SNR Estimation for M-ARY Non-Coherent Frequency Shift Keying Systems
In this paper, we provide an analytical performance comparison of various selection combining (SC) schemes for binary noncoherent frequency-shift keying (NCFSK) signals operating on independent and nonidentically distributed (i.n.d.) Rayleigh fading channels. With this motivation, we first derive the receiver structure for the optimum SC scheme, which combines one out of the available diversity branches so as to minimize the probability of bit error. We show that the optimum SC scheme chooses the diversity branch having the largest magnitude of the logarithm of the a posteriori probability ratio (LAPPR) of the transmitted information bit. We also show that: 1) the optimum noncoherent diversity receiver, for binary NCFSKsignals, is equivalent to combiningthe LAPPRs of all the diversity branches; 2) the SC scheme proposed by Neasmith and Beaulieu is a special case of the optimum SC scheme for independent and identically distributed (i.i.d.) Rayleigh fading; and 3) for i.n.d. fading with dual diversity (i.e., = 2), the performance of the optimum SC scheme is the same as that of the optimum noncoherent diversity receiver, whereas the Neasmith and Beaulieu SC scheme gives the performance of the suboptimum equal gain combining receiver. Bit-error rate (BER) results show that, at 10 4 BER, for i.i.d. Rayleigh fading, the proposed optimum SC scheme performs better than the existing SC schemes by 0.5–0.9 dB for = 3 and by 0.8–1.5 dB for = 5, and performs within 0.3 dB of the optimum noncoherent diversity receiver for = 3 5, and for i.n.d. Rayleigh fading with = 5, the optimum SC scheme gives an additional gain of about 2.0 dB over the Pierce SC scheme. Further, for i.i.d. Rayleigh fading, we derive the bit-error probability expression for a (3 ) selection scheme which combines three branches whose LAPPR magnitudes are the largest among the available branches. Numerical results for this (3 ) selection scheme show that for = 5 at a BER of 10 , combining the three branches with the largest LAPPR magnitudes yields almost the full performance of the optimum noncoherent diversity receiver, whereas, for = 7, it is just about 0.2 dB away from the latter.