信号与系统
信号分类
卷积
1.与单位冲激信号和单位阶跃信号的卷积
①[math]\displaystyle{ f\left ( t \right ) *\delta \left ( t \right ) =f\left ( t \right ) }[/math]
②[math]\displaystyle{ f\left ( t \right ) *\delta \left ( t -t_{0} \right ) =f\left ( t -t_{0} \right ) }[/math]
③[math]\displaystyle{ f\left ( t\right ) *\delta '\left ( t \right )=f'\left ( t \right ) }[/math]
④[math]\displaystyle{ f\left ( t\right ) *\delta ^{\left ( n \right ) } \left ( t \right )=f^{\left ( n \right ) } \left ( t \right ) }[/math]
⑤[math]\displaystyle{ f\left ( t\right ) *\varepsilon \left ( t \right ) =\int_{-\infty}^{t} f\left ( \tau \right )d\tau }[/math]
⑥[math]\displaystyle{ \varepsilon \left ( t \right )*\varepsilon \left ( t \right )=t\varepsilon \left ( t \right )=r\left ( t \right ) }[/math]
⑦[math]\displaystyle{ f\left ( t \right )*r\left (t\right)=\int_{-\infty}^{t}\left [\int_{-\infty}^{k}f\left(\tau\right )d\tau\right ] dk }[/math]
2.与单位冲激序列和单位阶跃序列的卷积
①[math]\displaystyle{ f\left ( k \right ) *\delta \left ( k \right ) =f\left ( k \right ) }[/math]
②[math]\displaystyle{ f\left ( k \right ) *\delta \left ( k-k_{0} \right ) =f\left ( k-k_{0} \right ) }[/math]
③[math]\displaystyle{ f\left ( k \right ) *\varepsilon \left ( k \right ) =\sum_{m=0}^{+\infty} f\left ( k-m \right ) =\sum_{m=-\infty}^{n}f\left ( m \right ) }[/math]
傅里叶变换
常见信号的傅里叶变换
(1)冲激函数[math]\displaystyle{ \delta \left ( t \right ) }[/math]
[math]\displaystyle{ \mathscr{F}\left[\delta \left(t\right)\right]=\int_{-\infty}^{+\infty} \delta \left ( t \right ) e^{-j\omega t} dt=\int_{-\infty}^{+\infty} \delta \left ( t \right )=1 }[/math]
(2)冲击偶函数[math]\displaystyle{ \delta' \left ( t \right ) }[/math]
[math]\displaystyle{ \mathscr{F}\left[\delta' \left(t\right)\right]=\int_{-\infty}^{+\infty} \delta' \left ( t \right ) e^{-j\omega t} dt=-\frac{d}{dt}\left ( e^{-j\omega t}\right )\bigg|_{t=0} =j\omega }[/math]
(3)门函数[math]\displaystyle{ g_{\tau }\left(t\right) }[/math]
[math]\displaystyle{ \mathscr{F}\left [ g_{\tau }\left(t\right)\right] =\int_{-\infty}^{+\infty} g_{\tau }\left(t\right)e^{-j\omega t}dt=\int_{-\frac{\tau }{2} }^{\frac{\tau}{2}} e^{-j\omega t} dt=\frac{e^{-j\omega t} }{-j\omega} \bigg|^{\frac{\tau }{2} }_{-\frac{\tau }{2} }= \frac{e^{-j\omega \frac{\tau }{2} }-e^{j\omega \frac{\tau }{2} } }{-j\omega } = \frac{2}{\omega }\cdot \frac{e^{j\omega \frac{\tau }{2} }-e^{-j\omega \frac{\tau }{2} } }{2j} =\frac{2\sin \frac{\omega \tau }{2} }{\omega } =\tau Sa\left ( \frac{\omega \tau }{2} \right ) }[/math]
(4)单边指数信号[math]\displaystyle{ e^{-at} \varepsilon \left ( t \right ) }[/math](a为正实数)
[math]\displaystyle{ \mathscr{F}\left [ e^{-at} \varepsilon \left ( t \right ) \right ]=\int_{-\infty}^{+\infty} e^{-at} \varepsilon \left ( t \right )\cdot e^{-j\omega t} dt=\int_{0}^{+\infty}e^{-\left ( a+j\omega \right ) } dt=\frac{e^{-\left ( a+j\omega \right )t } }{-\left ( a+j\omega \right )}\bigg|^{+\infty}_{0}=\frac{1}{a+j\omega } =\frac{a-j\omega }{a^{2}+\omega ^{2} } }[/math]
[math]\displaystyle{ \left|F\left(\omega \right )\right |=\sqrt{\frac{1^{2} }{a^{2}+\omega^{2}}}=\frac{1}{\sqrt{{a^{2}+\omega^{2}}}} }[/math]
常用信号的傅里叶变换表
时域 | 频域 | |
冲激信号 | [math]\displaystyle{ \delta \left ( t \right ) }[/math] | [math]\displaystyle{ 1 }[/math] |
冲激偶信号 | [math]\displaystyle{ \delta '\left ( t \right ) }[/math] | [math]\displaystyle{ j\omega }[/math] |
阶跃信号 | [math]\displaystyle{ \varepsilon \left ( t \right ) }[/math] | [math]\displaystyle{ \pi \delta \left (\omega \right ) +\frac{1}{j\omega } }[/math] |
斜升信号 | [math]\displaystyle{ r\left ( t \right ) =t\varepsilon \left ( t \right ) }[/math] | [math]\displaystyle{ j\pi \delta'\left ( \omega \right ) -\frac{1}{\omega ^{2} } }[/math] |
常数 | [math]\displaystyle{ C(直流分量) }[/math] | [math]\displaystyle{ 2\pi C\delta \left ( \omega \right ) }[/math] |
门函数 | [math]\displaystyle{ g_{\tau} \left ( t \right ) }[/math] | [math]\displaystyle{ \tau Sa\left ( \frac{\omega \tau}{2} \right ) }[/math] |
符号函数 | [math]\displaystyle{ sgn\left ( t \right ) }[/math] | [math]\displaystyle{ \frac{2}{j\omega } }[/math] |
单边指数信号 | [math]\displaystyle{ e^{-at} \varepsilon \left ( t \right ) \;\;\left ( a\gt 0 \right ) }[/math] | [math]\displaystyle{ \frac{1}{a+j\omega } }[/math] |
双边指数信号 | [math]\displaystyle{ e^{-a\left | t \right | } \varepsilon \left ( t \right ) \;\;\left ( a\gt 0 \right ) }[/math] | [math]\displaystyle{ \frac{2a}{a^{^{2} } +\omega ^{2} } }[/math] |
复指数信号 | [math]\displaystyle{ e^{-j\omega _{0}t } }[/math] | [math]\displaystyle{ 2\pi \delta \left ( \omega +\omega _{0} \right ) }[/math] |
余弦信号 | [math]\displaystyle{ \cos \left ( w_{0}t \right ) }[/math] | [math]\displaystyle{ \pi \left [ \delta \left ( \omega +\omega _{0} \right )+\delta \left ( \omega -\omega _{0} \right ) \right ] }[/math] |
正弦信号 | [math]\displaystyle{ \sin \left ( w_{0}t \right ) }[/math] | [math]\displaystyle{ j\pi \left[\delta\left(\omega+\omega_{0}\right)-\delta\left(\omega -\omega _{0} \right)\right] }[/math] |
抽样信号 | [math]\displaystyle{ Sa\left ( at \right ) }[/math] | [math]\displaystyle{ \frac{\pi }{a} g_{2a} \left ( \omega \right ) }[/math] |
标准一次函数 | [math]\displaystyle{ t }[/math] | [math]\displaystyle{ j2\pi \delta '\left ( \omega \right ) }[/math] |
标准反比例函数 | [math]\displaystyle{ \frac{1}{t} }[/math] | [math]\displaystyle{ -j\pi sgn\left( \omega \right ) }[/math] |
余弦函数 | [math]\displaystyle{ \cos \left ( \omega_{0}t+\varphi \right ) }[/math] | [math]\displaystyle{ \pi \left [ \delta \left ( \omega +\omega _{0} \right ) e^{-j\varphi }+ \delta \left ( \omega -\omega _{0} \right )e^{j\varphi }\right ] }[/math] |
正弦函数 | [math]\displaystyle{ \sin \left ( \omega_{0}t+\varphi \right ) }[/math] | [math]\displaystyle{ j\pi \left [ \delta \left ( \omega +\omega _{0} \right ) e^{-j\varphi }-\delta \left ( \omega -\omega _{0} \right )e^{j\varphi }\right ] }[/math] |
冲激序列 | [math]\displaystyle{ \delta _{\tau }\left ( t \right ) =\sum_{n=-\infty}^{\infty} \delta \left ( t-nT_{1} \right ) }[/math] | [math]\displaystyle{ \omega _{1} \sum_{n=-\infty}^{\infty} \delta \left ( \omega -n\omega _{1} \right ) }[/math] |
拉普拉斯变换
常用信号的拉氏变换
(1)阶跃信号\varepsilon \left ( t \right )