A systematic approach to a class of problems in the theory of noise and other random phenomena--III: Examples

The method of Part I is applied to the problem of finding the characteristic function for the probability distribution of , where denotes the th component of a stationary n-dimensional Markoffian Gaussian process. The problem is reduced to the problem of solving first-order linear differential equations with initial conditions only. For the case of constant , the explicit solution is given in terms of the eigenvalues and the first powers of a constant matrix. For the case of a symmetric correlation matrix which commutes with , the problem is reduced to the one-dimensional case treated in Part II. For the case , 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.