论文标题
当Ramanujan在复杂的时间序列分析中符合时频分析时
When Ramanujan meets time-frequency analysis in complicated time series analysis
论文作者
论文摘要
为了处理具有复杂振荡结构的时间序列，我们提出了一种新型的时频（TF）分析工具，该工具融合了短时间傅立叶变换（STFT）和周期性变换（PT）。由于许多时间序列随时间变化的频率，振幅和非鼻腔振荡模式振荡，因此PT或STFT的直接应用可能不合适。但是，我们表明，通过以正确的方式组合它们，我们可以获得强大的TF分析工具。我们首先将Ramanujan总和和$ L_1 $罚款结合起来，以实施PT。我们称为算法Ramanujan PT（RPT）。 RPT对其他应用具有其自身感兴趣，例如分析由具有整数周期的组件组成的简短信号，但这不是本文的重点。其次，将RPT应用于修改STFT并生成复杂时间序列的新型TF表示，该表示忠实地反映了每个振荡性成分的瞬时频率信息。我们在提出的TF分析中造成了Ramanujan De-Shape（RDS）和Vectorized RDS（VRD）。除了显示复杂生物医学信号的一些初步分析结果外，我们还提供了有关RPT的理论分析。具体来说，我们表明RPT对于通常遇到的三个噪音，包括信封波动，抖动和添加剂噪声是可靠的。
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.
