Statistical properties of two sine waves in Gaussian noise

A detailed study is presented of some statistical properties of the stochastic process, that consists of the sum of two sine waves of unknown relative phase and a normal process. Since none of the statistics investigated seem to yield a closed-form expression, all the derivations are cast in a form that is particularly suitable for machine computation. Specifically, results are presented for the probability density function (pdf) of the envelope and the instantaneous value, the moments of these distributions, and the relative cumulative density function (cdf). The analysis hinges on expanding the functions of interest in a way that allows computation by means of recursive relations. Specifically, all the expansions are expressed in terms of sums of products of Gaussian hypergeometric functions and Laguerre polynomials. Computer results obtained on a CDC 6600 are presented. If and are the amplitudes of the two sine waves, normalized to the rms noise level, the expansions presented are useful up to values of of about 17 dB, in double precision on the CDC 6600. A different approximation is also given for the case of very high SNR.