# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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where the contribution to the controlling variable due to the noise sources has been ignored (low level noise is assumed). Once the controlling variable u2ll(0 has been determined the response rгt) can be determined as

210

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

ri{t) — t^ifO ± ± ± ^i(0 + n2(t)\ R

+ 02 [si(t) + ni(t) + n2(t)]2 4- a3[s](t) + m(t) + п2(г)13

(8.146)

As a special case it is quite interesting to note that the autocorrelation in the situation where the system is linear— but including the (non-linear) modulation of noise source 2 — is given by

which can be derived by setting a2 = a-j = 0.

Simulation 1 The autocorrelation (j'rI,.il(£p)|2)f with p = 3 is simulated versus the number of iterations Г. The data and result of the simulation are shown in Figure 8.12. This simulation is for a linear system, i.e. a2 = a3 = 0, but where noise source 2 is being modulated by the applied deterministic signal. The simulated result = 1.7523 X 10_1° agrees with the theoretical

result (|гп,1(£з)|2) = 1.7596 X 10_1° with a deviation of -0.42 %.

Simulation 2 The autocorrelation {|, 1 (cfp)|2)здэо? with p = 3 is simulated versus tlie amplitude 0\t\ of the deterministic input sinusoidal signal. The data and result of the simulation are shown in Figure 8.13. All simulated values (|гад(£з)|2)зд9§° agree with the theoretical result (|rnil(f3)|2) in Equation

(8.144) with a deviation less than ±0.39 %. As seen from Figure 8.12 the noise level (|^i(£t)|2) increases with the amplitudes = 0-1 9 of the deterministic excitation signals. For very small only noise source 1 is of importance since it, as opposed to noise source 2, is independent of a modulating signal.

Simulation 3 The autocorrelation (|?л.1(£р)|2)з°яп? p = 3 is simulated versus the standard deviations 0\ — <j2. The data and result of the simulation are shown in Figure 8.14. All simulated values (|г*,|({з)|г)я99о? 'ri the range of ai = rr< from 10-9 to 10-2 agree with the theoretical result (|г,д((з)|2) in Equation (8.144) with deviations less than ±0.14 %. The simulation for н\ = n2 — 10-i deviates from the theoretical result by u.ii %. Tins rather large deviation is due to the fact that the mixing of noise with noise for increasing standard deviations is not considered in the theoretical prediction but is included in the numerical simulation.

(I^a(?p)|2) = (Rl + R)2C'i (ШЫ\3)

(8.147)

8.3. Example 3

■211

1.85

1.80

1.75 1.70 1.65 1.60 1.55

0 5000 10000 15000 20000 25000 30000 35000 40000

Number of iterations, Г

Figure 8.12: Example 3: A numerical experiment to illustrate the properties of noise response rn,i(£p). The full line gives the simulated values {\rnд(^)|2)г versus number of iterations, and the dotted line is the theoretical result (|гаД(^р))2) predicted from Equation

(8.144). The simulated (^Р)12)зэ§о? — 1-7523 x 10"iO with a standard deviation of 2.7810 x 10'19. The theoretical result is (!ni(£i7)P) = 1.7596 x 10-15. This corresponds to a deviation of the simulated result from the theoretical result of —0.42 %. The data for the experiment are: Л - 64, = 5.0 x 10-9. «т3 — 7.0 x if)-*, C\ ~ 0.3, C~ ~ 0.2, /?, 13.0,

R = 10.3. a2 = 0, a3 = 0; p = 3; r = 0 5, <?•.. = 0.2, - 4.0; and ^ , = 0.0.

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

Amplitude, Qi

Figure 8.13: Example 3: A numerical experiment ro illustrate the properties of noise response rnil(^). The full line gives the theoretical values for (!^n,iUp)|2) versus the amplitude of the input signal, and the circles give the simulated values {|гпд(чр)|2)зээо1 predicted from Equation (8.144). The simulated values agree with the theoretical predictions with a deviation less than ±0.39 % for the amplitudes seen from the figure. The data for the experiment are: Л — 64, = 5.4 x 10““, <t2 = 7.3 x 10~e. C\ = 0.3o, Co = 0/27, R{ = 13.1,

R — 10.3, 0.2 — 27.4, аз — 76.9, p ~ 3, r — 0.5, = 4.0, -md pi i — 0.0

8.3. Example 3

213

Standard deviation, — сг2

Figure 8.14: Example 3: A numerical experiment to illustrate the properties of noise response гпд(£Р)- The full lino gives the theoretical vaiues for (|гпд(£р)|2) versus the s-tandard deviations of the noise sources cri = rr2) and the circles give the simulated vaiues \|гпд(чр)|2)зэ9о? predicted from Equation (8.144). The simulated values agree with the theoretical predictions with deviations less than ±0.17 % for standard deviations up to 10~г. For <7\ = cr2 = 10”1 the deviation between simulation and theory is 6.0 %. The data for the experiment are: A = 64. C\ — 0.33. C-j. ~ 0.27. Ft-. ~ 13.1. Ft — 10.3. a 2 = 27 4, = 76-9,

p — 3, r = 0.5, t9i(i = 4.0, and = 0.0.

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

8.4 Conclusion

A method has been presented to analyze noise in non-autonomous non-linear multi-port networks and systems where low level noise can be assumed. This means that the systems must be small signal linear which is true lor many types of systems.

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