Quickest Sequence Phase Detection

A phase detection sequence is a length-n cyclic sequence, such that the location of any length-k contiguous subsequence can be determined from a noisy observation of that subsequence. In this paper, we derive bounds on the minimal possible k in the limit of n → ∞, and describe some sequence constructions. We further consider multiple phase detection sequences, where the location of any length-k contiguous subsequence of each sequence can be determined simultaneously from a noisy mixture of those subsequences. We study the optimal trade-offs between the lengths of the sequences, and describe some sequence constructions. We compare these phase detection problems to their natural channel coding counterparts, and show a strict separation between the fundamental limits in the multiple sequence case. Both adversarial and probabilistic noise models are addressed.