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We provide a theoretical foundation for further analysis and optimization of the $M$ th-power (MP) carrier phase estimator for $M$ PSK modulation. Also known as the Viterbi-Viterbi (VV) estimator, it is commonly used in practice because it leads to low-latency receiver implementations. The MP carrier phase estimator first raises the received noisy signal samples to the $M$ th-power to remove the unknown phase modulation, and then extracts the unknown carrier phase of the mid-symbol using a weighted sum of these modulation-wiped-off received signal samples over a symmetrical observation window. Our starting point is the single-term, complex exponential expression for a complex sinusoid received in complex, additive, white, Gaussian noise (AWGN), which leads to a great deal of simplicity in dealing with arbitrary powers of the noisy received signal sample when compared with the conventional approach of raising the sum of signal plus noise to higher powers. The single-exponential expression enables us to first optimize the weighting coefficients of the MP carrier phase estimator with respect to the statistics of the AWGN, in a manner much simpler than previous approaches. Then, it enables us to apply the linear minimum mean square error (LMMSE) criterion to optimize the MP estimator with respect to both the statistics of the AWGN and the carrier phase noise that we model here as a Wiener process. Although the LMMSE MP estimator is computationally intensive for online implementation, a much less complex version is suggested that can be efficiently implemented in real time. Extensive simulation results are presented to demonstrate the improved performance of the LMMSE MP estimator over the conventional MP estimator. By using a sufficiently long symmetrical observation window, the LMMSE estimator does not suffer from the block length effect, which leads to much performance gain over the VV/MP estimator especially at high signal-to-noise ratio (SNR) and high phase noise. A phase unwrapping algorithm is also presented for accurate unwrapping of the estimated carrier phase before it is used in data detection. The proposed LMMSE carrier phase estimator is suitable for implementing a coherent receiver at all SNRs.

A Refinement to the Viterbi-Viterbi Carrier Phase Estimator and an Extension to the Case With a Wiener Carrier Phase Process