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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 Previous << 1 .. 51 52 53 54 55 56 < 57 > 58 59 60 61 62 63 .. 85 >> Next ┬Żi.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 ŌĆö 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
ŌĆö0.09
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 ┬Ż1,1 = 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 %Ō¢Ā The data for the experiment are: ąø = 32, ui = 0.3, eri = 5.2. p = 7, ą│ ~ 1.0, čīąø ą╗ = 2.0, oi i = 0.0, ^i,2 = 'j-0, >^1,2 ŌĆ£ 0.U, OŌĆÖo ŌĆö 1-4, and ąĪąĘ ŌĆö 0.9.
8.1. Example 1
191
17
16
15
14
13
ą× 5000 10000 15000 20000 25000 30000
Number of iterations, ąō
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: ąø ŌĆö 3*2, /Ji = 0.8. <T\ ŌĆö 5.2, ŌĆö 4. p? ŌĆö (>j r ŌĆö 1 -0, ┬Żi,; = 2.1, ŌĆö 2.0,
ld\ \ ŌĆö 0.0. <)ŌĆÖ ąŠ ~ 1.0 ?ą╗ n = ą» 0, -pi ąŠ ŌĆö ą¤ ąŠ, 0^ ŌĆö 1.4, and ąĪąĘ O Q. Tlie simulated Re[{┬½i(C0 ŌĆØi(^))2950iJ ŌĆö 14.37 with a standard deviation of O.OOlo. The theoretical result is ąØ,ąĄ[{ąóą│1(čå4) ą│ą│T(čćą▒)/j ŌĆö 14.ŌĆś26. 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, ąō
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(┬ŻP3))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: ąø = 32, jj.\ = 0.8, Ci = 5.2, pi ŌĆö 4. po == 6, čé = 1.ą¤. on i ŌĆö 2.1, = 2.0; ┬ź1,1 = 0.0. = 1-9. ^1,2 ŌĆö 3-0,
Lpi v = 0-0, C-2 ŌĆö 1.4, and C3 = 0.9. The simulated value (so))29901] ŌĆ£ 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 ąÜ = 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 (rŌĆ×,i(┬Żpi) where [h,[>2 ┬Ż Z are integers.
*Af)( j) ą£ą»(ą¤ Ri
ą¤
'V
Ō¢Ā: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
ŌĆósi (/) = ąŻ2*\(ąż1,ą╗)ą¦/-ążąĖ1)
(8.33)
where čäi i = i?i,i > 0 and čä 1,2 = ŌĆö and
Ji(A,i) = ^ Pi.i exp[j
, . 1 ą│ .
Ō¢ĀSl(-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 ąø = 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) = čā2 i'2(t) (8.38)
where ą┤2,ą┤ąĘ ┬Ż 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(┬Żp) is
ąöą╗.Ōäó
LiLL>OV,ll ąĖ
where KitK2 ┬Ż H+ are positive real constants. The frequency sets So, Si, S2, S3, 54, S5, 5e, and 57 can be determined as
-So = {┬½o,ib ┬Ż0 = 1 (3.40)
= {0} (8.41)
Si = {┬½i.i, ┬½1.2) , Ei = 2 (8.42)
- {iJi.b-^.i} (8.43)
Si = {*2,1,┬½2,2,┬½2,3} , ┬Ż2=3 (8.44)
= {2ą│)1ąö, ŌĆö2^1,1, 0} (8.45)
53 = {┬½ąĘ,čŖ ┬½3,2, ┬½ąĘ,ąĘ> ┬½3,4} , -┬ŻąĘ = 4 (8.46)
= {3'7i,i, ŌĆö3i7i,i, 1.1} (8-47)
54 = ^4,1,┬½4,2,┬½4.3,┬«4,4,┬½4.5} , ┬Ż4 = 5 (8.48)
~ {4ą│?1 ą┤, -4^i,i, 21?i,i, -201ąø, 0} (8.-19)
<Ss = {┬½5,1, *5.2, *5,3, ┬½5,4. *5.5, ąż5,ą▓} , = 6 (8.50) Previous << 1 .. 51 52 53 54 55 56 < 57 > 58 59 60 61 62 63 .. 85 >> Next 