Abstract Intra-wave modulations are a class of inherent nonlinear characteristics, exhibiting a fast oscillating instantaneous frequency and/or amplitude of responses. With the aid of the well-known Hilbert-Huang transform (HHT), such phenomena have been observed and utilized in many practical nonlinear systems including mechanical, power, ocean and even human biological systems. However, the empirical nature of the HHT makes the results physically uninterpretable and sensitive to perturbations of noise. Variational nonlinear chirp mode decomposition (VNCMD) is a recently proposed tool for analyzing wideband multicomponent signals, including intra-wave modulated responses. On the other hand, the VNCMD has strict requirements on the priori information of the signal. In this paper, we combine the framework of the VNCMD with that of the HHT, by replacing the joint-optimization scheme of the VNCMD with a recursive procedure adopted in the HHT. In this way, the new method becomes more adaptive without losing the rigorous mathematical foundation. This construction leads to a descendant of VNCMD, named the iterative nonlinear chirp mode decomposition (INCMD). Through dynamic simulations and applications to real data, it is demonstrated that the INCMD considerably outperforms state-of-the-art techniques of the same class. Using the INCMD, intra-wave modulations can be captured with high accuracy and strong noise-robustness. Extracted modulation features by the INCMD greatly help to detect and identify nonlinear systems.

中文翻译：

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迭代非线性啁啾模式分解：一种类似希尔伯特-黄变换的方法，用于捕捉非线性响应的波内调制

摘要 波内调制是一类固有的非线性特性，表现出快速振荡的瞬时频率和/或响应幅度。借助著名的希尔伯特-黄变换 (HHT)，这种现象已经在许多实际的非线性系统中得到观察和利用，包括机械、动力、海洋甚至人类生物系统。然而，HHT 的经验性质使得结果在物理上无法解释并且对噪声扰动很敏感。变分非线性啁啾模式分解 (VNCMD) 是最近提出的用于分析宽带多分量信号（包括波内调制响应）的工具。另一方面，VNCMD 对信号的先验信息有严格的要求。在本文中，我们将 VNCMD 的框架与 HHT 的框架相结合，通过用 HHT 中采用的递归程序替换 VNCMD 的联合优化方案。这样，新方法在不失严谨的数学基础的情况下变得更具适应性。这种构造导致了 VNCMD 的后代，称为迭代非线性啁啾模式分解 (INCMD)。通过动态模拟和对真实数据的应用，证明 INCMD 大大优于同类的最先进技术。使用 INCMD，可以以高精度和强噪声鲁棒性捕获波内调制。INCMD 提取的调制特征极大地有助于检测和识别非线性系统。新方法在不失严谨的数学基础的情况下变得更具适应性。这种构造导致了 VNCMD 的后代，称为迭代非线性啁啾模式分解 (INCMD)。通过对真实数据的动态模拟和应用，证明 INCMD 大大优于同类的最先进技术。使用 INCMD，可以以高精度和强噪声鲁棒性捕获波内调制。INCMD 提取的调制特征极大地有助于检测和识别非线性系统。新方法在不失严谨的数学基础的情况下变得更具适应性。这种构造导致了 VNCMD 的后代，称为迭代非线性啁啾模式分解 (INCMD)。通过对真实数据的动态模拟和应用，证明 INCMD 大大优于同类的最先进技术。使用 INCMD，可以以高精度和强噪声鲁棒性捕获波内调制。INCMD 提取的调制特征极大地有助于检测和识别非线性系统。事实证明，INCMD 的性能明显优于同类的最先进技术。使用 INCMD，可以以高精度和强噪声鲁棒性捕获波内调制。INCMD 提取的调制特征极大地有助于检测和识别非线性系统。事实证明，INCMD 的性能明显优于同类的最先进技术。使用 INCMD，可以以高精度和强噪声鲁棒性捕获波内调制。INCMD 提取的调制特征极大地有助于检测和识别非线性系统。