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The solution of a homogeneous Wiener-Hopf integral equation occurring in the expansion of second-order stationary random functions

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In many of the applications of probability theory to problems of estimation and detection of random functions an eigenvalue integral equation of the type begin{equation} phi(x) = lambda int_0^T K(x - y)phi(y) dy, qquad 0 leq x leq T, end{equation} is encountered whereK(x)represents the covariance function of a continuous stationary second-order process possessing an absolutely continuous spectral density. In this paper an explicit operational solution is given for the eigenvalnes and eigenfunctions in the special but practical case when the Fourier transform ofK(x)is a rational function ofomega^2, i.e., begin{equation} K(x) doteqdot G(s^2) = frac{N(s^2)}{D(s^2)}, qquad s=iomega, end{euation} in whichN(s^2)andD(s^2)are polynomials ins^2. It is easy to show by elementary methods that the solutions are of the form begin{equation} phi(x)= sum C_r e^{-alpha_r x} cos (beta_r x + gamma_r), end{equation} the constantsC_r, alpha_r, beta_r, andgamma_rbeing linked together by the integral equation. It is precisely the labor involved in their determination that in practice often causes the problem to assume awesome proportions. By means of the results given herein, this labor is diminished to the irreducible minimum-the solving of a transcendental equation.

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Information Theory, IRE Transactions on  (Volume:3 ,  Issue: 3 )