Probabilities of Error for Adaptive Reception of M-Phase Signals

Probabilities of error are derived for 2- and 4-phase signaling over an
L
-diversity branch, time-invariant, additive Gaussian noise channel and for
M
-phase signaling over an
L
diversity branch, Rayleigh fading, additive Gaussian noise channel. Both channels corrupt the signaling waveforms transmitted through them by introducing an additive Gaussian noise and an unknown or random multiplicative gain and phase shift in the transmitted signal. The receiver processing consists of cross correlating the signal plus noise received over each diversity branch by a noisy reference signal, which is derived either from the previously received informationbearing signals or from the transmission and reception of a pilot signal, and adding the outputs from all
L
-diversity branches to form the decision variable. The novel aspects of this paper are: 1) the result for the phase distribution of the decision variable for
M
-phase signaling over an
L
-diversity branch, Rayleigh fading channel, 2) the explicit expressions for the probabilities of error for
M
-phase signaling systems operating over an
L
-diversity branch, Rayleigh fading channel, and 3) the expression for the probability of error for 4-phase signaling over a time-invariant channel. Previously published error probabilities for 2- and 4-phase signaling over a Rayleigh fading channel are obtained as special cases.