# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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£i.i = 2.1, i?i,i = 2.0, ^i,i = 0.0, o\'2 = 1-9, ^i,: - 3.0, ^i,2 = 0.0, C2 = 1.4, and C3 = 0.9. The simulated (i^i(so)i2)299Qi — 23.74 with a standard deviation of 0.0039. The theoretical result is = 23,76. This correspond.! to a, deviation from -he th^orpr.iral. result, of

—0.09

190

8. Noise in non-linear systems: Examples and Conclusion

Q 1,1 = 01,2

Figure 8.4: Example 1: A numerical experiment to illustrate the properties of the modulated noise source. The full line gives the theoretical values for (j^i(s7)|2) versus the amplitudes £1,1 = gi,-» f°r the modulating sinusoidals predicted from Equation (8.30), and circles give the simulated numerical values. The simulated values are the average of the ia-st 100 iterations of a total of 30000 iterations (ensembles'). Ail the simulated values agree with the theoretical predictions with a deviation less than ±1.1)5 %■ The data for the experiment are: Л = 32, ui = 0.3, eri = 5.2. p = 7, г ~ 1.0, ьЛ л = 2.0, oi i = 0.0, ^i,2 = 'j-0, >^1,2 “ 0.U, O’o — 1-4, and Сз — 0.9.

8.1. Example 1

191

17

16

15

14

13

О 5000 10000 15000 20000 25000 30000

Number of iterations, Г

Figure 8.5: Example 1: A numerical experiment to illustrate the properties of the modulated noise source. The full line gives the simulated values Re[(ni(^0l )*KsC,3)> 2390 l] with pi = 4 and p? = 6 versus number of iterations, and the dotted line is the theoretical result predicted from Equations (8.'26'l and (8.31)- The data for the experiment are: Л — 3*2, /Ji = 0.8. <T\ — 5.2, — 4. p? — (>j r — 1 -0, £i,; = 2.1, — 2.0,

ld\ \ — 0.0. <)’ о ~ 1.0 ?л n = Я 0, -pi о — П о, 0^ — 1.4, and Сз O Q. Tlie simulated Re[{«i(C0 ”i(^))2950iJ — 14.37 with a standard deviation of O.OOlo. The theoretical result is Н,е[{Тг1(ц4) ггT(чб)/j — 14.‘26. inis corresponds to a deviation from i.he i.heor^tical result of 0.78 %.

192

8. Noise in non-linear systems: Examples and Conclusion

0 5000 10000 15000 20000 25000 30000

Number of iterations, Г

Figure 8.6: Example 1: A numerical experiment to illustrate the properties of the modulated noise source. The full line gives the simulated values lm[(ni(^Pl) nJ(£P3))r] with p\ = 4 and pi = 6 versus number of iterations, and the dotted line is the theoretical result predicted from Equations (8.26) and (8.31). The data for the experiment are: Л = 32, jj.\ = 0.8, Ci = 5.2, pi — 4. po == 6, т = 1.П. on i — 2.1, = 2.0; ¥1,1 = 0.0. = 1-9. ^1,2 — 3-0,

Lpi v = 0-0, C-2 — 1.4, and C3 = 0.9. The simulated value (so))29901] “ 0.00095

witn a standard deviation oi *).00036. The theoretical result is lm[(ni(^4j nTf^a))j = 0.

8.2. Example 2

193

8.2 Example 2

Problem : As a simple example of the theory, consider the non-linear time invariant noisy system in Figure 8.7. For this system the number of signal input ports is К = 1, the number of controlling variables is Q = 1. and the number of output ports is L = 1. The deterministic input signal is a single sinusoidal signal, and the single noise source is unmodulated. The objective of the example is to determine the cross-correlation (r„,i(£pi) where [h,[>2 £ Z are integers.

*Af)( j) МЯ(П Ri

П

'V

■:Q

l(v)

lif)

Figure 8.7: Example 2: Non-linear noisy Van der Pool system with one noise sources and one non-linear clement-.

THEORY: The system in Figure 8.7 is excited by the deterministic signal s\(f) given by Equation (7.8) as

•si (/) = У2*\(Ф1,л)Ч/-Фи1)

(8.33)

where фi i = i?i,i > 0 and ф 1,2 = — and

Ji(A,i) = ^ Pi.i exp[j

, . 1 г .

■Sl(-l>I,l) = - ?!,! exp[-j Jt,!)

ind thus J\ = 2. The time domain noise signal ni(t) is given by

(S.'i-i)

(8.35)

where W[(t) is a coloured noise source. Equations :v ( :1 [ i and ir.ob) lead to Л = 0 and

(GOiffio.i) = Ci (8.3V)

The non-linear element is a current/voltage (non-linear conductance) element given

194

8. Noise in nan-linear systems: Examples and Conclusion

i[v) = i{v(t).t) = у2 i'2(t) (8.38)

where д2,дз £ 72 are real constants. For the system in Figure 8.7 the maximum order considered is chosen to be M = 8. Note from Figure 8.7 that tile Volterra transfer functions relating inputs s\(f) and ni(/) to the output 7'i(/) of arbitrary orders are non-zero even though the non-linear element described by Equation (8.38) is of second order. The autocorrelation for the fundamental noise source iJi(£p) is

Дл.™

LiLL>OV,ll и

where KitK2 £ H+ are positive real constants. The frequency sets So, Si, S2, S3, 54, S5, 5e, and 57 can be determined as

-So = {«o,ib £0 = 1 (3.40)

= {0} (8.41)

Si = {«i.i, «1.2) , Ei = 2 (8.42)

- {iJi.b-^.i} (8.43)

Si = {*2,1,«2,2,«2,3} , £2=3 (8.44)

= {2г)1Д, —2^1,1, 0} (8.45)

53 = {«з,ъ «3,2, «з,з> «3,4} , -£з = 4 (8.46)

= {3'7i,i, —3i7i,i, 1.1} (8-47)

54 = ^4,1,«4,2,«4.3,®4,4,«4.5} , £4 = 5 (8.48)

~ {4г?1 д, -4^i,i, 21?i,i, -201Л, 0} (8.-19)

<Ss = {«5,1, *5.2, *5,3, «5,4. *5.5, Ф5,в} , = 6 (8.50)

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