On the use of space diversity in the square-law reception of fading on-off keyed signals

In this paper the total bit-error probability using square-law detection of an on-off keyed carrier received through a fading medium is calculated. It is shown that the use of diversity will recover much of the performance loss caused by the fading, provided that an optimum decision threshold is used. Consideration is given to the performance obtained with a fixed optimum threshold (depending only on the mean-square signal strength) and to that obtained with two forms of variable threshold, assumed to follow the instantaneous fading signal strength. Compared to the fixed optimum threshold, the best variable threshold results in a 5- or 2.5-dB reduction in the required rms SNR for a 10^-4error rate with two- or four-branch diversity, respectively; as compared to the no fading case, use of the fixed threshold requires 3 dB more SNR for a 10^-4error rate with four-branch diversity, and requires 32 dB more SNR if only a single branch is used. A calculation of the average fade duration of the diversity-combiner output-signal component shows that diversity reduces the average time that the signal component spends below the threshold according to N^-1/2for N branches. This is true for both the fixed or the best variable threshold, but the average duration with the fixed threshold is larger by about [log (1+σρ²)]^1/2, where σρ²is the mean-square SNR per branch.