=i,i] (8.4)
•si(-^i,i) = ^ £>1,1 expf-j^i] (8.5)
1 г ■ ,
.«1(!>1,2) = - 01,2 exp[? ^>1,2] (8.6,1
1
•5l(-Vl,-2) = ^ <?!,2 expi-j^i.ii (8.7)
and thus = 4. The time domain noise signal щ[1) is given by
ni(t) — -г ^з и1л{1.) Ш1\1) [Ъ-Ъ)
where the fundamental (unmodulated) noise source ioi(t) is a white Gaussian noise source with mean p\ and standard deviation <7!. Equations (7.24) and (S.8) lead to /1 = 1 and
8.1. Example 1
183
(b)
Figure 8.1: Example i: Non-linear noisy system with a modulated noise source, (a) Basic system where the noise source ni is modulated by the controlling variable u\ (b) Equivalent representation of the modulated noise rcur^e in the same form as in Fiffim» 7 '■vhere 7/’» is the fundamental funmodulaled) noise source
184
8. Noise in non-linear systems: Examples and Conclusion
{Ci for mi — 1
C3 for ГЩ = 2 (8.9)
0 otherwise
The frequency sets So, Si and J>-> can be determined as
S0 = E0 = l (8.10)
= {0} (8.11)
5! = {Ф1,1,Ф1,2,Ф1,з.Ф1,4}> Ei = 4 (8.12)
= R.i,-tfi.i, ^1,2, (8.13)
S-2 = {«2.1, «2.2, «2.3, «2,4, *2,5, «2,6, «2.7, «2,S. «2,9} . E2 = 9
(8.14)
= {2^1,1, -2i?i,i,20]>2,-2l)1i2,0, i?u + tfli2, t>i,i - ^1,2,
>\i + tfi.2, -^1,1 - ^1,2} (8.15)
and thus E = 13, and
{«1, . . ., Ф13} = #1,2, — #1,2» 2£>i,i, -2tf |,i, 2^1,2, -2l?i,2,
^1.1 + $1,2, 01,1 _ 01,2,—#1,1 + ^1,2> —T^l.l — #1,2,0}
(8.16)
To determine the noise conversion vector ti(£p) it is observed that '(i,i(/) = si(/) and thus
»ltl(/) = %.l(tl.l)f(/- tl,l) + l«l.l(«l,*)*(/- *u)
+ 1«,л(«1.зЖ/- »1,з) + 1»1д(*м)^(/-«1,4) (8-17)
•tii.i(’P|,i) = -n^i.i) (8.18)
l’ii i(«i 2) - 1) (8.19)
'Si.ii ^1.3) = «i(t>i,2) (8.20)
1“l,l(«l,4) = (8-21)
using
8.1. Example 1
185
. [ 1 for Cli=l
= 1 n ■ (8.22)
[ U otherwise From Equations (7.57)-(7.59) and f.S.S) it is seen that
"i(fp) = (sp)*!^) (8.23)
Сз S’l('L?i.i)3’i( l?l,l), СзЗ\(- 1?1,1 ) 51 ( — 1?!^ ),
Сз Jl ( l5l,2 ).sl(^1,2)7 C,3Si(-^i,2)«i( —^1,2). 2C33i(ii?i,i)3j(t>it2), 2Cs-S\(
2CVsi( — 1,1 )si( 1^1,2)- 2C’3Si(-i/ili)-s 1 (— !?u),
2C3S1 (i?i,i)ii(— ,1) + 26з-<1 (^1,2)si(— ;.*i,2 )j : S.‘24 >
[wi(s? - #i,i),m[ZP + 0i.i),»i(£„ - ^1.2). wi(iP + ^1,2): 5i(?P ~ '^1,1). “(чр + 2^1,1),
W\(tp ~ 2l?l,2), Wl(£P + 2^,2), 3>i(f„ - I?!.! - 1^1,2),
®i(fp ~ $1,1 + ^1,2), ®i(fP + I?i,i - 1^1,2):
Snip + Л.1 + ^1,2), iui(fp)] (8.25)
Thus the cross-correlation between two arbitrary frequency domain Fourier series coefficients ni(fpi) and n\(t,P2) where pi.p2 6 3 is given by
inl(Ap'l) ™i(sp: )) = ^[ ' Sii ) ?L’ 11 SDi ) w[1 so. i) ^i1 m;.j ' (8.26)
The result for (гц(!;Р1) n“(!;P2)} remains unchanged for any choice of maximum order as long as M > 3. The vectors ti(Jp) and usi(£p) increase in dimension with increasing AI > 3 but merely introduce new zero elements. The cross-correlation (’*i(£j>!) j*I(£pj)) where pi,p2 £ 2 for the modulated noise source is generally different from 0 if
If -f I с / 0 '■), 2j?i . 3iJ, 4’), , ,! !;>, , - 'Ы, ,!
‘>/•*2! ^ ^ ~ 1 1 ^ t ' L ,jL'. ! c . . I ' L , 2 ’ I'll i . J ! -
if^.l — 01 ■-> i, r/' i j. i/| i -f Vi i, д 4- 20\ ,2>
^i.i 4- 3wI(2, \2vlti - 2i>1|2|, i‘21/1,1 - ^l,2i’
2^i,i 4- ^i,2, 2^1д 4- 2^1^, |3^i,i — '$1,2!;
+ J,.4Tj, , I (S.271
where
ti«p) =
Wl(tp) =
186
8. Noise in non linear systems: £xamp/es and Conclusion
The cross-correlation (ni(£Pl )n'(cp3)) may be 0 even if |£pi - £pJ fulfils Equation
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