Analytically solving the Fokker-Planck equation for the statistical characterization of the phase noise in envelope detection

Although envelope detection is an old technique, the analysis theory for it is by no means complete, especially when the signal to be detected is corrupted by strong phase noise. In optical communication, of which the carrier source generally has strong phase noise, envelope detection has wide application. So this kind of analysis is important and many studies have been reported. In this paper, the detailed analysis of getting an analytical solution for the Fokker-Planck equation which describe the statistical distribution of the detected envelope for the signal with strong phase noise is presented. The result is expressed as a series of summation. Theoretically the distribution of any precision can be obtained by careful numerical summation. Besides that, the time of calculation is relatively short as long as the intensity of the phase noise is not too weak. So the formula is also useful in the system design and optimisation