A unified approach to the probability of error for noncoherent and differentially coherent modulations over generalized fading channels

We present a unified approach to determine the exact bit error rate (BER) of noncoherent and differentially coherent modulations with single and multichannel reception over additive white Gaussian noise and generalized fading channels. The multichannel reception results assume independent fading in the channels and are applicable to systems that employ post-detection equal gain combining. Our approach relies on an alternate form of the Marcum Q-function and leads to expressions of the BER involving a single finite-range integral which can be readily evaluated numerically. Aside from unifying the past results, the new approach also allows for a more general solution to the problem in that it includes many situations that in the past defied a simple solution. The best example of this occurs for multichannel reception where the fading on each channel need not be identically distributed nor even distributed according to the same family of distributions.