Bounds on joint phase and frequency estimation for pilot-assisted PSK/QAM modulation

This paper derives the true Cramer-Rao lower bounds (CRLB) on the individual phase estimation and the coupled carrier phase and frequency estimation for transmission bursts with pilot-assisted PSK/QAM modulation. The reduction in frequency and phase CRLB due to introduction of embedded reference symbols increases as the modulation order increases. It is found that the better CRLB may be obtained if the reference symbols are distributed symmetrically in the burst, especially using a preamble and postamble. At low SNR, 5% pilot symbols in the burst reduces the bound on the phase standard deviation by a factor of 10 for QPSK or higher order modulated signals. The new CRLB for distributed pilot symbols are also presented. The comparison of the true CRLB and the CRLB of the pilot symbols has revealed that using only pilot symbols for estimation is an effective approach only if more than 5% of pilot symbols are in the burst and SNR ≤ 0 dB.