In this paper, we study Bayesian and hybrid Cramé r-Rao bounds (BCRB and HCRB) for the code-aided (CA), the data-aided (DA) and the non-data-aided (NDA) dynamical phase estimation of QAM modulated signals. We address the bounds derivation for both the off-line scenario, for which the whole observation frame is used, and the on-line which only takes into account the current and the previous observations. For the CA scenario we show that the computation of the Bayesian information matrix (BIM) and of the hybrid information matrix (HIM) is NP hard. We then resort to the belief-propagation (BP) algorithm or to the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm to obtain some approximate values. Moreover, in order to avoid the calculus of the inverse of the BIM and of the HIM, we present some closed form expressions for the various CRBs, which greatly reduces the computation complexity. Finally, some simulations allow us to compare the possible improvements enabled by the off-line and the CA scenarios.

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This paper proposes a near-maximum a posteriori smoothing algorithm for the dynamical carrier phase estimation of coded quadrature amplitude modulation (QAM) signals. This low-complexity near-optimum smoothing is obtained by averaging phase-locked loops (PLLs) with possibly the aid of the decoded a posteriori information. The proposed code-aided smoothing algorithm performs near the off-line Bayesian and hybrid Cramer-Rao bounds (BCRBs and HCRBs) of interest. It has a gain of several decibels compared to the conventional on-line loop and is able to track frequency offsets.

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