Error Probability for Binary Signaling Through an Multipath Channel

This paper considers error probability for the optimum reception of binary signals transmitted through a multipath channel having 1601 1602 T H E BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1907 P paths.* One of two possible signals is transmitted; the received waveform is the sum of P Rayleigh-faded, time-delayed, frequencyshifted versions of the transmitted signal, plus white Gaussian noise. That is to say, if the complex signal s/2Em xm(t) is transmitted, m = 1,2, the contribution to the received waveform from the pth path is VM]P) = V2E~m apxm(t - t,,) exp [i(2*jpt + #>,,)], where ap,
» having been selected from a set of two distinct functions by chance according to the prior probabilities {or,,,}, m = 1, 2, and the other second-moment function (z{s)z(t))m being zero. The receiver is to be designed so that its decision upon one of the two possible hypotheses is made with minimum average error probability Pe, where P c = 2a,,,Pc(m) and P e (ra) is the probability, when covariance indexed vi is true, of deciding otherwise.