# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

**Download**(direct link)

**:**

**54**> 55 56 57 58 59 60 .. 85 >> Next

4- f'l ■ I ) rr„ f 7.108 '

where l\,h 6 {L,2,...,Z} and pi,p2 6 Z. In many cases the ensemble average of a Fourier series coefficient is zero in which case the latter two terms in Equation (7.108) vanish. The expression for (г„ ^(^р1) г" :J^P2)) is given by Equation (7.105).

ITS

7. Noise in non-linear systems: Theory

It is recalled that (rni/,(fPl) г*^(£и)) where {1,2,..., £} and pbp2 £ - is a two-siued Fourier series coefficient cross-correlation. Thus, if ior example the average signal power is determined for the one-sided case with positive frequencies,

i.e. p\,p-2 £ 2o+, then a factor 2 must be multiplied on the right-hand side of Equation (7.108) if the time domain response signals are real.

7.4.4 Some special cases

Consider the following situations:

• If (i) all the noise sources ..., ng(/)} are unmodulated, and (ii) the

system is linear, then Equation (7.104) leads to

Q Q

(rn.idtv,)K,i2(ip.)) = £!>• *.(*„. K.(U

71 = 1 72 = 1

x(w„(p,) «»(?«)) Th,ntfn) (7Л09)

Q Q

= У , У ! (-^i'i )mi =0,...,771д —0,0.0,o71 = 1,0,...,0(£pi )

(ji — 1 q2 = l

x(G,nmP1) <ги(еР1)ад2)) (G«)i(seM)

)mi =0,...,mK=0,0..0,0,2 =1,0.0^P2 ) (7.110)

In this special case the analysis is very closely related to a nodal noise analysis of a linear network.

• If (i) all the noise sources {ггi(/),. . ., TiQ(f)} are unmodulated, (ii) the

system is linear, and (iii) the fundamental (unmodulated) noise sources {№;(/), . ,.,wQ(J)}

are all uncorrelated, then

3 I 12

(!^*n,tf(sp)!^ — У ] j ( Hi )mi =0,....7/1 r,- =0.0.....0,о,г = 1,0.. ...о(чр ) j

xj(G,)i(sp)|2 (|E-,(Wi'J> (7.111)

7.5. References

1.79

7.5 References

[1] Chua, L. O. & Tang, Y.-S.: “Nonlinear oscillation via Volterra series”. IEEE Trans. Circuits arid Systems, vol. 29, no. 3, pp. 150-168. 1982

[2j Hu, Y., Obregon, J. J. & Molher, J.-C.: "Nonlinear analysis of microwave FET oscillators using Volterra series’*, IEEE Trans. Microwave Theory and Techniques, vol. 37. no. 11, pp. 1689-1693, 1989.

[3] Haus, H. A. (Chairman of IRE Subcommittee 7.9 on Noise) et al.: “Representation of

^ r-. D_.. г D Г _} IO____i _____ r>t\ — I 1 .л <-• л

uuioi. in Ш11.Ш bHupwiia ] i <(/«.. i JLij, vui. -to, UU. 1, pp. uy-1 ;1, lyUU.

[4] Lighthill, M. J.: “Introduction to Fourier analysis and generalized functions". Cambridge University Press, 1953.

[5] Papoulis, A.: “Probability, random variables, and stochastic processes’’, McGraw-Hill,

1965.

[6] Held, D. N. & Kerr, A. R.: ‘'Conversion loss and noise of microwave and millimeter-wave mixers: part I - theory”, IEEE Trans. Microwave Theory and Techniques, vol. 26. no. 2, pp. 49-55, 197S.

[7] Larsen, Т.: ‘‘Determination of Volterra transfer functions of non-linear multi-port networks”, Ini. J. Circuit Theory and Applications, vol. 21, no. 2, pp. 107-131, 1993.

[8] Chua, L. O. k. Ng, C.-Y.: “Frequency domain analysis of nonlinear systems: formulation of transfer functions”, IEE Journal on Electronic Circuits and Systems, vol. 3, no.

6, pp. 257-269, 1979.

[9] Chua, L. O. h Ng, C.-Y.: “Frequency domain analysis of nonlinear systems: general theory”, IEE Journal on Electronic Circuits and Sysiem.s, vol. 3, no. 4, pp. 165-185, 1979.

[10] Maas, S. A.: t:A generai-purpose computer program for the Volterra-series analysis of nonlinear microwave circuits”, IEEE Microwave Theory and Techniques Symposium Digest, pp. 311-314, 1988.

[11] Steer, М. B., Chang, C.-R. Rhyne, G. W.: “Computer-aided analysis of nonlinear microwave circuits using frequency-domain nonlinear analysis techniques: the state of the art”, Int. I. Microwave and Millimeter-wave Computer-aided Engineering, vol. 1, no. 2, pp. 181-200, 1991.

8

Noise in non-linear systems: Examples and Conclusion

This chapter contains three examples for the analysis of low-level noise in non-linear networks and systems. The first example illustrates the properties of a modulated

noise source. The second and third examples illustrate the analysis of noise in two types of networks. The examples have been constructed to facilitate comparative numerical simulations of the networks. The numerical simulations turn out to agree very well with the predicted theoretical results.

Generally, the simulations are performed as follows. First a number of 2Л + 1 fundamental noise samples, which are in accordance with the specified statistical properties, are generated for each noise source by a pseudo random number generator. Then the modulated time domain noise samples are determined from the modulating function, the fundamental noise samples are processed by the network equation(s), and next these time domain samples are Fourier series transformed. The Fourier series coefficients for the frequency points of interest are saved. Then new noise samples are generated for the next: iteration and so forth. Thus, the number of iterations is the same as the number of ensembles in the stochastic process for the noise. Denoting the number of iterations by Г, the value for where rj and zj are stochastic variables and p\,pi 6 { —Л,. . ., - I, 0, 1,..., A} is given by

**54**> 55 56 57 58 59 60 .. 85 >> Next