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When Ramanujan meets time-frequency analysis in complicated time series analysis

Ziyu Chen and Hau-Tieng Wu

Vol. 4 (2022), No. 4, 629–673

To handle time series with complicated oscillatory structure, we propose a novel time-frequency (TF) analysis tool that fuses the short-time Fourier transform (STFT) and periodic transform (PT). As many time series oscillate with time-varying frequency, amplitude and nonsinusoidal oscillatory pattern, a direct application of PT or STFT might not be suitable. However, we show that by combining them in a proper way, we obtain a powerful TF analysis tool. We first combine the Ramanujan sums and l1 penalization to implement the PT. We call the algorithm Ramanujan PT (RPT). The RPT is of its own interest for other applications, like analyzing short signals composed of components with integer periods, but that is not the focus of this paper. Second, the RPT is applied to modify the STFT and generate a novel TF representation of the complicated time series that faithfully reflects the instantaneous frequency information of each oscillatory component. We coin the proposed TF analysis the Ramanujan de-shape (RDS) and vectorized RDS (vRDS). In addition to showing some preliminary analysis results on complicated biomedical signals, we provide theoretical analysis about the RPT. Specifically, we show that the RPT is robust to three commonly encountered noises, including envelop fluctuation, jitter and additive noise.

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periodicity transform, Ramanujan sums, $l^1$ regularization, time-frequency analysis, de-shape, Ramanujan de-shape
Mathematical Subject Classification
Primary: 42C20, 62M10, 68P01, 92C55
Received: 11 February 2021
Revised: 26 January 2022
Accepted: 22 March 2022
Published: 21 January 2023
Ziyu Chen
Department of Mathematics
Duke University
Durham, NC
United States
Hau-Tieng Wu
Department of Mathematics and Department of Statistical Science
Duke University
Durham, NC
United States
Mathematics Division
National Center for Theoretical Sciences