On the use of Laguerre polynomials in treating the envelope and phase components of narrow-band Gaussian noise

The joint probability density of the envelope of a Gaussian process at two different times is expanded by the use of Hardy's identity into a series involving Laguerre polynomials. It is shown how this result may be used to estimate the cross-correlation function of the output of two quite general envelope-distorting filters. A generalization of this result, involving the use of the associated Laguerre polynomials, is obtained and applied to the calculation of a cross-correlation function which involves both the phase and envelope of the process at two points in time.