Sequential testing of sorted and transformed data as an efficient way to implement long GLRTs

It is often required to detect a long weak signal in Gaussian noise, and frequently, the exact form of that signal is parameterized but not known. A bank of matched filters provides an appropriate detector. However, in some practical applications, there are very many matched filters, and most are quite long. The consequent computational needs may render the classical bank-of-filters approach infeasibly expensive. One example, and our original motivation, is the detection of chirp gravitational waves by an Earth-based interferometer. In this paper, we provide a computational approach to this problem via sequential testing. Since the sequential tests to be used are not for constant signals, we develop the theory in terms of average sample number (ASN) for this case. Specifically, we propose two easily calculable expressions for the ASN: one a bound and the other an approximation. The sequential approach does yield moderate computational savings, but we find that by preprocessing the data using short/medium fast Fourier transforms (FFTs) and an appropriate sorting of these FFT outputs such that the most informative samples are entered to a sequential test first, quite high numerical efficiency can be realized. The idea is simple but appears to be quite successful: Examples are presented in which the computational load is reduced by several orders of magnitude. The FFT is an example of an energy-agglomerating transform, but of course, there are many others. The point here is that the transform need not match the sought signal exactly in the sense that all energy becomes confined to a single sample; it is enough that the energy becomes concentrated, and the more concentrated the better.