True Cramer-Rao Lower Bound for NDA Phase Estimation from General M-QAM Modulated Signals over Flat Rayleigh Fading Channel

In this paper, we derive for the first time the analytical expression of the CramerRao lower bound (CRLB) for non-data-aided (NDA) carrier phase estimation from quadrature amplitude modulated (QAM) signals over flat Rayleigh fading channel (FRFC), where the fading gain is not constant during the observation window. The second derivative of the neperian logarithm of the probability density function (PDF) p(x|φ) of the observation vector x parametrized by the carrier phase error φ is analytically developed. Then numerical Monte Carlo (MC) method is used to find out the true CRLB from square M-QAM modulation signals over FRFC with complex additive white Gaussian noise (CAWGN). It is through the assessment of the true CRLB from QAM modulation signals for various observation window lengths that our method is proven to be trustworthy. For sufficiently large observation window size, simulation results show that optimal NDA carrier phase recovery depends on the constellation density for low SNR range. For different modulation orders the CRLB decreases as far as the observation window length increases. The derived bound provides an efficient standard for measuring the performance of any unbiased NDA carrier phase estimator for square M-QAM modulation signals over FRFC. The method presented here stands useful for general linear modulations.