A Fast Multitaper Power Spectrum Estimation in Nonuniformly Sampled Time Series (2407.01943v3)

Abstract: Nonuniformly sampled signals are prevalent in real-world applications, but their power spectra estimation, usually from a finite number of samples of a single realization, presents a significant challenge. The optimal solution, which uses Bronez Generalized Prolate Spheroidal Sequence (GPSS), is computationally demanding and often impractical for large datasets. This paper describes a fast nonparametric method, the MultiTaper NonUniform Fast Fourier Transform (MTNUFFT), capable of estimating power spectra with lower computational burden. The method first derives a set of optimal tapers through cubic spline interpolation on a nominal analysis band. These tapers are subsequently shifted to other analysis bands using the NonUniform FFT (NUFFT). The estimated spectral power within the band is the average power at the outputs of the taper set. This algorithm eliminates the needs for time-consuming computation to solve the Generalized Eigenvalue Problem (GEP), thus reducing the computational load from $O(N^4)$ to $O(N \log N + N \log(1/\epsilon))$, which is comparable with the NUFFT. The statistical properties of the estimator are assessed using Bronez GPSS theory, revealing that the bias of estimates and variance bound of the MTNUFFT estimator are identical to those of the optimal estimator. Furthermore, the degradation of bias bound may serve as a measure of the deviation from optimality. The performance of the estimator is evaluated using both simulation and real-world data, demonstrating its practical applicability. The code of the proposed fast algorithm is available on GitHub (https://github.com/jiecui/mtnufft).