Han Ningning, Mhaskar H N, Chui Charles K
IEEE Trans Neural Netw Learn Syst. 2021 Feb 10;PP. doi: 10.1109/TNNLS.2021.3052966.
In the mathematical and engineering literature on signal processing and time-series analysis, there are two opposite points of view concerning the extraction of time-varying frequencies (commonly called instantaneous frequencies, IFs). One is to consider the given signal as a composite signal consisting of a finite number of subsignals that are oscillating, and the goal is to decompose the signal into the sum of the (unknown) subsignals, followed by extracting the IF from each subsignal; the other is first to extract from the given signal, the IFs of the (unknown) subsignals, from which the subsignals that constitute the given signal are recovered. Let us call the first the signal decomposition approach'' and the second the signal resolution approach.'' For the signal decomposition approach,'' rigorous mathematical theories on function decomposition have been well developed in the mathematical literature, with the most relevant one, called atomic decomposition'' initiated by R. Coifman, with various extensions by others, notably by D. Donoho, with the goal of extracting the signal building blocks, but without concern of which building blocks constitute any of the subsignals, and consequently, the subsignals along with their IFs cannot be recovered. On the other hand, the most popular of the decomposition approach is the empirical mode decomposition (EMD),'' proposed by N. Huang et al., with many variations by others. In contrast to atomic decomposition, all variations of EMD are ad hoc algorithms, without any rigorous mathematical theory. Unfortunately, all existing versions of EMD fail to resolve the inverse problem on the recovery of the subsignals that constitute the given composite signal, and consequently, extracting the IFs is not satisfactory. For example, EMD fails to extract even two IFs that are not far apart from each other. In contrast to the signal decomposition approach, the signal resolution approach'' has a very long history dated back to the Prony method, introduced by G. de Prony in 1795, for solving the inverse problem of time-invariant linear systems. On the other hand, for nonstationary signals, the synchrosqueezed wavelet transform (SST), proposed by I. Daubechies over a decade ago, with various extensions and variations by others, was introduced to resolving the inverse problem, by first extracting the IFs from some reference frequency, followed by recovering the subsignals. Unfortunately, the SST approximate IFs could not be separated when the target IFs are close to one another at certain time instants, and even if they could be separated, the approximation is usually not sufficiently accurate. For these reasons, some signal components could not be recovered, and those that could be recovered are usually inexact. More recently, we introduced and developed a more direct method, called signal separation operation (SSO), published in 2016, to accurately compute the IFs and to accurately recover all signal components even if some of the target IFs are close to each other. The main contributions of this article are twofold. First, the SSO method is extended from uniformly sampled data to arbitrarily sampled data. This method is localized as illustrated by a number of numerical examples, including components with different subsignal arrival and departure times. It also yields a short-term prediction of the digital components along with their IFs. Second, we present a novel theory-inspired implementation of our method as a deep neural network (DNN). We have proved that a major advantage of DNN over shallow networks is that DNN can take advantage of any inherent compositional structure in the target function, while shallow networks are necessarily blind to such structure. Therefore, DNN can avoid the so-called curse of dimensionality using what we have called the blessing of compositionality. However, the compositional structure of the target function is not uniquely defined, and the constituent functions are typically not known so that the networks still need to be trained end-to-end. In contrast, the DNN introduced in this article implements a mathematical procedure so that no training is required at all, and the compositional structure is evident from the procedure. We will disclose the extension of the SSO method in Sections II and III and explain the construction of the deep network in Section IV.
在信号处理与时间序列分析的数学及工程文献中，关于时变频率（通常称为瞬时频率，IFs）的提取存在两种相反观点。一种是将给定信号视为由有限数量振荡子信号组成的复合信号，目标是将该信号分解为（未知）子信号之和，然后从每个子信号中提取瞬时频率；另一种是先从给定信号中提取（未知）子信号的瞬时频率，再据此恢复构成给定信号的子信号。我们将第一种称为“信号分解方法”，第二种称为“信号解析方法”。对于“信号分解方法”，数学文献中已充分发展了关于函数分解的严格数学理论，其中最相关的是由R. 科伊夫曼发起的“原子分解”，其他人进行了各种扩展，特别是D. 多诺霍，其目标是提取信号构建块，但不关心哪些构建块构成任何子信号，因此，子信号及其瞬时频率无法恢复。另一方面，分解方法中最流行的是N. 黄等人提出的“经验模态分解（EMD）”，其他人又进行了许多变体。与原子分解不同，EMD的所有变体都是临时算法，没有任何严格的数学理论。不幸的是，EMD的所有现有版本都未能解决恢复构成给定复合信号的子信号的逆问题，因此，提取瞬时频率并不令人满意。例如，EMD甚至无法提取两个彼此相距不远的瞬时频率。与信号分解方法相反，“信号解析方法”有着悠久的历史，可以追溯到1795年G. 德普罗尼引入的用于解决时不变线性系统逆问题的普罗尼方法。另一方面，对于非平稳信号，十多年前I. 多贝西提出的同步挤压小波变换（SST），其他人进行了各种扩展和变体，用于解决逆问题，即先从某个参考频率中提取瞬时频率，然后恢复子信号。不幸的是，当目标瞬时频率在某些时刻彼此接近时，SST近似的瞬时频率无法分离，即使可以分离，近似通常也不够准确。由于这些原因，一些信号成分无法恢复，而那些可以恢复的通常也不准确。最近，我们在2016年引入并开发了一种更直接的方法，称为信号分离运算（SSO），即使某些目标瞬时频率彼此接近，也能准确计算瞬时频率并准确恢复所有信号成分。本文的主要贡献有两个方面。首先，SSO方法从均匀采样数据扩展到任意采样数据。如许多数值示例所示，该方法是局部化的，包括具有不同子信号到达和离开时间的成分。它还能对数字成分及其瞬时频率进行短期预测。其次，我们提出了一种受新理论启发的方法实现，即作为深度神经网络（DNN）。我们已经证明，DNN相对于浅层网络的一个主要优势是，DNN可以利用目标函数中任何固有的组合结构，而浅层网络必然对这种结构视而不见。因此，可以利用我们所谓的组合性优势来避免所谓的维度灾难。然而，目标函数的组合结构不是唯一确定的，组成函数通常也未知，因此网络仍需要端到端地进行训练。相比之下，本文中引入的DNN实现了一个数学过程，根本不需要训练，组合结构从该过程中显而易见。我们将在第二节和第三节中披露SSO方法的扩展，并在第四节中解释深度网络的构建。
