Sequential detection of almost-harmonic signals

To detect a purely harmonic signal, it is difficult to beat a fast Fourier transform (FFT). However, when the signal is very long and weak, Parker and White (1999) have shown that a sequential probability ratio test (SPRT) operating on magnitudesquare FFT data is far more efficient. Indeed, both from a numerical-error perspective and in terms of robustness against a deviation from a precisely tonal signal, the block-FFT/SPRT idea is very appealing. Here, the approach is extended to the case that the frequency is unknown, and expressions are developed for performance both in terms of detection and of sample number. The approach is applicable to a large number of practical problems, but particular attention is paid to the continuous gravitational wave (GW) example. The computational savings as compared with a fixed test vary as a function of signal strength, block length, bandwidth and operating point; however, gains of a factor of two are easy. That these gains are not more exciting relates mostly to the underlying FFT structure; although many SPRTs "end early," it is difficult to take advantage of that with an efficient FFT algorithm. However, the progressive reduction of the number of working SPRTs implies a substantial reduction of the ensemble of the candidate frequencies with time, which is an appealing feature, particularly in the GW case.