Time Synchronization of Turbo-Coded Square-QAM-Modulated Transmissions: Code-Aided ML Estimator and Closed-Form Cramér–Rao Lower Bounds

Abstract

This paper introduces a new maximum likelihood (ML) solution for the code-aided (CA) timing recovery problem in square-quadrature amplitude modulation (QAM) transmissions and derives, for the very first time, its CA Cramer–Rao lower bounds (CRLBs) in closed-form expressions. The channel is assumed to be slowly time varying so that it can be considered as constant over the observation interval. By exploiting the full symmetry of square-QAM constellations and further scrutinizing the Gray-coding mechanism, we express the likelihood function of the system explicitly in terms of the code bits’ 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 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 resynchronize the system, through feedback to the matched filter, 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 remarkable performance improvements of CA estimation against the traditional nondata-aided scheme, thereby highlighting the potential performance gains in time synchronization that can be achieved owing to the decoder assistance. Over a wide range of practical signal-to-noise ratios (SNRs), CA estimation becomes even equivalent to the completely data-aided scheme in which all the transmitted symbols are perfectly known to the receiver. 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 algorithm.