Analysis of sequential frame synchronizers in Gaussian noise channels

We present a framework for the analysis of frame synchronization based on Synchronization Words (SWs), where the detection is based on the common sequential algorithm: the received samples are observed over a window of length equal to the SW; over this window a metric (e.g. correlation) is computed; a SW is declared if the computed metric is greater than a proper threshold, otherwise the observation window is time-shifted of one sample. We assume a Gaussian channel, antipodal signalling and coherent detection, where soft values are provided to the frame synchronizer. We state the problem starting from the hypothesis testing theory, deriving the optimum metric (optimum likelihood ratio test (LRT)) according to the Neyman-Pearson lemma. When the data distribution is unknown, we design a simple and effective test based on the Generalized LRT (GLRT). %added - begin We also analyze the performance of the commonly used correlation metric, both in the "hard" and "soft" version. We show that synchronization by correlation can be greatly improved by the LRT and GLRT metrics, and also that, among correlation based tests, sometimes hard correlation is better than soft correlation. The obtained closed form expressions allow the derivation of the receiver operating characteristic (ROC) curves for the LRT and GLRT synchronizers, showing a remarkable gain with respect to synchronization based on correlation metric. The effect on the performance of non-equally distributed data is also shown.