A General Framework for Asynchronous Communication

We study a problem of sequential frame detection in an asynchronous framework, where a single frame of length <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> slots is transmitted uniformly in a large interval of known size <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> slots. In this setup, we seek to characterize the scaling needed of <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> and the channel (input) parameters for asynchronous optimal frame synchronization. We note that the framework permits a natural trade-off between <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> is the synchronization threshold of the channel (usually parameterized by the channel input). We present a general framework that permits this trade-off and then characterize the scaling needed of both <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> as a function of the asynchronism period <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula>. Finally, we apply our results to the AWGN channel as an illustration.