True Cramer-Rao lower bound for NDA carrier frequency estimation from general M-QAM modulated signals over flat Rayleigh fading channel

In this paper, we derive for the first time the analytical expressions of the Cramer-Rao lower bound (CRLB) for non-data-aided (NDA) carrier frequency estimation from linearly modulated signals over flat Rayleigh fading channel (FRFC), where the fading gain is not constant during the observation window. Timing delay and phase offset are assumed as perfectly known. The second derivative of the neperian logarithm of the probability density function (PDF) p(x|γ) of the observation vector x parametrized by the carrier frequency offset, is theoretically built up. Then numerical Monte Carlo (MC) method is used to find out the true CRLB and the modified CRB (MCRB) from linearly modulated signals over FRFC with complex additive white Gaussian noise (CAWGN). Through the assessment of the CRLB and the MCRB for various observation window size our method is proven to be trustworthy. The reliability of the method is made obvious from both square quadrature amplitude modulation (QAM) and phase shift keying (PSK) modulated signals. The simulation result has shown that optimal NDA carrier frequency recovery depends on the constellation density for low SNR range. For both QAM and PSK modulations and for different modulation orders, it is shown that the CRLB is as low as the observation window length is larger. We also show that CRLB to MCRB ratio tends asymptotically towards 1 for larger SNR range as far as the modulation density increases. The derived bound provides an efficient standard to compare the performance of any unbiased NDA carrier frequency estimator for linearly modulated signals over FRFC.