傅里叶变换

非周期函数

对于非周期函数，我们可以将其视为周期趋近于无穷大的周期函数。

不确定性原理 (Uncertainty Principle)

不确定性原理指出，一个信号的时域和频域的宽度满足以下关系：

$$\mathrm{\Delta }t\mathrm{\Delta }\omega \ge \frac{1}{2}$$

其中，$\mathrm{\Delta }t$ 是信号的时域宽度，$\mathrm{\Delta }\omega$ 是信号的频域宽度。

这个关系表明，一个信号的时域和频域的宽度不能同时任意小。

定义

正变换：

$$\boxed{{\displaystyle F(\omega )=\mathcal{F}[f(t)]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)e^{-j\omega t}\mathrm{d}t}}$$

逆变换：

$$\boxed{{\displaystyle f(t)={\mathcal{F}}^{-1}[F(\omega )]=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}F(\omega )e^{j\omega t}\mathrm{d}\omega }}$$

推导过程

考虑傅里叶级数的指数形式

$$f(t)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n{e}^{jn{\omega }_{0}t}$$

其中

$$c_n=\frac{1}{T}{\int }_{-T/2}^{T/2}f(t)e^{-jn{\omega }_{0}t}\mathrm{d}t$$

基频为

$${\omega }_0=\frac{2\pi }{T}$$

相邻频率的间隔为

$$\mathrm{\Delta }\omega =(n+1){\omega }_0-n{\omega }_0={\omega }_0=\frac{2\pi }{T}$$

于是有

$$\begin{array}{rl}f(t)& =\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}[\frac{1}{T}{\int }_{-T/2}^{T/2}f(t)e^{-jn{\omega }_{0}t}\mathrm{d}t]e^{jn{\omega }_{0}t}\\ & =\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}[\frac{\mathrm{\Delta }\omega }{2\pi }{\int }_{-T/2}^{T/2}f(t)e^{-jn{\omega }_{0}t}\mathrm{d}t]e^{jn{\omega }_{0}t}\\ & =\frac{1}{2\pi }\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}[{\int }_{-T/2}^{T/2}f(t)e^{-jn{\omega }_{0}t}\mathrm{d}t]e^{jn{\omega }_{0}t}\mathrm{\Delta }\omega \end{array}$$

如果令 $T\to \mathrm{\infty }$，则有

$$\begin{array}{rl}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}& \Rightarrow {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\\ \mathrm{\Delta }\omega & \Rightarrow \mathrm{d}\omega \\ n{\omega }_0& \Rightarrow \omega \end{array}$$

整理得到

$$f(t)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{{\displaystyle F(\omega )}}{\underbrace{[{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(\tau )e^{-j\omega \tau }\mathrm{d}\tau ]}}{e}^{j\omega t}\mathrm{d}\omega$$

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适用对象：非周期信号/能量有限信号（或满足狄利克雷条件的周期信号推广）

性质

Linearity

$$\mathcal{F}[a_1{f}_1(t)+a_2{f}_2(t)]=a_1{F}_1(\omega )+a_2{F}_2(\omega )$$

Time Scaling

$$\mathcal{F}[f(at)]=\frac{1}{|a|}F\left(\frac{\omega }{a}\right)$$

Time Shifting

$$\mathcal{F}[f(t-t_0)]=e^{-j\omega t_0}F(\omega )$$

Frequency Shifting

(or Amplitude Modulation)

$$\mathcal{F}[f(t)e^{j{\omega }_{0}t}]=F(\omega -{\omega }_0)$$

Time Differentiation

$$\mathcal{F}[\frac{\mathrm{d}}{\mathrm{d}t}f(t)]=j\omega F(\omega )$$

and

$$\mathcal{F}[\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}f(t)]=(j\omega {)}^{n}F(\omega )$$

Time Integration

$$\mathcal{F}[{\int }_{-\mathrm{\infty }}^{t}f(\tau )\mathrm{d}\tau ]=\frac{F(\omega )}{j\omega }+\pi F(0)\delta (\omega )$$

Reversal

$$\mathcal{F}[f(-t)]=F(-\omega )=F^{\ast }(\omega )$$

Duality

$$\mathcal{F}[f(t)]=F(\omega )\Rightarrow \mathcal{F}[F(t)]=2\pi f(-\omega )$$

Convolution

Recall:

$$y(t)=h(t)\ast x(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}h(\tau )x(t-\tau )\mathrm{d}\tau$$

If $X(\omega )$, $H(\omega )$, and $Y(\omega )$ are the Fourier transforms of $x(t)$, $h(t)$, and $y(t)$, respectively, then:

$$Y(\omega )=\mathcal{F}[h(t)\ast x(t)]=H(\omega )X(\omega )$$

Parseval's Theorem

Parseval's theorem states that the total energy delivered to a 1-ohm resistor equals the total area under the squared of f(t) or 1/2π times the total area under the squared of the magnitude of the Fourier transform of f(t).

$$\boxed{{\displaystyle W_{1\mathrm{\Omega }}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}^2(t)\mathrm{d}t=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|F(\omega )|}^2\mathrm{d}\omega }}$$

推导过程

$$\begin{array}{rl}{W}_{1\mathrm{\Omega }}& ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}^2(t)\mathrm{d}t\\ & ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)[\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}F(\omega )e^{j\omega t}\mathrm{d}\omega ]\mathrm{d}t\\ & =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)F(\omega )e^{j\omega t}\mathrm{d}\omega \mathrm{d}t\\ & =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}F(\omega )[{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)e^{-j(-\omega )t}\mathrm{d}t]\mathrm{d}\omega \\ & =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}F(\omega )F(-\omega )\mathrm{d}\omega \\ & =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}F(\omega )F^{\ast }(\omega )\mathrm{d}\omega \\ & =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|F(\omega )|}^2\mathrm{d}\omega \end{array}$$

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由于 $|F(\omega ){|}^2$ 是偶函数，所以有

$$W_{1\mathrm{\Omega }}=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}{|F(\omega )|}^2\mathrm{d}\omega$$

也可以计算任意频带 ${\omega }_1<\omega <{\omega }_2$ （频率为正）内的能量

$$W_{1\mathrm{\Omega }}=\frac{1}{\pi }{\int }_{{\omega }_1}^{{\omega }_2}{|F(\omega )|}^2\mathrm{d}\omega$$

查表法