Optimal detection of a repeated tuple of reference fragments in a quasi-periodic sequence

Abstract

An a posteriori (off-line) approach to the problem of maximum likelihood detection of a repeated tuple of reference fragments in a numerical quasi-per iodic sequence is studied. The problem is considered under the following assumptions The total amount of fragments in a sequence is unknown. The number of sequence members corresponding to the beginning of a fragment is a determined (not random) value. The sequence available for observation is distorted by an additive Gaussian uncorrelated noise. The problem can be reduced to a test of a single mean hypothesis about a random Gaussian vector. A notable aspect of the problem is the fact that the cardinality of the set of tested hypotheses grows exponentially as the vector dimension (the length of the observed sequence) is increased. The search for the maximum likelihood hypothesis is shown to be equivalent to the search for arguments that provide the maximum of a special objective function. The search for the maximum of the function is demonstrated to be accomplished by solving a basic extremal problem. The basic extremal problem is shown to be solvable in polynomial time. The exact solving algorithm is presented and proved to provide optimal detection of a repeated tuple. Results of numerical simulation that show the detection algorithm to be noise-proof are given.