Multitaper Statistical Tests for the Detection of Frequency-Modulated Signals
Detection of periodic signals in noise is an important problem in many scientific fields and there exist tools in the multitaper spectrum estimation and harmonic analysis framework for doing so, for example, the Harmonic F statistic. However, the Harmonic F statistic can lose effectiveness under certain types of frequency modulation, when the signal to noise ratio is low, and when the background spectrum is highly coloured. In his 2009 paper, "Polynomial Phase Demodulation in Multitaper Analysis," Thomson proposed methods for dealing with time series data (specifically, solar data) where these problems are present. In this thesis we propose an extension of this work to deal with the detection of frequency modulated signals. The method uses the Slepian sequences as projection filters to reconstruct the series based on a 2W band around a given carrier frequency and then tests the instantaneous frequency series for a low-degree polynomial form in that band using the Slepians combined with an associated family of polynomials in a variance ratio test statistic. Under the null hypothesis that there are no sinusoidal signals with polynomial frequency modulation at the given carrier frequency, these test statistics are approximately distributed according to an F distribution with degrees of freedom depending on the number of tapers used and the degree of the polynomial being tested. We compare several such test statistics via simulation studies and apply them to a solar time series from the GOLF SoHO instrument.