Code-Aided Time Synchronization of Turbo-Coded Square-QAM-Modulated Transmissions: Closed-Form Cramer-Rao Lower Bounds

This paper tackles the problem of code-aided (CA) timing recovery in turbo-coded square-QAM transmissions. Owing to a simple recursive construction process, some hidden properties of Gray-coded (GC) square-QAM constellations are demystified and the code bits´ a priori log- likelihood ratios (LLRs) are explicitly incorporated in the log-likelihood function (LLF). Then, by splitting the underlying LLF into the sum of two analogous terms, we derive for the very first time the closed-form expressions for the exact Cramer-Rao lower bounds (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 the CA scheme against the traditional non-data-aided (NDA) one. Over a wide range of practical SNRs, the new CA CRLBs reach those of the completely data-aided (DA) scheme in which all the transmitted symbols are perfectly known to the receiver.