The Error Performance of A Class of Binary Communications Systems in Fading and Noise

A recent approach to the problem of digital communication over fading transmission paths is the development of the frequency differential system. This method employs binary phase reversal keying of a sinusoid together with the simultaneous transmission of an additional unmodulated sinusoid at a closely spaced neighboring frequency. The additional steady tone is used to establish reference models at the receiver for the correlation detection of the modulated signal. This paper examines the performance of the frequency differential system in the presence of noise and fading wherein the model for the fading signal path assumes Gaussian statistics for the received signals and a certain proportionality among the auto- and cross-correlation functions of the modulated and unmodulated signals as they appear at the receiver. Particular emphasis is placed on the asymptotic error probability which is approached as received signal strength is indefinitely increased. An important part of the paper is the derivation of a conceptually "optimum" receiver (in the maximum likelihood sense) for the reception of the frequency differential binary signals. This optimum receiver is shown to be free of the "bottoming" or asymptotic error behavior at high signal-to-noise ratios. In addition, a procedure is outlined for the best diversity operation of the elementary binary channels using minimum error as the criterion. A number of numerical examples are presented for representative fading and noise conditions which give quantitative descriptions of the elementary channel performance in noise alone, noise and fading, and dual diversity with noise and fading. Finally the performance of a truncated form of the optimum receiver is similarly described quantitatively in the presence of noise and fading.