Optimal sequential frame synchronization

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 N starts being emitted at a random time within some interval of size A, where A 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 N → ∞, if the asynchronism level grows as O(_e^Nα), with α below the synchronization threshold, a constant that admits a simple expression depending on the channel. If α exceeds the synchronization threshold, any decoder, sequential or nonsequential, locates the sync pattern with an error that tends to one as N → ∞. 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 A+N-1, but with far fewer observations.