مرکز منطقه ای اطلاع رساني علوم و فناوري - Level crossings of nondifferentiable shot processes

Abstract :

An expression for the expectation of level crossings of a class of nondifferentiable shot processes that involve impulse responses having discontinuities is derived. Its first term is essentially the Rician formula [l, pp. 51-53] except that the random variables $(y_o, \\dot{y}_o)$ in the integrand are conditioned on nonoccurrence of discontinuities and can be interpreted as yielding the contribution of "smooth" crossings; the second term then describes the direct contribution of the jumps. The moments of $(y_o, \\dot{y}_o)$ reflect the influence, at the points of continuity of the process $y(t)$ , of the jumps at its points of discontinuity. Whereas in differentiable processes $(y, \\dot{y})$ are orthogonal, here $(y_o, \\dot{y}_o)$ are correlated; furthermore, although $y_o$ converges in distribution to $y$ and $E[\\dot{y}_o]$ is finite and nonzero in general, evidently $\\dot{y}$ has no finite moments. In the Gaussian limit of large densities of counts of the underlying Poisson process, an explicit formula is obtained in terms of known parameters. The Rician term is slightly altered by the correlation between $y_o$ and $\\dot{y}_o$ . On the other hand, the additional term has a component that increases as the square root of the average number of counts. In various practical cases either the bounded Rician or the diverging term can be dominant. An example of particular interest is examined in detail.