Hou Thomas Y, Shi Zuoqiang
Applied and Computational Mathematics, MC 9-94, Caltech, Pasadena, CA 91125, USA.
Mathematical Sciences Center, Tsinghua University, Beijing 100084, People's Republic of China
Philos Trans A Math Phys Eng Sci. 2016 Apr 13;374(2065):20150194. doi: 10.1098/rsta.2015.0194.
In this paper, we develop an effective and robust adaptive time-frequency analysis method for signals with intra-wave frequency modulation. To handle this kind of signals effectively, we generalize our data-driven time-frequency analysis by using a shape function to describe the intra-wave frequency modulation. The idea of using a shape function in time-frequency analysis was first proposed by Wu (Wu 2013 Appl. Comput. Harmon. Anal. 35, 181-199. (doi:10.1016/j.acha.2012.08.008)). A shape function could be any smooth 2π-periodic function. Based on this model, we propose to solve an optimization problem to extract the shape function. By exploring the fact that the shape function is a periodic function with respect to its phase function, we can identify certain low-rank structure of the signal. This low-rank structure enables us to extract the shape function from the signal. Once the shape function is obtained, the instantaneous frequency with intra-wave modulation can be recovered from the shape function. We demonstrate the robustness and efficiency of our method by applying it to several synthetic and real signals. One important observation is that this approach is very stable to noise perturbation. By using the shape function approach, we can capture the intra-wave frequency modulation very well even for noise-polluted signals. In comparison, existing methods such as empirical mode decomposition/ensemble empirical mode decomposition seem to have difficulty in capturing the intra-wave modulation when the signal is polluted by noise.
在本文中，我们针对具有波内频率调制的信号开发了一种有效且稳健的自适应时频分析方法。为了有效处理这类信号，我们通过使用形状函数来描述波内频率调制，对数据驱动的时频分析进行了推广。在时频分析中使用形状函数的想法最初是由吴提出的（吴，2013年，《应用计算与调和分析》，第35卷，第181 - 199页。（doi:10.1016/j.acha.2012.08.008））。形状函数可以是任何光滑的2π周期函数。基于此模型，我们提出求解一个优化问题来提取形状函数。通过探究形状函数相对于其相位函数是周期函数这一事实，我们可以识别信号的某些低秩结构。这种低秩结构使我们能够从信号中提取形状函数。一旦获得形状函数，就可以从形状函数中恢复具有波内调制的瞬时频率。我们将该方法应用于几个合成信号和真实信号，以证明其稳健性和效率。一个重要的观察结果是，这种方法对噪声扰动非常稳定。通过使用形状函数方法，即使对于受噪声污染的信号，我们也能很好地捕捉波内频率调制。相比之下，当信号受到噪声污染时，诸如经验模式分解/总体经验模式分解等现有方法似乎难以捕捉波内调制。
