Hou Thomas Y, Shi Zuoqiang
Applied and Comput. Math, MC 9-94, Caltech, Pasadena, CA 91125, USA.
Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, People's Republic of China
Philos Trans A Math Phys Eng Sci. 2016 Apr 13;374(2065):20150192. doi: 10.1098/rsta.2015.0192.
In this paper, we propose a time-frequency analysis method to obtain instantaneous frequencies and the corresponding decomposition by solving an optimization problem. In this optimization problem, the basis that is used to decompose the signal is not known a priori. Instead, it is adapted to the signal and is determined as part of the optimization problem. In this sense, this optimization problem can be seen as a dictionary adaptation problem, in which the dictionary is adaptive to one signal rather than a training set in dictionary learning. This dictionary adaptation problem is solved by using the augmented Lagrangian multiplier (ALM) method iteratively. We further accelerate the ALM method in each iteration by using the fast wavelet transform. We apply our method to decompose several signals, including signals with poor scale separation, signals with outliers and polluted by noise and a real signal. The results show that this method can give accurate recovery of both the instantaneous frequencies and the intrinsic mode functions.
在本文中，我们提出一种时频分析方法，通过求解一个优化问题来获得瞬时频率及相应的分解。在这个优化问题中，用于分解信号的基并非先验已知。相反，它会根据信号进行调整，并作为优化问题的一部分来确定。从这个意义上讲，这个优化问题可被视为一个字典自适应问题，其中字典是针对单个信号自适应的，而非字典学习中的训练集。通过迭代使用增广拉格朗日乘子（ALM）方法来解决这个字典自适应问题。在每次迭代中，我们通过使用快速小波变换进一步加速ALM方法。我们将我们的方法应用于分解多个信号，包括尺度分离不佳的信号、存在异常值且被噪声污染的信号以及一个真实信号。结果表明，该方法能够准确恢复瞬时频率和本征模函数。
