Optimal Sequential Frame Synchronization

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We consider the “one-shot frame synchronization problem,” where a decoder wants to locate a sync pattern at the output of a memoryless channel on the basis of sequential observations. The sync pattern of length <emphasis><formula formulatype="inline"><tex>$N$</tex></formula></emphasis> starts being emitted at a random time within some interval of size <emphasis><formula formulatype="inline"> <tex>$A$</tex></formula></emphasis>, where <emphasis><formula formulatype="inline"> <tex>$A$</tex></formula></emphasis> characterizes the asynchronism level. We show that a sequential decoder can optimally locate the sync pattern, i.e., exactly, without delay, and with probability approaching one as <emphasis><formula formulatype="inline"><tex>$N \rightarrow \infty$</tex></formula></emphasis>, if the asynchronism level grows as <emphasis><formula formulatype="inline"> <tex>$O(e^{N\alpha})$</tex></formula></emphasis>, with <emphasis><formula formulatype="inline"><tex>$\alpha$</tex></formula></emphasis> below the <emphasis emphasistype="boldital">synchronization threshold</emphasis>, a constant that admits a simple expression depending on the channel. If <emphasis><formula formulatype="inline"><tex>$\alpha$</tex></formula></emphasis> exceeds the synchronization threshold, any decoder, sequential or nonsequential, locates the sync pattern with an error that tends to one as <emphasis><formula formulatype="inline"> <tex>$N\rightarrow \infty$</tex></formula></emphasis>. Hence, a sequential decoder can locate a sync pattern as well as the (nonsequential) maximum-likelihood decoder that operates on the basis of output sequences of maximum length <emphasis><formula formulatype="inline"><tex>$A+N-1$</tex></formula></emphasis>, but with far fewer observations. </para>