By Alper Demir

In digital circuit and method layout, the notice noise is used to consult any undesired excitation at the procedure. In different contexts, noise can be used to consult indications or excitations which show chaotic or random habit. The resource of noise may be both inner or exterior to the procedure. for example, the thermal and shot noise generated inside of built-in circuit units are in ternal noise resources, and the noise picked up from the surroundings via electromagnetic interference is an exterior one. Electromagnetic interference may also take place among diversified parts of an analogous process. In built-in circuits (Ies), indications in a single a part of the process can propagate to the opposite components of an analogous process via electromagnetic coupling, energy provide traces and the Ie substrate. for example, in a mixed-signal Ie, the switching job within the electronic elements of the circuit can adversely impact the functionality of the analog portion of the circuit by way of touring during the strength offer traces and the substrate. Prediction of the influence of those noise assets at the functionality of an digital method is termed noise research or noise simulation. a strategy for the noise research or simulation of an digital approach often has the subsequent 4 elements: 2 NOISE IN NONLINEAR digital CIRCUITS • Mathematical representations or versions for the noise resources. • Mathematical version or illustration for the procedure that's less than the in fluence of the noise sources.

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A stochastic process X is called mean-square differentiable at t if there exists a random variable X'(t) such that E [lk[X(t) - X(t - h)]- X'(t)n -+ 0 as h -+ O. 63) If X is mean-square differentiable for all t, then the process X'(t) = d/dtX(t) (assuming that it is a well-defined process in the underlying probability space) is called the derivative of X. 64) exists. 65) exists. 64). Hence, the Wiener process is not mean-square differentiable although it is meansquare continuous. 69) These can be obtained by interchanging the order of the operations of differentiation and expectation (which should be justified).

In 1951, K. 207) to for a broad class of so-called "non anticipating" functionals of the Wiener process W (t), and in doing so, put the theory of stochastic differential equations on a solid foundation. This theory has its peculiarities. 209) which can not be derived by formal calculation according to the classical rules of calculus. 205) is a Markov process with continuous sample paths, which is called a 52 NOISE IN NONLINEAR ELECTRONIC CIRCUITS diffusion process. In a loose sense, diffusion processes are "smooth" Markov processes.

E. X(f) = F {x(t)}, then ll(x)(t) = i: H(f, t)X(f) exp (j27rft)df. 112) 34 NOISE IN NONLINEAR ELECTRONIC CIRCUITS For a memoryless LTV system with h(t, u) = c(t)J(t-u), we have H(f, t) = c(t), independent of f. 113) for all t, u E IR, and for some period T > O. 114) n=-oo where fc = liT is the fundamental frequency, and the Fourier coefficients hn (r) are given by 1 jT/2 hn(r) = T h(t+r,t)exp(-j27rnfct)dt. 118) If the input to an LPTV system 11. 111). e. X(f) = F{x(t)}, then the output is given by ll(x)(t) f~co H(f, t)X(f) exp (j27rft)df + nfc) exp (j27rnfc t )X(f) exp (j27rft)df F- 1 {I:~==-oo Hn (f)X(f + nfc)}.