A posteriori joint detection of a recurring tuple of reference fragments in a quasi-periodic sequence

The problem of joint detection of a recurring tuple of reference fragments in a noisy numerical quasi-periodic sequence is solved in the framework of the a posteriori (off-line) approach. It is assumed that (i) the total number of fragments in the sequence is known, (ii) the index of the sequence member corresponding to the beginning of a fragment is a deterministic (not random) value, and (iii) a sequence distorted by an additive uncorrelated Gaussian noise is available for observation. It is shown that the problem consists of testing a set of simple hypotheses about the mean of a random Gaussian vector. A specific feature of the problem is that the cardinality of the set grows exponentially as the vector dimension (i.e., the length of the observed sequence) and the number of fragments in the sequence increase. It is established that the search for a maximum-likelihood hypothesis is equivalent to the search for arguments that maximize a special auxiliary objective function with linear inequality constraints. It is shown that this function is maximized by solving the basic extremum problem. It is proved that this problem is solvable in polynomial time. An exact algorithm for its solution is substantiated that underlies an algorithm guaranteeing optimal (maximum-likelihood) detection of a recurring tuple of reference fragments. The results of numerical simulation demonstrate the noise stability of the detection algorithm.