The Influence of Fading Spectrum on the Binary Error Probabilites of Incoherent and Differentially Coherent Matched Filter Recievers

Previous derivations of the influence of fading on the error probabilities of binary data transmission systems have assumed that the fading rate is so slow that fluctuations within a bit may be ignored. This slow fading assumption is removed in the present paper which derives general expressions for the binary error probabilities of incoherent and differentially coherent matched filter receivers employing post-detection diversity combining. In the analysis it is assumed that the transmitted signals occupy a bandwidth much smaller than the coherence bandwidth of the medium so that "flat" fading may be assumed. In addition, it is assumed that the amplitude and phase fluctuations produced by the medium have the same statistical character as those of narrow-band Gaussian noise. The general analytical results are specialized to the cases of frequency shift keying using incoherent detection, and phase shift keying using differentially coherent detection, and to the cases of exponential and Gaussian fading correlation functions. For these special cases, signal-to-noise degradation curves are given as a function of fading bandwidth. The PSK system is degraded more rapidly with increasing fading bandwidth than is the FSK system. Curves are given which show the error probabilities and corresponding fading bandwidths for which the noncoherent FSK and the differentially coherent PSK systems break even. For lower error probabilities or higher fading bandwidths, the FSK system becomes superior to the PSK system in the sense of being able to provide the same error probability with less signal-to-noise ratio. The existence of an irreducible error probability is demonstrated for the incoherent and differentially coherent matched filter receivers. Thus, in general, an increase in transmitted signal power cannot reduce the error probability below a certain value depending upon the fading spectrum and order of diversity. Theoretical curves of irreducible error probability are given for- the incoherent FSK and differentially coherent PSK systems. An important result of the analysis is that the shape of the fading spectrum can make a significant difference in the amount of signal-to-noise degradation. The results of the analysis also indicate that care must be exercised in employing a "slow fading" assumption since, if low bit error probabilities are desired, significant degradations in performance can occur even though the fading rate is quite low relative to the data rate.