Time Synchronization of Turbo-Coded Square-QAM-Modulated Transmissions: Code-Aided ML Estimator and Closed-Form Cram\'er-Rao Lower Bounds

This paper introduces a new maximum likelihood (ML) solution for the code-aided (CA) timing recovery problem in square-QAM transmissions and derives, for the very first time, its CA Cram\'er-Rao lower bounds (CRLBs) in closed-form expressions. By exploiting the full symmetry of square-QAM constellations and further scrutinizing the Gray-coding mechanism, we express the likelihood function (LF) of the system explicitly in terms of the code bits' \textit{a priori} log-likelihood ratios (LLRs). The timing recovery task is then embedded in the turbo iteration loop wherein increasingly accurate estimates for such LLRs are computed from the output of the soft-input soft-output (SISO) decoders and exploited at a per-turbo-iteration basis in order to refine the ML time delay estimate. The latter is then used to better re-synchronize the system, through feedback to the matched filter (MF), so as to obtain more reliable symbol-rate samples for the next turbo iteration. In order to properly benchmark the new CA ML estimator, we also derive for the very first time the closed-form expressions for the exact CRLBs of the underlying turbo synchronization problem. Computer simulations will show that the new closed-form CRLBs coincide exactly with their empirical counterparts evaluated previously using exhaustive Monte-Carlo simulations. They will also show unambiguously the potential performance gains in time synchronization that can be achieved owing to the decoder assistance. Moreover, the new CA ML estimator almost reaches the underlying CA CRLBs, even for small SNRs, thereby confirming its statistical efficiency in practice. It also enjoys significant improvements in computational complexity as compared to the most powerful existing ML solution, namely the combined sum-product and expectation-maximization (SP-EM) algorithm