Introduction

Envelope Explanation

Envelope:
- 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.
Fine Structure:
- 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.

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。

[!hint] "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”

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

有关系：


\begin{align}
A^2 & = x^2+y^2 \\
\varphi & = \arctan{\frac{y}{x}}
\end{align}

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：


e^{i\varphi} = \cos{\varphi} + i\sin{\varphi}

1946年， Gabor先生定义了复函数更一般化的欧拉公式


Y(t) = u(t) + iv(t)

这里的$v(t)$是$\mu(t)$的希尔伯特变换

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

The mathematical description of the Hilbert transform is to rotate the Fourier components in complex area.


H(\mu)(t) = \frac{1}{\pi} \text{p.v.} \int_{\infty}^{\infty} \frac{\mu(t)}{t-\tau}d\tau

The Hilbert transform is given by the Cauchy principal value of the convolution with the function 1/(\pi t).