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Solution of an integral equation occurring in the theories of prediction and detection

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In many of the theories of prediction and detection developed during the past decade, one encounters linear integral equations which can be subsumed under the general formint_a^b R(t, tau) x(tau) dtau = f(t), a leqq t leqq b. This equation includes as special cases the Wiener-Hopf equation and the modified Wiener-Hopf equationint_0^T R(mid t - tau mid ) x(tau) dtau = f(t), 0 leqq t leqq T. The type of kernel considered in this note occurs when the noise can be regarded as the result of operating on white noise with a succession of not necessarily time-invariant linear differential and inverse-differential operators. For this type of noise, which is essentially a generalization of the stationary noise with a rational spectral density function, it is shown that the solution of the integral equation can be expressed in terms of solution of a certain linear differential equation with variable coefficients.

Published in:

Information Theory, IRE Transactions on  (Volume:2 ,  Issue: 2 )

Date of Publication:

June 1956

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