当Ramanujan在复杂的时间序列分析中遇到时频分析时
When Ramanujan meets time-frequency analysis in complicated time series analysis
摘要：
为了处理具有复杂振荡结构的时间序列，提出了一种融合短时傅里叶变换（STFT）和周期变换（PT）的时频分析工具。由于许多时间序列以时变的频率、幅度和非正弦振荡模式振荡，直接应用PT或STFT可能不合适。然而，我们表明，通过以适当的方式将它们结合起来，我们获得了一个强大的TF分析工具。我们首先结合Ramanujan和和$l_1$惩罚来实现PT。我们称该算法为Ramanujan PT（RPT）。RPT在其他应用中也有自己的优势，如分析由整数周期分量组成的短信号，但这不是本文的重点。其次，RPT被应用于修改STFT，并产生一个新的TF表示的复杂的时间序列，忠实地反映每个振荡分量的瞬时频率信息。我们提出的TF分析Ramanujan变形（RDS）和矢量化RDS（vRDS）。除了对复杂的生物医学信号进行初步分析外，本文还对RPT进行了理论分析。具体来说，我们表明，RPT是强大的三个常见的噪声，包括包络波动，抖动和加性噪声。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
