Abstract
In this paper we present a new approach for the transient noise simulation of electronic circuits with stochastic differential algebraic equations (SDAEs). The first part treats the modeling of noise in the time domain which is accomplished with generalized stochastic processes. This allows not only to model white noise like thermal and shot noise, but also 1/f-noise or flicker noise. It is shown that fractional Brownian motion reflects the properties of 1/f -noise, namely a spectrum proportional to 1/f with f denoting the frequency. Some consequences of this approach on the solvability of the circuit equations are presented. In the second part we give remarks on the implementation of numerical schemes for SDAEs. Besides the integration scheme itself the generation of appropriate random numbers is a major issue. Finally we present some numerical experiments.
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References
T.E. Duncan, Y.Z. Hu, B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion. I. Theory. SIAM J. Control Optim. 38 (2000), 582–612.
L. Decreusefond, A.S. Ustunel, Stochastic analysis of the fractional Brownian motion, Potential Anal. 10 (1999), 177–214.
A. Demir, A. Sangiovanni-Vincentelli, Analysis and Simulation of Noise in Nonlinear Electronic Circuits and Systems. Kluwer Academic Publishers, 1998.
P. Flandrin, On the spectrum of fractional Brownian motions. IEEE Trans. Inform. Theory 35 (1989), 197–199.
I. M. Gelfand, J. Wilenkin, Verallgemeinerte Funktionen. VEB Deutscher Verlag der Wissenschaften, 1964.
M. Günther, U. Feldmann, CAD-based electric-circuit modeling in industry I: Mathematical structure and index of network equations. Surv. Math. Ind. 8 (1999), 97–129.
D.E. Knuth, The Art of Computer Programming I. Addison-Wesley (1973).
B.B. Mandelbrot. J.W. van Ness, Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968), 422–437.
G.N. Milstein, M.V. Tretyakov, Mean-square numerical methods for stochastic differential equations with small noise. SIAM J. Sci. Comput. 18 (1997), 1067–1087.
Chr. Penski, Numerische Integration stochastischer differential-algebraischer Gleichungen in elektrischen Schaltungen. Shaker Verlag (2002) (Ph. D. thesis, Technische Universität München).
W. Römisch, R. Winkler, Stepsize control for mean-square numerical methods for SDEs with small noise. In preparation.
O. Schein, Stochastic differential algebraic equations in circuit simulation. Ph. D. thesis, Technische Universität Darmstadt (1999).
O. Schein, G. Denk, Numerical solution of stochastic differential-algebraic equations with applications to transient noise simulation of microelectronic circuits. J. Comput. Appl. Math. 100 (1998), 77–92.
L. Weill, W. Mathis, A thermodynamical approach to noise in nonlinear networks. Int. J. Circ. Theor. Appl. 26 (1998), 147–165.
R. Winkler, Stochastic DAEs in transient noise simulation. To appear in the Proceedings of Scientific Computing in Electrical Engineering, June, 23rd-28th, 2002, Eindhoven, Springer Series Mathematics in Industry. Proceedings of the SCEE (2002).
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Denk, G., Meintrup, D., Schäffler, S. (2003). Transient Noise Simulation: Modeling and Simulation of 1/f -Noise. In: Antreich, K., Bulirsch, R., Gilg, A., Rentrop, P. (eds) Modeling, Simulation, and Optimization of Integrated Circuits. ISNM International Series of Numerical Mathematics, vol 146. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8065-7_16
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DOI: https://doi.org/10.1007/978-3-0348-8065-7_16
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9426-5
Online ISBN: 978-3-0348-8065-7
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