The Generalized Wiener Process for Colored Noise

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We define the generalized Wiener process as the output of a first-order differential equation when the input is an arbitrary stochastic input. This is in contrast to the standard Wiener process, where the input is white Gaussian noise. We obtain a simple explicit result for any input wide sense stationary random process, namely, that the Wigner spectrum of the output random process is the product of the power spectrum of the input process times a simple universal function of time and frequency. We also obtain the impulse response function wherein the output of the generalized process is expressed as the impulse response function integrated with the input random process. Various limiting values are derived. We apply the method to the case where the driving stochastic process has an exponentially decaying autocorrelation function