Binary Error Probability Due to an Adaptable Fading Model

The received signal is assumed to consist of a fading replica of the original signal plus Gaussian noise. An adaptable fading model is constructed as follows. First, a certain short-term fading distribution (Rayleigh, singular or Rice) is assumed. Then the mean power of this short-term distribution is allowed to possess a long-term variation. The statistics of this variation are assumed to be describable by some gamma distribution. This suffices to derive the binary element error probabilities for a number of receiver types. Particular emphasis is placed on the noncoherent FSK. The performance of more significant ideal receivers is displayed in analytical and graphical forms. Let us select a fixed value of signal-to-noise ratio, possibly a large number. It is shown that the choice of a single parameter suffices to give any error probability up to the extreme value of one half. Such a channel, therefore, can be arbitrarily bad, no matter what the signal-to-noise ratio. It is conjectured that the effects of this peculiar fading, possibly more than the non-Gaussian character of atmospheric noise, may be chiefly responsible for intolerable error rates on fading high frequency radio circuits. For the purpose of utility, it should be noted that the nonfading shortterm model is mathematically most tractable.