查看“信号与系统”的源代码

== 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>f\left ( t \right ) *\delta \left ( t \right ) =f\left ( t \right ) </math>

②<math>f\left ( t \right ) *\delta \left ( t -t_{0} \right ) =f\left ( t -t_{0} \right )</math>

③<math>f\left ( t\right ) *\delta '\left ( t \right )=f'\left ( t \right )</math>

④<math>f\left ( t\right ) *\delta ^{\left ( n \right ) } \left ( t \right )=f^{\left ( n \right ) } \left ( 
t \right )</math>

⑤<math>f\left ( t\right ) *\varepsilon \left ( t \right ) =\int_{-\infty}^{t} f\left ( \tau  \right )d\tau</math>

⑥<math>\varepsilon \left ( t \right )*\varepsilon \left ( t \right )=t\varepsilon \left ( t \right )=r\left ( t \right )</math>

⑦<math>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>f\left ( k \right ) *\delta \left ( k \right ) =f\left ( k \right ) </math>

②<math>f\left ( k \right ) *\delta \left ( k-k_{0}  \right ) =f\left ( k-k_{0} \right ) </math>

③<math>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>\delta \left ( t \right ) </math>

<math>\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>\delta' \left ( t \right ) </math>

<math>\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>g_{\tau }\left(t\right)</math>

<math>\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>e^{-at} \varepsilon \left ( t \right ) </math>（a为正实数）

<math>\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>\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 ===
{| class="wikitable"
|-
|
|时域
|频域
|-
|冲激信号
|<math>\delta \left ( t \right ) </math>
|<math>1</math>
|-
|冲激偶信号
|<math>\delta '\left ( t \right ) </math>
|<math>j\omega </math>
|-
|阶跃信号
|<math>\varepsilon \left ( t \right ) </math>
|<math>\pi \delta \left (\omega \right ) +\frac{1}{j\omega } </math>
|-
|斜升信号
|<math>r\left ( t \right ) =t\varepsilon \left ( t \right ) </math>
|<math>j\pi \delta'\left ( \omega  \right ) -\frac{1}{\omega ^{2} } </math>
|-
|常数
|<math>C（直流分量）</math>
|<math>2\pi C\delta \left ( \omega  \right ) </math>
|-
|门函数
|<math>g_{\tau} \left ( t \right ) </math>
|<math>\tau Sa\left ( \frac{\omega \tau}{2}  \right ) </math>
|-
|符号函数
|<math>sgn\left ( t \right ) </math>
|<math>\frac{2}{j\omega } </math>
|-
|单边指数信号
|<math>e^{-at} \varepsilon \left ( t \right ) \;\;\left (  a>0 \right )</math>
|<math>\frac{1}{a+j\omega } </math>
|-
|双边指数信号
|<math>e^{-a\left | t \right | } \varepsilon \left ( t \right ) \;\;\left (  a>0 \right )</math>
|<math>\frac{2a}{a^{^{2} } +\omega ^{2} } </math>
|-
|复指数信号
|<math>e^{-j\omega _{0}t } </math>
|<math>2\pi \delta \left ( \omega +\omega _{0}  \right ) </math>
|-
|余弦信号
|<math>\cos \left ( w_{0}t  \right ) </math>
|<math>\pi \left [ \delta \left ( \omega +\omega _{0}  \right )+\delta \left ( \omega -\omega _{0}  \right )   \right ] </math>
|-
|正弦信号
|<math>\sin \left ( w_{0}t  \right ) </math>
|<math>j\pi \left[\delta\left(\omega+\omega_{0}\right)-\delta\left(\omega -\omega _{0} \right)\right]</math>
|-
|抽样信号
|<math>Sa\left ( at \right ) </math>
|<math>\frac{\pi }{a} g_{2a} \left ( \omega  \right ) </math>
|-
|标准一次函数
|<math>t</math>
|<math>j2\pi \delta '\left ( \omega  \right ) </math>
|-
|标准反比例函数
|<math>\frac{1}{t} </math>
|<math>-j\pi sgn\left( \omega  \right )</math>
|-
|余弦函数
|<math>\cos \left ( \omega_{0}t+\varphi  \right )</math>
|<math>\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>\sin \left ( \omega_{0}t+\varphi  \right )</math>
|<math>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>\delta _{\tau }\left ( t \right ) =\sum_{n=-\infty}^{\infty} \delta \left ( t-nT_{1} \right ) </math>
|<math>\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<math>\varepsilon \left ( t \right )</math>

<math>\mathscr{L}\left [ \varepsilon \left ( t \right )  \right ] =\int_{0}^{+\infty} \varepsilon \left ( t \right )e^{-st} dt=\int_{0}^{\infty} e^{-st} dt=\frac{e^{-st} }{-s} \bigg|^{\infty} _{0} =\frac{1}{s} </math>

=== 常用信号的拉普拉斯变换表 ===
{| class="wikitable"
|-
|
|时域
|s域
|-
|冲激信号
|<math>\delta \left ( t \right ) </math>
|1
|-
|阶跃信号
|<math>\varepsilon \left ( t \right ) </math>
|<math>\frac{1}{s} </math>
|-
|单边指数信号
|<math>e^{-at} \varepsilon \left ( t \right ) </math>
|<math>\frac{1}{s+a} </math>
|-
|延时冲激信号
|<math>\delta \left ( t-t_{0}  \right )</math>
|<math>e^{-st_{0} } </math>
|-
|冲激信号求导
|<math>\delta ^{\left ( n \right ) } \left ( t \right )</math>
|<math>s^{n} </math>
|-
|幂函数
|<math>t^{n} \varepsilon \left ( t \right )</math>
|<math>\frac{n!}{s^{n+1} } </math>
|-
|正弦信号
|<math>\sin t\;\varepsilon \left ( t \right )</math>
|<math>\frac{\omega }{s^{2+}\omega ^{2}  } </math>
|-
|余弦信号
|<math>\cos t\;\varepsilon \left ( t \right )</math>
|<math>\frac{s}{s^{2+}\omega ^{2}  } </math>
|-
|
|<math>t\sin t\;\varepsilon \left ( t \right )</math>
|<math>\frac{2\omega s}{\left ( s^{2}+\omega ^{2} \right )^{2}  } </math>
|-
|
|<math>t\cos t\;\varepsilon \left ( t \right )</math>
|<math>\frac{s^{2}-\omega ^{2}}{\left ( s^{2}+\omega ^{2}   \right )^{2}  } </math>
|-
|
|<math>\sum_{n=0}^{\infty} \delta \left ( t-nT \right ) </math>
|<math>\frac{1}{1-e^{-sT} } </math>