信号公式汇总之傅里叶变换

傅里叶级数
$$f(t)=a_0+\sum _{n=1}^{\infty }[a_{n}cos(n{\omega }_{1}t)+b_{n}sin(n{\omega }_{1}t)]$$

其中：$a_0=\frac{1}{T_1}{\int }_{t_0}^{t_0+T_1}f(t)dt$

      $a_n=\frac{2}{T_1}{\int }_{t_0}^{t_0+T_1}f(t)\cdot cos(n{\omega }_{1}t)dt$

      $b_n=\frac{2}{T_1}{\int }_{t_0}^{t_0+T_1}f(t)\cdot sin(n{\omega }_{1}t)dt$

傅里叶系数推导

合并同频项：$f(t)=A_0+{\sum }_{n=1}^{\infty }{A}_{n}sin(n{\omega }_{1}t+{\phi }_n)$

其中：$A_0=a_0,A_n=\sqrt{a_n^2+b_n^2},{\phi }_n=arctan\frac{a_n}{b_n}$

指数形式：$f(t)={\sum }_{n=-\infty }^{\infty }{F}_n\cdot e^{jn{\omega }_{1}t}$

其中：$F_n=\frac{1}{T_1}{\int }_{t_0}^{t_0+T_1}f(t)e^{-jn{\omega }_{1}t}dt=$

$\frac{1}{2}(a_n-j{b}_n)=\frac{1}{2}\sqrt{a_n^2+b_n^2}{e}^{j{\phi }_n},{\phi }_n=arctan(-\frac{b_n}{a_n})$

帕塞瓦尔定理：$P={\sum }_{n=-\infty }^{\infty }|F_n{|}^2$

傅里叶变换
正变换：$F(j\omega )=\mathcal{F}[f(t)]={\int }_{-\infty }^{\infty }f(t)e^{-j\omega t}dt$

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


基本信号的傅里叶变换：

性质：

线性：  $a_1{f}_1(t)+a_2{f}_2(t)$     $\leftarrow \to$   $a_1{F}_1(\omega )+a_2{F}_2(\omega )$

对称性：  $F(t)$     $\leftarrow \to$   $2\pi f(-\omega )$

尺度变换：  $f(at)$     $\leftarrow \to$   $\frac{1}{|a|}F(j\frac{\omega }{a})$

虚实特性：  $F(\omega )=R(\omega )+jX(\omega ),R(\omega )偶,X(\omega )奇$     
            $f(t)实偶，F(\omega )实偶；f(t)实奇，F(\omega )虚奇$

位移：时移：$f(t-t_0)$$\leftarrow \to$ $F(j\omega )e^{-j\omega t_0}=>$ $\delta (t-t_0)$$\leftarrow \to$ $e^{-j\omega t_0}$
      频移：$f(t)e^{j{\omega }_{0}t}$ $\leftarrow \to$ $F[j(\omega -{\omega }_0)]=>$ $e^{j{\omega }_{0}t}$ $\leftarrow \to$ $2\pi \delta (\omega -{\omega }_0)$

==>$\{\begin{array}{l}cos{\omega }_{0}t=\frac{1}{2}(e^{j\omega t}+e^{-j\omega t})\\ sin{\omega }_{0}t=\frac{1}{2j}(e^{j\omega t}-e^{j\omega t})\end{array}=>\{\begin{array}{l}f(t)cos({\omega }_{0}t)\leftarrow \to \frac{1}{2}[F(\omega -{\omega }_0)+F(\omega +{\omega }_0)]\\ f(t)sin({\omega }_{0}t)\leftarrow \to \frac{1}{2j}[F(\omega -{\omega }_0)+F(\omega +{\omega }_0)]\end{array}$

==>$\{\begin{array}{l}cos{\omega }_{0}t\leftarrow \to \pi [\delta (\omega -{\omega }_0)+\delta (\omega +{\omega }_0))]\\ sin{\omega }_{0}t\leftarrow \to \frac{\pi }{j}[\delta (\omega -{\omega }_0)-\delta (\omega +{\omega }_0))]\end{array}$

$调制定理：若信号f(t)乘以cos{\omega }_{0}t或sin{\omega }_{0}t，等效于f(t)的频谱一分为二，沿数轴向左或向右各平移{\omega }_0$

推导：

$f(t)\cdot Cos({\omega }_{0}t)$
傅里叶变换：
$\frac{1}{2\pi }F(j\omega )\ast \pi [\delta (\omega +{\omega }_0)+\delta (\omega -{\omega }_0))]$
=$\frac{1}{2}F(j\omega )\ast \delta (\omega +{\omega }_0)+\frac{1}{2}F(j\omega )\ast \delta (\omega -{\omega }_0)$
由冲激函数性质 $f(t)\ast \delta (t-t_0)=f(t-t_0)$
=$\frac{1}{2}F(j(\omega +{\omega }_0))+\frac{1}{2}F(j(\omega -{\omega }_0)$

卷积：时域卷积：$f_1(t)\ast f_2(t)\leftarrow \to F_1(\omega )\cdot F_2(\omega )$

      频域卷积：$f_1(t)\cdot f_2(t)\leftarrow \to \frac{1}{2\pi }{F}_1(\omega )\ast F_2(\omega )$

微积分：时域：$\{\begin{array}{l}微分:\frac{df(t)}{dt}\leftarrow \to j\omega F(j\omega )\\ 积分:{\int }_{-\infty }^{t}f(\tau )d\tau \leftarrow \to \frac{1}{j\omega }F(j\omega )+\pi F(0)\delta (ik\omega )\end{array}$

        频域：$\{\begin{array}{l}微分:-jtf(t)\leftarrow \to \frac{dF(j\omega )}{d\omega }\\ 积分:\frac{f(t)}{-jt}\leftarrow \to {\int }_{-\infty }^{\omega }F(x)dx\end{array}$

欧拉公式:
$$e^{j\theta }=cos\theta +jsin\theta$$

$$==>\{\begin{array}{l}{e}^{j\omega t}=cos\omega t+jsin\omega t\\ e^{-j\omega t}=cos\omega t-jsin\omega t\end{array}$$

$$==>\{\begin{array}{l}cos(n\omega t)=\frac{e^{jn\omega t}+e^{-jn\omega t}}{2}\\ sin(n\omega t)=\frac{e^{jn\omega t}-e^{-jn\omega t}}{2j}\end{array}$$

阶跃信号和冲激信号:

冲激函数：
   加权特性：$f(t)\delta (t)=f(0)\delta (t),f(t)\delta (t-t_0)=f(t_0)\delta (t-t_0)$
   抽样特性：${\int }_{-\infty }^{\infty }f(t)\delta (t)dt={\int }_{-\infty }^{\infty }f(0)\delta (t)dt=f(0)$
          ${\int }_{-\infty }^{\infty }f(t)\delta (t-t_0)dt=f(t_0)$
   尺度变换：$\delta (at)=\frac{1}{|a|}\delta (t),\delta (at-t_0)=\frac{1}{|a|}\delta (t-\frac{t_0}{a})$

冲激偶：
   抽样特性：${\int }_{-\infty }^{\infty }f(t)\delta \prime (t)dt={f(t)\delta (t)|}_{-\infty }^{\infty }-{\int }_{-\infty }^{\infty }\delta (t)f\prime (t)dt=$$-{\int }_{-\infty }^{\infty }\delta (t)f\prime (0)dt=-f\prime (0)$
   加权特性：$f(t)\delta \prime (t)=f(0)\delta \prime (t)-f\prime (0)\delta (t)$
             $f(t)\delta \prime (t-t_0)=f(t_0)\delta \prime (t-t_0)-f\prime (t_0)\delta (t-t_0)$

卷积运算:$f(t)=f_1(t)\ast f_2(t)={\int }_{-\infty }^{\infty }{f}_1(\tau )f_2(t-\tau )d\tau$

性质：微分：$f(t)=f_1\prime (t)\ast f_2(t)=f_1(t)\ast f_2\prime (t)$
      积分：$f^{-1}(t)=f_1^{-1}(t)\ast f_2(t)=f_1(t)\ast f_2^{-1}(t)$
      微积分：$f(t)=f_1^{-1}(t)\ast f_2\prime (t)=f_1(t)\ast f_2(t)$
      =======>$f^{(i)}(t)=f_1^{(j)}(t)\ast f_2^{(i-j)}(t)$
      交换律：$f_1(t)\ast f_2(t)=f_2(t)\ast f_1(t)$
      分配律：$f_1(t)\ast [f_2(t)+f_3(t)]=f_1(t)\ast f_2(t)+f_1(t)\ast f_3(t)$
       $==>$$并联系统h(t)=h_1(t)+h_2(t)$
      结合律：$[f_1(t)\ast f_2(t)]\ast f_3(t)=f_1(t)\ast [f_2(t)\ast f_3(t)]$
       $==>$$串联系统h(t)=h_1(t)\ast h_2(t)$

与冲激函数、冲激偶、阶跃函数
      冲激函数$\delta (t)$：$f(t)\ast \delta (t)=f(t)$
                      $f(t)\ast \delta (t-t_0)=f(t-t_0)$
                           $f(t-t_1)\ast \delta (t-t_2)=f(t-t_2)\ast \delta (t-t_1)=f(t-t_1-t_2)$
      冲激偶$\delta \prime (t)$：$f(t)\ast \delta \prime (t)=f\prime (t)\ast \delta (t)=f\prime (t)$
                  $f(t)\ast \delta \prime \prime (t)=f\prime \prime (t)\ast \delta (t)=f\prime \prime (t)$
      阶跃函数$\epsilon (t)$：$f(t)\ast \epsilon (t)=f(t)\ast {\delta }^{-1}(t)=f^{-1}(t)={\int }_{-\infty }^{t}f(\tau )d\tau$
                  $f(t)\ast \epsilon (t-t_0)={\int }_{-\infty }^{t}f(\tau -{\tau }_0)d\tau ={\int }_{-\infty }^{t-t_0}f(\tau )d\tau$
（与阶跃函数卷积就是变上限积分，阶跃函数是个理想的积分器）

时移特性：$f_1(t-t_1)\ast f_2(t-t_2)=f(t-t_1-t_2)$