Some problems in the analysis of possibly cyclostationary data

Cyclostationary processes are encountered in fields ranging from climate to cognitive radio. They are important as an intermediate step between stationary and Loève´s harmonizable processes, perhaps the most general case that can be described with second-order statistics. However, when the cyclostationary period is unknown, or there are several periods, one must compute the Loève spectrum to isolate the relevant period or periods. This entails a high “false discovery” rate because, for a series of N samples, such spectra includes ~ N² individual cross-spectra or coherence estimates. Moreover, most processes encountered from both natural and engineering sources contain many weak periodic, or almost periodic, components. For example, the RF spectrum is littered with weak signals from distant sources, intermodulation products, radiation from poorly-shielded digital circuits, etc. Similarly, processes such as daily temperature records contain evidence of many discrete solar normal modes. Because one sinusoid looks much like another, these cause spurious peaks in a Loève two-frequency coherence estimate. I show examples from daily temperature series and dropped calls in cellular phone systems. This paper describes a process for estimating the Loève spectrum via a singular value decomposition of multitaper eigencoefficients that is considerably faster than the direct method. Because it involves O(N) individual hypothesis tests the false alarm rate is lower than the naive approach. I also describe a multivariate, T², version of the harmonic F-test for simultaneous periodic components, and this has proven useful both for reducing the false alarm rate and for identifying some unexpected signals.