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

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). Since many time series oscillate with time-varying frequency, amplitude and non-sinusoidal 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 $l_1$ 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 signal 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 reflect the instantaneous frequency information of each oscillatory components. 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 RPT. Specifically, we show that the RPT is robust to three commonly encountered noises, including envelop fluctuation, jitter and additive noise

Dictionary approaches for identifying periodicities in data

We propose several dictionary representations for periodic signals and use them for estimating their periodicity. This includes estimating concurrent multiple periodicities. These are inspired from the recently proposed DFT based Farey dictionary, where period estimation was cast as a sparse vector recovery problem. We show that this can instead be framed as an l2 norm based data-fitting problem with closed form solutions and much faster computations. We also generalize the complex valued Farey dictionary to simpler integer valued dictionaries. We find that dictionaries constructed using the recently proposed Ramanujan Periodicity Transforms provide the best trade-off between complexity and noise immunity