Closed-Form CRLBs for CFO and Phase Estimation From Turbo-Coded Square-QAM-Modulated Transmissions

In this paper, we consider the problem of joint phase and carrier frequency offset (CFO) estimation for turbo-coded systems. We derive for the first time the closed-form expressions for the exact Cramér-Rao lower bounds (CRLBs) of these estimators over turbo-coded square-QAM-modulated single- or multi-carrier transmissions. In the latter case, the derived bounds remain valid in the general case of adaptive modulation and coding (AMC) where the coding rate and modulation order vary from one subcarrier to another depending on the corresponding channel quality information (CQI). In particular, we introduce a new recursive process that enables the construction of arbitrary Gray-coded square-QAM constellations. Some hidden properties of such constellations will be revealed, owing to this recursive process, and carefully handled to decompose the system's likelihood function (LF) into the sum of two analogous terms. This decomposition makes it possible to carry out analytically all the statistical expectations involved in the Fisher information matrix (FIM). The new analytical CRLB expressions corroborate the previous attempts to evaluate the underlying bounds empirically. In the low-to-medium signal-to-noise ratio (SNR) region, the CRLB for code-aided (CA) estimation lies between the bounds for completely blind [non-data-aided (NDA)] and completely data-aided (DA) estimation schemes, thereby highlighting the effect of the coding gain. Most interestingly, in contrast to the NDA case, the CA CRLBs start to decay rapidly and reach the DA bounds at relatively small SNR thresholds. It will also be shown that contrary to the CRLB of the phase shift, the CRLB of the CFO improves in a multi-carrier system as compared to its counterpart in a single-carrier system. The derived bounds are also valid for LDPC-coded systems and they can be evaluated in the same way when the latter are decoded using the turbo principal.