Channel knowledge and optimal performance for two-wave Rayleigh fading channels

The optimal detector structures and error probability performances for the two-wave Rayleigh fading channel with known delay are compared for different levels of channel knowledge. A number of different detectors are examined, for equal energy signals having equal complex autocorrelation magnitudes. Envelope orthogonal frequency-shift keying and variants of chirp signals are considered, so that the complex cross correlation of the signals is unconstrained. The performance of the optimal detector, when the fading in neither wave is tracked, is obtained. Two other, suboptimal, quadratic detectors are also considered for this case. Optimal detection, when the fading in only one of the waves is tracked, while a statistical knowledge of the other wave is available, is examined. The optimal performance that can be achieved by a time-varying matched-filter detector that makes use of complete knowledge of the channel fading in both waves is also determined. These detectors, for all the different levels of channel information considered, are studied in a unified framework, the probability of error being expressed as the probability that a quadratic form in Gaussian random variables is less than zero. It is found that the power gain that can be derived from partial or complete tracking is small. All these detectors exhibit a diversity-like effect for all nonzero values of the delay and for most values of the signal parameters. Signals with larger dispersion factors, such as chirp signals and variants, perform well on the channel, enhancing the diversity effect, even at small delays