Aspects on single symbol signaling on the frequency flat Rayleigh fading channel

The optimal single symbol detector on the Rayleigh fading channel computes a functional quadratic form in the time continuous received signal. A drawback is that closed-form solutions of integral equations based on the channel statistics are required. This makes simplified discrete receivers attractive. A class of suboptimal receivers that transforms the received random process to a set of discrete observables is derived. The set of observables constitutes a random vector in a finite dimensional receiver signal space. Given this vector, the maximum likelihood detector computes a quadratic form in the received vector. The discretization implies a loss of information, therefore such a detector is not, in general, optimal given the received time continuous signal. The purpose is, however, to achieve close to optimal performance when the number of observables becomes large. This class of detectors is analyzed using exact error probability calculations, which reveal several interesting properties. The length of the observation interval and the number of discrete observables have significant influences on the error probability when the time variations of the fading process are rapid compared with the symbol duration. By increasing the number of observables, the error floor is lowered, and the implicit diversity order is increased. This implicit diversity arises as soon as more than one observable per symbol interval is used and is a consequence of the information bearing signal being a random process. Matched filter receivers use few discrete observables per symbol interval, and thus suffer from high error floors and low implicit diversity orders on fast fading channels. The error probability is highly dependent on the shapes and durations of the modulator waveforms. For instance, pulses of long duration give lower error probabilities than shorter pulses, and for a certain type of orthogonal waveforms there is no error floor.