Comparison of erasure tests with diversity reception for noncoherent M-ary FSK signaling

A l-Bayesian erasure test (BET) based on order statistics with diversity reception for noncoherent M-ary frequency shift keying (FSK) signaling is examined. In the l-BET, the M-ary FSK symbol is erased whenever the a posteriori probability of correct decision, which is a function of the l-largest sums of energy detector outputs, is less than a threshold. For binary FSK signaling in Rayleigh flat fading channels, the 2-BET is found to be the same as the difference threshold test (DTT), which declares an erasure whenever the difference between the largest and the second largest sums of energy detector outputs is less than a threshold. The output threshold test (OTT), which declares an erasure whenever the largest sum of energy detector outputs is less than a threshold, is proved equivalent to the 1-BET for the Rayleigh flat fading channel. Closed-form expressions for the probabilities of correct decision and symbol error for the DTT, Viterbi's ratio threshold test (RTT), and the OTT are derived when square-law combining (SLC) diversity is used. Several erasure tests including the l-BET, the RTT, the OTT, and the DTT are compared in terms of probability of codeword error. The 2-BET provides almost the same performance as the BET that requires all the sums of energy detector outputs. The 2-BET provides power gains of 0.9 dB and 0.7 dB over the RTT for (7, 3) and (15, 7) Reed-Solomon (RS) codes, respectively, when the probability of codeword error is 10^-3 and the diversity order is one. Higher-order diversity has reduced gains.