[论文解读] Theory inspired deep network for instantaneous-frequency extraction and signal components recovery from discrete blind-source data

Charles K. Chui, Ningning Han+1 more|arXiv (Cornell University)|Jan 31, 2020

Blind Source Separation Techniques参考文献 16被引用 4

一句话总结

该论文提出了一种理论驱动的深度神经网络，能够从非均匀采样、盲源复合信号中提取瞬时频率并恢复各个信号分量，而无需传统训练。基于严格的数学框架，该方法能精确识别分量数量，并实现超分辨率分解，在准确性和鲁棒性方面优于经验模态分解（EMD）、同步压缩变换（SST）以及先前的SSO方法，即使在存在噪声或超出数据边界的外推情况下也表现优异。

ABSTRACT

This paper is concerned with the inverse problem of recovering the unknown signal components, along with extraction of their instantaneous frequencies (IFs), governed by the adaptive harmonic model (AHM), from discrete (and possibly non-uniform) samples of the blind-source composite signal. None of the existing decomposition methods and algorithms, including the most popular empirical mode decomposition (EMD) computational scheme and its current modifications, is capable of solving this inverse problem. In order to meet the AHM formulation and to extract the IFs of the decomposed components, called intrinsic mode functions (IMFs), each IMF of EMD is extended to an analytic function in the upper half of the complex plane via the Hilbert transform, followed by taking the real part of the polar form of the analytic extension. Unfortunately, this approach most often fails to resolve the inverse problem satisfactorily. More recently, to resolve the inverse problem, the notion of synchrosqueezed wavelet transform (SST) was proposed by Daubechies and Maes, and further developed in many other papers, while a more direct method, called signal separation operation (SSO), was proposed and developed in our previous work published in the journal, Applied and Computational Harmonic Analysis, vol. 30(2):243-261, 2016. In the present paper, we propose a synthesis of SSO using a deep neural network, based directly on a discrete sample set, that may be non-uniformly sampled, of the blind-source signal. Our method is localized, as illustrated by a number of numerical examples, including components with different signal arrival and departure times. It also yields short-term prediction of the signal components, along with their IFs. Our neural networks are inspired by theory, designed so that they do not require any training in the traditional sense.

研究动机与目标

• 解决从盲源复合信号的离散、可能非均匀采样中恢复信号分量及其瞬时频率的逆问题。
• 克服现有方法（如EMD、SST及传统SSO）在一般条件下无法可靠求解逆问题的局限性。
• 开发一种数学基础坚实的深度网络架构，无需依赖经典反向传播训练。
• 实现对信号分量的高精度短期预测与外推，延伸至观测时间区间之外。
• 实现超分辨率分解，精确识别真实分量数量，且无需事先知晓信号结构。

提出的方法

• 该方法基于自适应谐波模型（AHM）推导出的理论框架，将非平稳信号建模为幅值与频率调制的正弦波之和。
• 将信号分离操作（SSO）扩展为一种直接从离散、非均匀采样中构建的深度神经网络架构。
• 通过希尔伯特变换构造解析信号，并利用解析函数的极坐标表示，提取瞬时频率。
• 基于理论洞察预先构建网络架构，无需依赖反向传播训练。
• 通过在网络结构中嵌入数学约束，确保局部化与稳定性，实现鲁棒的分量分离。
• 支持信号分量的重建与短期预测，包括对未观测时间区间的外推。

实验结果

研究问题

• RQ1 能否构建一种无需传统训练的深度神经网络，以准确从非均匀采样的盲源数据中恢复信号分量及其瞬时频率？
• RQ2 所提出的方法是否在求解信号分解的逆问题方面优于EMD、SST及先前的SSO方法？
• RQ3 该网络能否在不事先知晓复合信号结构的前提下，自动确定正确的信号分量数量？
• RQ4 在存在噪声或外推至观测时间区间之外时，该方法的性能如何？
• RQ5 该方法能否实现超分辨率分解，特别是在瞬时频率紧密相邻时？

主要发现

• 所提方法成功从非均匀采样数据中恢复了所有信号分量及其瞬时频率，蝙蝠回声定位示例中第五个IMF的均方误差（MSE）为 $1.74 \times 10^{-2}$。
• 在外推示例中，该方法在区间 $[-0.1, 0]$ 和 $[30, 30.1]$ 内准确预测了信号分量与瞬时频率，展示了强大的短期预测能力。
• 对于蝙蝠回声定位信号，该方法正确识别出五个IMF，且各IMF的MSE均低于 $1.74 \times 10^{-2}$，证实了高保真度的恢复效果。
• 网络在无需先验知识的情况下实现了精确的分量数量检测，验证了其以系统化方式求解逆问题的能力。
• 与EMD和SST相比，该方法在分辨紧密相邻的瞬时频率及处理非均匀采样方面表现出更优性能。
• 理论分析与数值实验表明，该方法在噪声条件下仍具稳定性和有效性，如定理2.1所示，并在第5节得到验证。