Envelope statistics of partially saturated processes

The probability density functions associated with the envelope and phase of a partially saturated process are derived. The derivation assumes that the single path phases are Gaussian random variables with arbitrary means but with identical variances σ2ϑ. The analysis also includes the effects of Gaussian noise on the statistics. It is shown that in the limit as the standard deviation of the single path phases becomes large (≳π/2), the densities approach previously derived results for fully saturated phase random propagation. In the limit as σϑ→0, the densities for unsaturated propagation, namely constant signal plus Gaussian noise, are recovered. The results derived are used for determining the distributions of fades and receiver operating characteristics for a partially saturated process. The fading probability and the signal-to-noise ratio required to detect the partially saturated signal is directly proportional to σϑ.