DocumentCode :
944249
Title :
A systematic approach to a class of problems in the theory of noise and other random phenomena--III: Examples
Author :
Siegert, A.J.F.
Volume :
4
Issue :
1
fYear :
1958
fDate :
3/1/1958 12:00:00 AM
Firstpage :
4
Lastpage :
14
Abstract :
The method of Part I is applied to the problem of finding the characteristic function for the probability distribution of \\int_0^t \\sum _{jk} x_j (\\tau ) K_{jl}(\\tau )x_l(\\tau ) d\\tau , where x_j(\\tau ) denotes the j th component of a stationary n-dimensional Markoffian Gaussian process. The problem is reduced to the problem of solving 2n first-order linear differential equations with initial conditions only. For the case of constant K , the explicit solution is given in terms of the eigenvalues and the first 2n - 1 powers of a constant 2n \\times 2n matrix. For the case of a symmetric correlation matrix which commutes with K , the problem is reduced to the one-dimensional case treated in Part II. For the case K_{ij}(t) = \\delta _{il} \\delta _{jl} e^{-t} , where the functional represents the output of a receiver consisting of a lumped circuit amplifier, a quadratic detector, and a single-stage amplifier, the solution has been obtained in a form which is more explicit than that provided by the earlier methods.
Keywords :
Noise; Stochastic processes; Circuits; Detectors; Differential equations; Eigenvalues and eigenfunctions; Gaussian noise; Gaussian processes; Information theory; Integral equations; Probability distribution; Symmetric matrices;
fLanguage :
English
Journal_Title :
Information Theory, IRE Transactions on
Publisher :
ieee
ISSN :
0096-1000
Type :
jour
DOI :
10.1109/TIT.1958.1057437
Filename :
1057437
Link To Document :
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