信号与系统：希尔伯特变换

令f(t)=Amcos⁡(ωmt),那么y(t)=h(t)∗f(t)=H[f(t)]令f(t) =A_{m} \cos \left(\omega_{m} t\right),那么y(t) = h(t)*f(t) = H[f(t)]令f(t)=Amcos(ωmt),那么y(t)=h(t)∗f(t)=H[f(t)]

所以y(t)=Amsin⁡(ωmt)所以y(t) = A_{m} \sin \left(\omega_{m} t\right)所以y(t)=Amsin(ωmt)

对于一个实信号x(t)，其希尔伯特变换为：

x~(t)=x(t)∗1πt\tilde{x}(t)=x(t) * \frac{1}{\pi t} x~(t)=x(t)∗πt1

性质：

相移−π2rad- \frac{\pi}{2} rad−2πrad,幅度不变。即理想−π2rad- \frac{\pi}{2} rad−2πrad全通相移器的频率响应特性定义为

1πt\frac{1}{\pi t}πt1傅里叶变换

这里需要修改一下，因为引用的文章好像有点点问题：

考虑符号函数的傅里叶变换：
sign⁡(t)⇔2jω\operatorname{sign}(t) \Leftrightarrow \frac{2}{j \omega} sign(t)⇔jω2
利用傅里叶变换的性质得：
2jt⇔2π⋅sign⁡(−ω)\frac{2}{j t} \Leftrightarrow 2 \pi \cdot \operatorname{sign}(- \omega) jt2⇔2π⋅sign(−ω)
运用线性性质，于是有：
1πt⇔−j⋅sign⁡(ω)\frac{1}{\pi t} \Leftrightarrow -j \cdot \operatorname{sign}(\omega) πt1⇔−j⋅sign(ω)

希尔伯特变换性质

x^(t)=x(t)∗1πt\hat{x}(t) = x(t) * \frac{1}{\pi t} \\[10pt]x^(t)=x(t)∗πt1

正变换

x^(t)=∫−∞∞x(τ)h(t−τ)dτ=1π∫−∞∞x(τ)t−τdτ\hat{x}(t) =\int_{-\infty}^{\infty} x(\tau) h(t-\tau) d \tau=\frac{1}{\pi} \int_{-\infty}^{\infty} \frac{x(\tau)}{t-\tau} d \tau x^(t)=∫−∞∞x(τ)h(t−τ)dτ=π1∫−∞∞t−τx(τ)dτ

反变换

x(t)=H−1[x^(t)]=−1π∫−∞∞x^(τ)t−τdτx(t)=\mathrm{H}^{-1}[\hat{x}(t)]=-\frac{1}{\pi} \int_{-\infty}^{\infty} \frac{\hat{x}(\tau)}{t- \tau} d \tau x(t)=H−1[x^(t)]=−π1∫−∞∞t−τx^(τ)dτ

H[H[x(t)]]=−x(t)H−1[x(t)]=−H[x(t)]H[cos(ω0t)]=sin(ω0t)H[sin(ω0t)]=−cos(ω0t)H[x(t)∗cos(ω0t)]=x(t)∗sin(ω0t)H[x(t)∗sin(ω0t)]=−x(t)∗cos(ω0t)H[奇函数]=偶函数H[偶函数]=奇函数−j⋅sign⁡(ω)⋅−j⋅sign⁡(ω)=−1H[{H[x(t)]}] = -x(t) \\[4pt] H^{-1}[x(t)] = -H[x(t)] \\[4pt] H[cos(\omega_0 t)] = sin(\omega_0 t) \\[4pt] H[sin(\omega_0 t)] = -cos(\omega_0 t) \\[4pt] H[x(t)*cos(\omega_0 t)] = x(t)*sin(\omega_0 t) \\[4pt] H[x(t)*sin(\omega_0 t)] = -x(t)*cos(\omega_0 t) \\[4pt] H[奇函数] = 偶函数 \\[4pt] H[偶函数] = 奇函数 \\[4pt] -j \cdot \operatorname{sign}(\omega) \cdot -j \cdot \operatorname{sign}(\omega) = -1 H[H[x(t)]]=−x(t)H−1[x(t)]=−H[x(t)]H[cos(ω0t)]=sin(ω0t)H[sin(ω0t)]=−cos(ω0t)H[x(t)∗cos(ω0t)]=x(t)∗sin(ω0t)H[x(t)∗sin(ω0t)]=−x(t)∗cos(ω0t)H[奇函数]=偶函数H[偶函数]=奇函数−j⋅sign(ω)⋅−j⋅sign(ω)=−1

常用希尔伯特变换对：

f(t)f^(t)cos⁡ω0tsin⁡ω0tsin⁡ω0t−cos⁡ω0tejω0t−jejω0tm(t)ejω0t−jm(t)ejω0t\begin{array}{|l|l|} \hline \boldsymbol{f}(\boldsymbol{t}) & \hat{\boldsymbol{f}}(\boldsymbol{t}) \\ \hline \cos \omega_{0} t & \sin \omega_{0} t \\ \hline \sin \omega_{0} t & -\cos \omega_{0} t \\ \hline \mathrm{e}^{\mathrm{j} \omega_{0} t} & -\mathrm{j} \mathrm{e}^{\mathrm{j} \omega_{0} t} \\ \hline m(t) \mathrm{e}^{\mathrm{j} \omega_{0} t} & -\mathrm{j} m(t) \mathrm{e}^{\mathrm{j} \omega_{0} t} \\ \hline \end{array} f(t)cosω0tsinω0tejω0tm(t)ejω0tf^(t)sinω0t−cosω0t−jejω0t−jm(t)ejω0t

参考：

[1] https://blog.csdn.net/qq_37083038/article/details/108308162
[2] https://www.cnblogs.com/xingshansi/p/6498913.html
[3] https://www.cnblogs.com/xingshansi/p/6904215.html