Non-decision-aided detection and SNR estimation of weak BPSK signals in AWGN

This paper provides additional insight into non-decision-aided detection and SNR estimation of BPSK signals in white noise with a focus on low SNR environments. The likelihood ratio test (LRT) statistic, the maximum likelihood (ML) estimates and Cramer-Rao lower bounds (CRLBs) for this problem are all derived and asymptotic analyses of them are carried out for both high and low SNR regimes. These show that in both regimes the ML estimates are naturally approximated by estimators based on sample moments. While the high SNR results confirm standard practice, the low SNR results demonstrate unexpectedly high sensitivity: for an SNR ρ close to zero the analyses of the LRT statistic and CRLB show O(ρ^-4) samples are needed to either reliably detect the presence of a signal or to accurately estimate its parameters. The paper also reviews the iterative method proposed by Li, DiFazio and Zeira (IEEE Commun. Lett. 6(11), pp. 469-471, 2002) for calculating the true ML estimate of SNR: it shows the method to be an instance of the standard bisection algorithm for root finding, and it derives a condition on the sample moments that guarantees existence of a solution and convergence of the algorithm. Finally the paper presents some simulations that compare the iterative scheme with the two standard moment-based estimators and the CRLB: these indicate the widely used M2-M4 estimator based on second and fourth order moments is a good practical choice in many situations.