Performance Analysis of Coded $M$-ary Orthogonal Signaling Using Errors-and Erasures Decoding Over Frequency-Selective Fading Channels

The performance of $M$-ary orthogonal signaling schemes employing Reed–Solomon (RS) codes and redundant residue number system (RRNS) codes is investigated over frequency-selective Rayleigh fading channels. “Errors-and-erasures” decoding is considered, where erasures are judged based on two low-complexity, low-delay erasure insertion schemes—Viterbi’s ratio threshold test (RTT) and the proposed output threshold test (OTT). The probability density functions (PDF) of the ratio associated with the RTT and that of the demodulation output in the OTT conditioned on both the correct detection and erroneous detection of $M$-ary signals are derived, and the characteristics of the RTT and OTT are investigated. Furthermore, expressions are derived for computing the codeword decoding error probability of RS codes or RRNS codes based on the above PDFs. The OTT technique is compared to Viterbi’s RTT, and both of these are compared to receivers using “error-correction only” decoding over frequency-selective Rayleigh-fading channels. The numerical results show that by using “errors-and-erasures” decoding, RS or RRNS codes of a given code rate can achieve higher coding gain than that without erasure information, and that the OTT technique outperforms the RTT, provided that both schemes are operated at the optimum decision thresholds. Index Terms—“Errors-and-erasures” decoding, $M$-ary orthogonal signaling, Rayleigh fading, redundant residue number system codes, Reed–Solomon codes