# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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[8.27) but then it depends on the conversion vector ti(£p) given by Equation (8.2 1). Note that (rci(fpi) nf(£P2)) may be different from 0 if pi ф p2. This is due to the modulation of the fundamental white noise source. When there are pi ф pn which fulfils Equation (8.27) then \W<x{£pi) iuj (f.J2)} is a non-zero matrix with non-zero oif-diagonal elements, e.g. for = 3 and 1912 = 7 it can be shown that

0 0 vf 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 () 0 0 0 0 0 4$ 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 Vo 0

0 0 0 0 0 0 Vo 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 Vo

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 Vo 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 Vo 0 0

(8.28)

where

2

Tlf = ~ + atf, a 6 {0,1} (8.29)

The superscript ct for rf* denotes that the coefficient is used for the continuous rime case.

Numerical Experiment: The frequency domain autocorrelation function for the Fourier series coefficient of the modulated noise source n\(£p) at an arbitrary frequency (p where p £ Z is given by Equation (7.66). To investigate the correctness of Equation (3.26) numerical simulations are performed according to the description in Section 7.3. In this case it can he shown that the autocorrelation is given by

с ■)! 2 \ _ j ° L <iii/ ' T 1 ’’ iL\ iSi vsp, Ф 13 у г ” ЮГ P = 1J /ч .,n,

м''ПчМ ' \ rr'r !jt;(fp)!!J/(2A + 1} for p/о

The following numerical simulations have been performed to illustrate the properties of the modulated noise source:

8.1. Example 1

187

Simulation 1 The autocorrelation (jni(f,,)|2)r with p = 17 is simulated versus the number of iterations Г. The data and result oi the simulation are shown in Figure 8.2. For this situation only the standard deviation and not the mean is of importance as predicted from the theoretical result in Equation (8-30). The simulated result = L5.-48 agrees with the theoretical result

(I^i(fi7)j2) = 15.42 with a deviation of 0.37 %.

Simulation 2 The autocorrelation {|Hx)j2)p with p = 0 is simulated versus the number of iterations Г. The data and result of the simulation are shown in Figure 8.3. For this situation both the standard deviation and the mean are of importance as predicted from the theoretical result in Equation (8.30). The simulated result {]«!(<f0)!2)'2аэо? = 23.74 agrees with the theoretical result (|Hi(£o)l2) = 23.76 with a deviation of -0.09 %.

Simulation 3 The autocorrelation with p = 7 is simulated versus

the amplitudes ju = oi 2 of the input sinusoidais. The data and result of the simulation are shown in Figure 8.1. All simulated valves (|Hi(£r)l2)?99oi agree with the theoretical result (|Si(fr)J2) in Equation (8.30) with a deviation less than ±1.25 %. As seen from Figure 8.4 the noise level (|«i!^7)|2) increases with the amplitudes 0\ \ — o\ 2 the deterministic excitation signals.

Simulation 4 The cross-correlation ^(spj ))r with pi = 4 and po = 6 for

r = 1.0, = 2.0 and 2 = 3-0 is simulated versus the number of iterations

Г (thus, (pi = f-4 = $i,i and £P2 = = 0i,2)- The data and results of the

simulations for Re[(ni(£p,) Hj(f?2)}r] and 1т[(Й1(£р,) пК£п))rj are shown in Figures 8.5 and 8.6. The theoretical result for this situation can be predicted from Equation (8.26) observing that

) 0 m 0 nf 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 rl t 0 0 0

0 0 0 0 Vo 0 0 0 0 0 0 0 0

о Чо о о 0 1) о n о о 0 n n

0 0 0 0 0 0 0 0 0 0 0 0

U и и По 1} 0 0 (J 0 0 0 0 0

0 0 0 0 0 0 0 0 Q 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 Ho 0

0 0 0 0 0 0 'lo 0 0 0 0 0 0

S! f! n 0 n 0 n f! n n n 0 'to

0 0 0 и 0 0 0 i) 0 0 0 0

0 0 0 0 0 По 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 n§‘ 0 0

188

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

where

9

^ (S.32)

The superscript dt lor 7?“‘ denotes that the coefficient is used for the discrete time case. The simulated value Re[(ni(£,i) — 1-4.37 agrees with

the theoretical result Re[{«lff-i) ^ifse))] = 14.26 with a deviation of 0.78 %. The simulated value Im[(ni(fpi) ^(чрг)) 29901] = 0.00095 and the theoretical result is 1т[(Й1(£Р1)г^(£и)>] = 0.

Number of iterations, Г

Figure 8.2: Exarnnle 1* A numerical experiment to illustrate th<=* properties of the mod-ulaced noise source. The full line gives the simulated values {j«if^)i“)r versus nurnber of iterations, and the dotted line is the theoretical result (J гг i ) S “) predicted from Equation Г8.30). The data for the experiment; any Л — 32, jji — 0.8. — 5.2, p = 17. r = 1.0,

ipi л = 2.1, 1/1 1 = 2.0, yi 1 = 0.0. i/1.2 — 1-^. 2 — yi .2 — O.U, 0'2 — i.4. anu C';j — U.y.

The simulated (lnif£-)l2)29g£9 — lh.48 with a standard deviation of 0 0015. The theoretical result is (|rti(si7)|2) = 15.42. This corresponds to a deviation of the simulated result from the theoretical result of 0.37 %.

8.1. Example 1

189

Number of iterations, Г

Figure 8.3: Example 1: A numerical experiment to illustrate the properties of the modulated noise source. The full line gives the simulated values (|Н1(^р)|2)р versus number of iterations, and the dotted line is the theoretical result (|fii(sp)|2) predicted from Equation (3.30j. The data tor the experiment are: A = 32r = 0.8, <ti — 5.2, p = 0, r = 1.0,

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