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97 lines
4.1 KiB
Markdown
97 lines
4.1 KiB
Markdown
---
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title: Hilbert Transform Envelope
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tags:
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- signal-processing
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- algorithm
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- envelope
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date: 2024-02-28
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---
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# Introduction
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# Envelope Explanation
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## Envelope and Fine Structure
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* Envelope:
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* The envelope of a signal represents the slowly varying amplitude or outline of the signal. It provides a smooth curve that encapsulates the main shape of the signal, ignoring the rapid oscillations or fluctuations. The envelope is typically associated with the low-frequency components of a signal.
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* Fine Structure:
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* The fine structure of a signal refers to the detailed, high-frequency components or rapid oscillations present in the signal. It captures the fast variations that occur on a shorter time scale compared to the envelope.
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# Algorithm Detail
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## History
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* 1905年---Hilbert在研究黎曼-希尔伯特问题时提出希尔伯特变换，而他关于离散希尔伯特变换的早期工作可追溯到他在哥根廷的讲课。
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* Hermann Weyl在他的学位论文中发表了离散希尔伯特变换的结论。
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* Schur改进了离散希尔伯特变换的结果，并将其扩展到了积分条件下。
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**而将Hilbert变换运用到信号处理中还得追溯到解析信号表达的建立。**
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> [!hint]
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> "传统经典的信号研究方法主要概括为基于傅里叶变换的谱分析、基于概率分布的统计分析和其它随机信号表示方法，同时还有起源于很早的典型谱、相关和分布特征，而这些分析方法研究的**一个基本考虑是将随机信号表达为两个独立函数的乘积**”
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>
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早期关于包络和瞬时相位的研究都是基于笛卡尔坐标系x-y
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有关系：
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$$
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\begin{align}
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A^2 & = x^2+y^2 \\
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\varphi & = \arctan{\frac{y}{x}}
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\end{align}
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$$
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这样的表达被引入傅里叶序列中，$x_k = \sum a_k\cos\varphi_k + b_k\sin\varphi_k$, 其幅度和相位均可由上面笛卡尔坐标系中的两个关系得到，此时坐标$(x,y)$就是$(a_k,b_k)$。 用这种方法研究调制信号的包络和瞬时相位依赖于一个伟大的公式：
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$$
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e^{i\varphi} = \cos{\varphi} + i\sin{\varphi}
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$$
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1946年， Gabor先生定义了复函数更一般化的欧拉公式
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$$
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Y(t) = u(t) + iv(t)
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$$
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这里的$v(t)$是$\mu(t)$的希尔伯特变换
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1998年，Huang在现代希尔伯特变换研究领域做出了显著性工作 —— EMD、HHT，使得希尔伯特变换理论在现代信号分析中遍地开花
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## Analytical Signal
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## Mathematical description
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The mathematical description of the Hilbert transform is to **rotate the Fourier components in complex area**.
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$$
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H(\mu)(t) = \frac{1}{\pi} \text{p.v.} \int_{\infty}^{\infty} \frac{\mu(t)}{t-\tau}d\tau
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$$
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The Hilbert transform is given by the [Cauchy principal value](math/real_analysis/cauchy_principal_value.md) of the convolution with the function $1/(\pi t)$.
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## Geometrical meaning of HT
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# Reference
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* [Mathuranathan. “Extract Envelope, Phase Using Hilbert Transform: Demo.” _GaussianWaves_, 24 Apr. 2017, https://www.gaussianwaves.com/2017/04/extract-envelope-instantaneous-phase-frequency-hilbert-transform/.](https://www.gaussianwaves.com/2017/04/extract-envelope-instantaneous-phase-frequency-hilbert-transform/)
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* [_CFC: What Does the Hilbert Transform Do? (V9)_. _www.youtube.com_, https://www.youtube.com/watch?v=-CjnFEOopfw. Accessed 2 Jan. 2024.](https://www.youtube.com/watch?v=-CjnFEOopfw)
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* [_Extract Envelope and Fine Structure in Praat Using the Hilbert Transform_. _www.youtube.com_, https://www.youtube.com/watch?v=qp1G3a2g8r0. Accessed 2 Jan. 2024.](https://www.youtube.com/watch?v=qp1G3a2g8r0)
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* [“希尔伯特变换与瞬时频率问题--连载（一）.” 知乎专栏, https://zhuanlan.zhihu.com/p/25213895. Accessed 2 Jan. 2024.](https://zhuanlan.zhihu.com/p/25213895)
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* [_The Hilbert Transform_. _www.youtube.com_, https://www.youtube.com/watch?v=VyLU8hlhI-I. Accessed 3 Jan. 2024.](https://www.youtube.com/watch?v=VyLU8hlhI-I)
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