Cramér–Rao lower bound for non-data-aided carrier phase estimation from general M-ary phase-shift keying modulation signals over flat Rayleigh fading channel

Abstract

The authors derive the true Cramér–Rao lower bound (CRLB) for a non-data-aided (NDA) carrier phase estimation from general M-ary phase-shift keying (M-PSK) modulation signal over flat Rayleigh fading channels (FRFC). The numerical Monte Carlo method is used to compute the true CRLB from general M-PSK modulation signals over FRFC with complex additive white Gaussian noise. The efficiency of the proposed method is made obvious through the assessment of the true CRLBs from both symmetric and asymmetric 8-PSK modulation signals for various observation window sizes. The simulation results show that optimal NDA carrier phase recovery is harder to acquire from symmetric modulation signals for signal-to-noise ratio (SNR) range up to 5 dB, but depends on the constellation density, only for SNR range superior to 5 dB. For both constellations the CRLB decreases as long as the observation window size increases. The derived bound provides an efficient standard for evaluating the performance of any unbiased NDA carrier phase estimator from general M-PSK modulation signals over FRFC. The method presented here stands useful for general M-ary quadrature amplitude modulation signals as well.