Carrier phase and frequency estimation for pilot-symbol assisted transmission: bounds and algorithms

In this paper we consider the Cramer-Rao lower bound (CRB) for the joint estimation of the carrier phase and the frequency offset from a noisy linearly modulated burst signal containing random data symbols (DSs) as well as known pilot symbols (PSs). We point out that the CRB depends on the location of the PSs in the burst, the number of PSs, the number of DSs, the signal-to-noise ratio (SNR), and the data modulation scheme. Distributing the PSs symmetrically about the center of the burst and estimating the carrier phase in the center of the burst interval decouples the frequency and phase estimation, making the CRB for phase estimation independent of the specific location of the PSs. At low and moderate SNR, the CRBs for both phase and frequency estimation decrease as the fraction of the PSs in the burst increases. In addition, the CRB for frequency estimation decreases as the PSs are separated with more DSs. Numerical evaluation of the CRB indicates that the carrier phase and frequency of a "hybrid" burst (i.e., containing PSs and DSs) can be estimated more accurately when exploiting both the presence of the DSs and the a priori knowledge about the PSs, instead of using only the knowledge about the PSs (and ignoring the DSs), or considering all the received symbols (PSs and DSs) as unknown (and ignoring the knowledge about the PSs). Comparison of the CRB with the performance of existing carrier synchronizers shows that the iterative soft-decision-directed (sDD) estimator with data-aided (DA) initialization performs very closely to the CRB and provides a large improvement over the classical non-data-aided (NDA) estimator at lower SNR.