Error probabilities in binary angle modulation

It is shown that many signal-detection problems involving binary phase and frequency modulation can be solved by considering the problem as the convolution of two independent phase distributions. When the two distributions are similar and each represents the probability density function of the phase of a sinusoid in noise, the probability of error is described in terms of modified Bessel functions by the expression obtained recently by Jain and Blachman. The probability of error can also be expressed in terms of Rice'sIefunction. By appropriate definition of the signal-to-noise ratios, this general expression can be used to determine the error probability in a number of cases of practical interest, such as detection of a hard-limited phase-shift keyed (PSK) signal, PSK detection with noisy reference, DPSK detection, frequency-shift keying (FSK) detection, binary FM with discriminator detection, and binary pulse-position modulation (PPM).