Optimum selection combining of binary NCFSK signals on independent Rayleigh fading channels

In this paper, we derive a new selection combining (SC) scheme for noncoherent binary FSK signals on independent (but not necessarily identically distributed) Rayleigh fading channels with L-antenna diversity reception. With this combining scheme, we choose the diversity branch having the largest magnitude of the logarithm of the ratio of the a posteriori probabilities (log-APP ratio - LAPPR) of the transmitted information bit. We show that this scheme minimizes the probability of bit error, thus proving the optimality. We also show that a) the traditional square-law combining of all L diversity branches is equivalent to combining the LAPPR of all the L diversity branches, and b) the SC scheme proposed by Neasmith and Beaulieu (1998) is a special case of the proposed optimum SC scheme for independent and identically distributed (i.i.d) Rayleigh fading. Bit error probability results show that, at 10^-4 BER, a) for i.i.d Rayleigh fading, the proposed optimum SC combining scheme performs better than the existing SC schemes by 0.5 dB for L = 3 and 1.5 dB for L = 5, and performs within 0.5 dB of the scheme which square-law combines all the L diversity branches, and b) for independent Rayleigh fading, the proposed optimum SC scheme gives an additional gain of 2.0 dB over the SC schemes of Pierce (1958) and Chyi et al. (1989).