Detection of a PSK signal transmitted through a hard-limited channel

This paper considers the problem of the detection of a binary phase-modulated carrier which has been transmitted along with noise through a hard-limiting repeater, corrupted by additional noise, and demodulated by cross correlation and sampling at the receiver. Three equivalent expressions are obtained for the error probability of the receiver output. Two of these expressions take the form of an infinite series involving either confluent hypergeometric functions or modified Bessel functions. A third form allows representation of the error probability in terms of Rice's Ie function. In the special case where the signal-to-noise ratios \rho_1^2 and \rho_2^2 at the limiter and receiver inputs are equal, the error probability is simply \begin{equation} P_e = \frac{1}{2} \exp (-p_1^2). \end{equation} Asymptotic expressions for the error probability for large and small signal-to-noise ratios are also derived. The error probability is found to be smaller than that of a linear system for all practical values of limiter and receiver input signal-to-noise ratios. The optimum repeater non-linearity is investigated and is shown to be a limiter at such signal-to-noise ratios.