Analytical evaluation of performance for harmonic and biharmonic methods of blind phase offset estimation

We presented a new algorithm for blind phase offset estimation for signals with quadrature amplitude modulation (QAM). The algorithm is based on a circular harmonic expansion of log-likelihood function (LLF). Retaining one or two most significant terms in this series gives a harmonic or biharmonic circular decomposition of the LLF, this approach leads to notable improvement of the estimation quality comparing to known versions of popular 4th power phase estimation algorithm. Computer simulation results justified the advantages of the proposed method. It was also shown that the simplified implementation of the algorithm with weighting functions independent of signal-to-noise ratio (SNR) does not lead to any notable performance loss. In this paper we present analytical evaluation of estimation variance for both harmonic and biharmonic methods. Computation is based on quadratic approximation of objective function in the neighborhood of the phase offset true value, Taylor expansion of analytical expression for estimation error, and retaining statistical moments of objective function derivatives up to second order. Analytical results demonstrate good agreement with computer simulation for moderate and high SNR values.