Error Probabilities for Binary Symmetric Ideal Reception through Nonselective Slow Fading and Noise

One of two correlated, equal energy, equiprobable waveforms is transmitted through a channel during a given time interval. The signal is corrupted in the channel by slowly-varying, frequency-nonselective fading and by additive, Gaussian noise. On reception, the corrupted signal is processed by an ideal receiver, which guesses that the transmitted waveform was the one which it computes to be a posteriori most probable. Expressions for the probability of committing an error in making such a guess are derived for both coherent and noncoherent receivers; these are studied in detail, and some general trends and system design considerations are noted. In an illustrative example, the results are applied to binary frequency-shift keyed (FSK) systems with various pulse shapes and frequency separations.