SEP calculations for coherent M‐ary FSK in different fading channels with MRC diversity

Abstract

SUMMARYIn this paper, the authors derive symbol error probability (SEP) expressions for coherent M-ary frequency shift keying (MFSK) modulation schemes in multipath fading channels. The multipath or small-scale fading process is assumed to be slow and frequency non-selective. In addition, the channel is also subjected to the usual degradation caused by the additive white Gaussian noise (AWGN). Different small-scale fading statistics such as Rayleigh, Rician (Nakagami-n), Hoyt (Nakagami-q), and Nakagami-m have been considered to portray diverse wireless environments. Further, to mitigate fading effects through space diversity, the receiver front-end is assumed to be equipped with multiple antennas. Independent and identically distributed (IID) as well as uncorrelated signal replicas received through all these antennas are combined with a linear combiner before successive demodulation. As the detection is coherent in nature and thus involves phase estimation, optimum phase-coherent combining algorithms, such as predetection maximal ratio combining (MRC), may be used without any added complexity to the receiver. In the current text, utilizing the alternate expressions for integer powers (1 n 4) of Gaussian Q function, SEP values of coherent MFSK are obtained through moment generating function (MGF) approach for all the fading models (with or without MRC diversity) described above. The derived end expressions are composed of finite range integrals, which can be numerically computed with ease, dispenses with the need of individual expressions for different M, and gives exact values up to M = 5. When the constellation size becomes bigger (M 6), the same SEP expressions provide a quite realistic approximation, much tighter than the bounds found in previous literatures. Error probabilities are graphically displayed for each fading model with different values of constellation size M, diversity order L, and for corresponding fading parameters (K , q, or m). To validate the proposed approximation method extensive Monte-Carlo simulations were also performed, which show a close match with the analytical results deduced in the paper. Both these theoretical and simulation results offer valuable insight to assess the efficacy of relatively less studied coherent MFSK in the context of the optimum modulation choice in wireless communication.