Observability of Spectral Components beyond Nyquist Limit in Nonuniformly Sampled Signals

Abstract

Identification of a signal component with the frequency exceeding the Nyquist limit is a challenging problem in signal theory as well as in some specific applications areas like astronomy and biosciences. A consequence of the well-known sampling theorem for a uniformly sampled signal is that the spectral component above the Nyquist limit is aliased into lower frequency range, making a distinction between true and aliased components impossible. The nonuniform sampling, however, offers a possibility to reduce aliased components and uncover information above the Nyquist limit. In this paper, we provide a theoretical analysis of the aliased components reduction in the nonparametric periodogram for two sampling schemes: the random sampling pattern and the sampling pattern generated by the integral pulse frequency modulation (IPFM), the latter widely accepted as the heart rate timing model. A general formula that relates the variance of timing deviations from a regular scheme with the aliased component suppression is proposed. The derived relation is illustrated by Lomb-Scargle periodograms applied on simulated data. Presented experimental data consisting of the respiration signal derived from the electrocardiogram and the heart rate signal also support possibility to detect frequencies above the Nyquist limit in the condition known as the cardiac aliasing.