Double symbol error rates ofM-ary differentially coherent phase shift keying (DPSK) systems in a satellite-aircraft multipath channel are evaluated. A joint probability density function (pdf) of two consecutive differential phase errors in the DPSK outputs is newly derived. Numerically calculated results are shown in figures for binary, quaternary, and octonary DPSK systems. It is further shown that the analytical values agree well with the data observed by ATS-5 experimental tests.

The average error probabilities and outage rates of error probability for theM-phase CPSK signal through the Nakagami'sm-distributed fading channel are exactly evaluated both for nondiversity reception and for diversity reception. The probability density functions of the composite phase of fading signal and noise and those of the diversity-combined signal envelopes are newly derived in this paper. The results are generally obtained including the digital phase modulation component, the fading figure as a measure of fading depth, the average carrier-to-noise power ratio, the power correlation coefficient between two diversity branches, etc. The diversity improvements are also verified. Additionally some useful approximate formulas are briefly shown. This study will add a newly widened view to the considerations of system performances and designs for digital radio communications via fading channels.