# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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Simulation 2 The autocorrelation for the output response (|Т7!(sp)(2)з§2о? with p = 5 is simulated versus the amplitude of the input sinusoidal for various maximum orders M 6 {2,4, 6, 8}. The data and results of the simulation are shown in Figure 8.9. As seen from Figure 8.9 the noise level increases with amplitude Oij. Quite as expected from the theory, the agreement between theory and experiments improves when the maximum order M for the theoretical prediction increases. When M = 8 the experimental results (!?1(£п)|2)чэ9о? and the theoretical results (S?i(£n)j2) agree with deviations less than ±0.7 % for 0!д 6 {0.0, 0.2, 0.4, . .., 1.6} as suggested from Figure 8.9.

Simulation 3 The autocorrelation for the output response (|ri(fp)|2)39901 is simulated versus the frequency point p £ { — A,..., -1,0,1,..., Л}. The data and results of the simulation are shown in Figure 8.10. All simulated values (Ir 1 (^p)12)з§§8? where |p| G {1,3,5,7,9,11,13,15} agree with the theoretical results (!?f,,i(fp)!2) with deviations less than ±1.6 %. To avoid making assumptions on the simulation it is only possible to make direct noise comparisons for |pj S {1,3,5,7,9,11.13,15} since the simulated result for \pj 6 {0,2,4,6,8,10,12,14} contains both a noise contribution and a deterministic contribution which can not be separated without making some assumptions.

8.3 Example 3

Problem: As an example of the theory consider the non-linear time invariant noisy system in Figure 8.11. For this system [\ — 1, Q = 2 and L = 1. The objective of ь ii о example is to determine the autocorrelation ' I ?r- ; [ c,, where p 6 2 is an

202

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

Frequency point, p

Figure 8.8: Example 2: A numerical experiment to illustrate the properties of the fundamental noise source. The full line gives the theoretical values for (|ич(£р)|“) versus the frequency point p predicted from Equation (8.39), and circles give the simulated numerical values. The simulated values are the average of the last 100 iterations of a total of 40000 iterations (ensembles). The simulated values agree with the theoretical predictions with deviations less than ±1.3 %. The data for the experiment are: A = 16. ex; = 1.0. к-i _ 5 x 10-'\ л.? = 15, and r = 1.0.

8.3. Example 3

203

Amplitude, оi,i

Figure 8.9: Example 2: A numerical experiment; to illustrate the properties of the fundamental noise source. The full and dotted lines give the theoretical values for (!(ч»)I2) with p = 5 versus the amplitude of the input signal for various choices of the maximum order Л/, and the circles give the results for the experiments. The simulated values are the average of the last 100 iterations of a total of 40000 iterations (ensembles). The simulated values agree with the theoretical predictions for А/ = 8 with a deviation less than +0 7 7o for amplitudes о^д £ {0.0, 0.2, 0.4, . .. , 1.6}. The data for the experiment are: Л = 10, ,гт — I u, (J\ — u.07. = 5 x 10“\ к-i — 15.0, r — l.u, 0\ i ~ 1.0, 91 д = 0.0, g-> = U.l. and

/4 - 10.

204

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

Frequency point, p

Figure 8.10: Example 2; A numerical experiment to illustrate the properties of the output noise frequency spectrum. The full and dotted lines give the theoretical values for (|rn.iKP)|2> with = 1.6 versus the frequency point p and for various maximum order-s M £ {1,4,8} predicted from Equation (8.39), and circles give the simulated numerical values. The simulated values are the average of the last 100 iterations of a total of 40000 iterations (ensembles). The simulated values agree with the theoretical predictions with deviations less than ±1.6 %. The data for the experiment are: Л = 16, &\ = 1.0, Ci = 0.07. «1 = 5 x 10“’\ «2 = 15.0, r = 1.0, 'Лд = 1.0, о?] д — 0.0, g2 ~ 0.1, and R{ — 1.0.

8.3. Example 3

integer, and to compare the result with a direct analytical time domain simulation.

■5i (/)

Figure 8.11: Example 3: Non-linear noisy system with two noise sources and one non-linear element.

THEORY: The system in Figure 3.11 is excited by the deterministic signal s\(f) given by Equation (T.S) as

si(f) = Л -3i(01,л) '5(/ - Vi.n)

л=1

where ipi:i = !?1д > 0 and V'i^ = — ’^1,1, and

(8.95)

■si(i?i,i) = 2 "i-1 exp[J V31,1]

3"i( —= £ "i,i exp[-J>i,i]

and thus Ji = '2. The time domain noise signal rai(i) is given by

nl(‘) = C\ 'Wl(t)

(8.96)

(8.97)

(8.98)

where wi(t) is a white noise source. Thus the noise source ni{t) is unmodulated. Equation (8.98) leads to Ix = 0 and

The tim* dc

bv

(Gi)iiSlo.i) - Ci

l) in Fie’,ire 8.! ! is

f'8.99'1

( Cl for

1 к

2 ЮГ TO] —

otherwise

(8.101)

206

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

Note that Equation (8.100) corresponds to the noise source n2(f) in a sense being

For the system in Figure 8.11 the maximum order considered is chosen to be M = 4. All Volterra transfer functions relating s\(/), n\(f) and njf/) to the response r^f) of orders higher than 3 are zero, even though, due to the modulation of noise source

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