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


During this paper, we consider the matter of joint part and carrier frequency offset (CFO) estimation for turbo-coded systems. We have a tendency to derive for the primary time the closed-type expressions for the exact Cramér-Rao lower bounds (CRLBs) of those estimators over turbo-coded square-QAM-modulated single- or multi-carrier transmissions. In the latter case, the derived bounds stay 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 info (CQI). In explicit, we tend to introduce a replacement recursive process that allows the construction of arbitrary Gray-coded sq.-QAM constellations. Some hidden properties of such constellations will be revealed, owing to the current recursive method, and rigorously handled to decompose the system's likelihood operate (LF) into the total of two analogous terms. This decomposition makes it attainable to carry out analytically all the statistical expectations concerned within the Fisher info matrix (FIM). The new analytical CRLB expressions corroborate the previous tries to judge 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 fully blind [non-data-aided (NDA)] and fully information-aided (DA) estimation schemes, thereby highlighting the result 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 comparatively small SNR thresholds. It will additionally be shown that contrary to the CRLB of the phase shift, the CRLB of the CFO improves during a multi-carrier system as compared to its counterpart in an exceedingly single-carrier system. The derived bounds also are valid for LDPC-coded systems and they will be evaluated in the identical approach when the latter are decoded using the turbo principal.

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