信号与系统：修订间差异
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== 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>
3.一些卷积积分
①<math>\varepsilon \left ( t \right ) \ast \varepsilon \left ( t \right ) =t\varepsilon \left ( t \right )=r\left ( t \right ) </math>
②<math>e^{at}\varepsilon \left ( t \right ) *\varepsilon \left ( t \right ) =\frac{1}{a} \left ( e^{at}-1 \right )\varepsilon \left ( t \right ) </math>
③<math>e^{at}\varepsilon \left ( t \right ) *e^{at}\varepsilon \left ( t \right ) =te^{at}\varepsilon \left ( t \right ) </math>
④<math>e^{a_{1} t}\varepsilon \left ( t \right ) *e^{a_{2}t}\varepsilon \left ( t \right )=\frac{1}{a_{1}-a_{2}} \left ( e^{a_{1}t }-e^{a_{2}t}\right ) \varepsilon \left ( t \right ) </math>
4.一些卷积和
①<math>\varepsilon \left ( k \right ) \ast \varepsilon \left ( k \right ) =\left ( k+1 \right ) \varepsilon \left ( k \right )</math>
②<math>a^{k}\varepsilon \left ( k \right )*a^{k}\varepsilon \left ( k \right )=\left ( k+1 \right )a^{k}\varepsilon \left ( k \right )</math>
③<math>a^{k}\varepsilon \left ( k \right )*\varepsilon \left ( k \right )=\frac{1-a^{k+1} }{1-a} </math>
④<math>a_{1}^{k}\varepsilon \left ( k \right )*a_{2}^{k}\varepsilon \left ( k \right )=\frac{a_{2}^{k+1}-a_{1}^{k+1}}{a_{2}-a_{1}} \varepsilon \left ( k \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"
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|<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>
|}
== z变换 ==
=== 常用序列的z变换表 ===
{| class="wikitable"
|-
|
|时域
|z区域
|-
|冲激序列
|<math>\delta \left ( k \right ) </math>
|<math>1</math>
|-
|阶跃序列
|<math>\varepsilon \left ( k \right ) </math>
|<math>\frac{z}{z-1}\;\;\;\left | z \right | >1</math>
|-
|幂函数序列
|<math>a^{k} \varepsilon \left ( k \right ) </math>
|<math>\frac{z}{z-a} \;\;\;\left | z \right | > \left | a \right | </math>
|-
|斜升序列
|<math>k\varepsilon \left ( k \right ) </math>
|<math>\frac{z}{\left ( z-1 \right )^{2} } \;\;\;\left | z \right | >1</math>
|-
|
|<math>\frac{k\left ( k-1 \right ) }{2} \varepsilon \left ( k \right ) </math>
|<math>\frac{z}{\left ( z-1 \right )^{3} }\;\;\;\left | z \right | >1 </math>
|-
|
|<math>\frac{k\left ( k-1 \right )\left ( k-2 \right ) }{3!} \varepsilon \left ( k \right )</math>
|<math>\frac{z}{\left ( z-1 \right )^{4} }\;\;\;\left | z \right | >1</math>
|-
|
|<math>\frac{k\left ( k-1 \right )\cdots \left ( k-m+1 \right ) }{m!} \varepsilon \left ( k \right )</math>
|<math>\frac{z}{\left ( z-1 \right )^{m+1} }\;\;\;\left | z \right | >1</math>
|-
|
|<math>\frac{1}{a^{m} } \cdot a^{k }\cdot \frac{k\left ( k-1 \right )\cdots \left ( k-m+1 \right ) }{m!} \varepsilon \left ( k \right )</math>
|<math>\frac{z}{\left ( z-a \right )^{m+1} }\;\;\;\left | z \right | > \left | a \right |</math>
|-
|指数序列
|<math>e^{jk\omega _{0} } \varepsilon \left ( k \right )</math>
|<math>\frac{z}{z-e^{j\omega _{0} } }\;\;\;\left | z \right | >1</math>
|-
|正弦序列
|<math>\sin \left ( \omega _{0} k \right ) \varepsilon \left ( k \right )</math>
|<math>\frac{2\sin \omega _{0} }{z^{2}-2z\cos \omega _{0}+1 } \;\;\;\left | z \right | >1</math>
|-
|余弦序列
|<math>\cos \left ( \omega _{0} k \right ) \varepsilon \left ( k \right )</math>
|<math>\frac{z^{2}-z\cos \omega _{0} }{z^{2}-2z\cos \omega _{0}+1 } \;\;\;\left | z \right | >1</math>
|-
|
|<math>-\varepsilon \left ( -k -1\right )</math>
|<math>\frac{z}{z-1}\;\;\;\left | z \right | <1</math>
|-
|
|<math>-k\varepsilon \left ( -k -1\right )</math>
|<math>\frac{z}{\left ( z-1 \right )^{2} } \;\;\;\left | z \right | <1</math>
|}
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