信号与系统：修订间差异
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== 信号分类 == |
== 信号分类 == |
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== 卷积 == |
== 卷积 == |
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1.与单位冲激信号和单位阶跃信号的卷积 |
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①<math>f\left ( t \right ) *\delta \left ( t \right ) =f\left ( t \right ) </math> |
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②<math>f\left ( t \right ) *\delta \left ( t -t_{0} \right ) =f\left ( t -t_{0} \right )</math> |
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③<math>f\left ( t\right ) *\delta '\left ( t \right )=f'\left ( t \right )</math> |
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④<math>f\left ( t\right ) *\delta ^{\left ( n \right ) } \left ( t \right )=f^{\left ( n \right ) } \left ( |
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t \right )</math> |
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⑤<math>f\left ( t\right ) *\varepsilon \left ( t \right ) =\int_{-\infty}^{t} f\left ( \tau \right )d\tau</math> |
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⑥<math>\varepsilon \left ( t \right )*\varepsilon \left ( t \right )=t\varepsilon \left ( t \right )=r\left ( t \right )</math> |
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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> |
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2.与单位冲激序列和单位阶跃序列的卷积 |
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①<math>f\left ( k \right ) *\delta \left ( k \right ) =f\left ( k \right ) </math> |
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②<math>f\left ( k \right ) *\delta \left ( k-k_{0} \right ) =f\left ( k-k_{0} \right ) </math> |
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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> |
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3.一些卷积积分 |
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①<math>\varepsilon \left ( t \right ) \ast \varepsilon \left ( t \right ) =t\varepsilon \left ( t \right )=r\left ( t \right ) </math> |
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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> |
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③<math>e^{at}\varepsilon \left ( t \right ) *e^{at}\varepsilon \left ( t \right ) =te^{at}\varepsilon \left ( t \right ) </math> |
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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> |
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4.一些卷积和 |
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①<math>\varepsilon \left ( k \right ) \ast \varepsilon \left ( k \right ) =\left ( k+1 \right ) \varepsilon \left ( k \right )</math> |
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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> |
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③<math>a^{k}\varepsilon \left ( k \right )*\varepsilon \left ( k \right )=\frac{1-a^{k+1} }{1-a} </math> |
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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> |
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== 傅里叶变换 == |
== 傅里叶变换 == |
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=== 常见信号的傅里叶变换 === |
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（1）冲激函数<math>\delta \left ( t \right ) </math> |
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<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> |
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（2）冲击偶函数<math>\delta' \left ( t \right ) </math> |
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<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> |
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（3）门函数<math>g_{\tau }\left(t\right)</math> |
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<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> |
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（4）单边指数信号<math>e^{-at} \varepsilon \left ( t \right ) </math>（a为正实数） |
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<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> |
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<math>\left|F\left(\omega \right )\right |=\sqrt{\frac{1^{2} }{a^{2}+\omega^{2}}}=\frac{1}{\sqrt{{a^{2}+\omega^{2}}}} </math> |
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=== 常用信号的傅里叶变换表 === |
=== 常用信号的傅里叶变换表 === |
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{| class="wikitable" |
{| class="wikitable" |
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|<math>\delta _{\tau }\left ( t \right ) =\sum_{n=-\infty}^{\infty} \delta \left ( t-nT_{1} \right ) </math> |
|<math>\delta _{\tau }\left ( t \right ) =\sum_{n=-\infty}^{\infty} \delta \left ( t-nT_{1} \right ) </math> |
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|<math>\omega _{1} \sum_{n=-\infty}^{\infty} \delta \left ( \omega -n\omega _{1} \right ) </math> |
|<math>\omega _{1} \sum_{n=-\infty}^{\infty} \delta \left ( \omega -n\omega _{1} \right ) </math> |
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|} |
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== 拉普拉斯变换 == |
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=== 常用信号的拉氏变换 === |
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（1）阶跃信号<math>\varepsilon \left ( t \right )</math> |
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<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> |
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=== 常用信号的拉普拉斯变换表 === |
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{| class="wikitable" |
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|- |
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| |
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|时域 |
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|s域 |
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|- |
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|冲激信号 |
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|<math>\delta \left ( t \right ) </math> |
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|1 |
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|- |
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|阶跃信号 |
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|<math>\varepsilon \left ( t \right ) </math> |
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|<math>\frac{1}{s} </math> |
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|- |
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|单边指数信号 |
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|<math>e^{-at} \varepsilon \left ( t \right ) </math> |
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|<math>\frac{1}{s+a} </math> |
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|- |
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|延时冲激信号 |
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|<math>\delta \left ( t-t_{0} \right )</math> |
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|<math>e^{-st_{0} } </math> |
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|- |
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|冲激信号求导 |
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|<math>\delta ^{\left ( n \right ) } \left ( t \right )</math> |
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|<math>s^{n} </math> |
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|- |
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|幂函数 |
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|<math>t^{n} \varepsilon \left ( t \right )</math> |
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|<math>\frac{n!}{s^{n+1} } </math> |
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|- |
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|正弦信号 |
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|<math>\sin t\;\varepsilon \left ( t \right )</math> |
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|<math>\frac{\omega }{s^{2+}\omega ^{2} } </math> |
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|- |
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|余弦信号 |
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|<math>\cos t\;\varepsilon \left ( t \right )</math> |
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|<math>\frac{s}{s^{2+}\omega ^{2} } </math> |
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|- |
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| |
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|<math>t\sin t\;\varepsilon \left ( t \right )</math> |
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|<math>\frac{2\omega s}{\left ( s^{2}+\omega ^{2} \right )^{2} } </math> |
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|- |
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| |
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|<math>t\cos t\;\varepsilon \left ( t \right )</math> |
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|<math>\frac{s^{2}-\omega ^{2}}{\left ( s^{2}+\omega ^{2} \right )^{2} } </math> |
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|- |
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| |
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|<math>\sum_{n=0}^{\infty} \delta \left ( t-nT \right ) </math> |
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|<math>\frac{1}{1-e^{-sT} } </math> |
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|} |
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== z变换 == |
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=== 常用序列的z变换表 === |
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{| class="wikitable" |
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|- |
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| |
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|时域 |
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|z区域 |
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|- |
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|冲激序列 |
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|<math>\delta \left ( k \right ) </math> |
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|<math>1</math> |
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|- |
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|阶跃序列 |
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|<math>\varepsilon \left ( k \right ) </math> |
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|<math>\frac{z}{z-1}\;\;\;\left | z \right | >1</math> |
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|- |
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|幂函数序列 |
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|<math>a^{k} \varepsilon \left ( k \right ) </math> |
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|<math>\frac{z}{z-a} \;\;\;\left | z \right | > \left | a \right | </math> |
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|- |
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|斜升序列 |
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|<math>k\varepsilon \left ( k \right ) </math> |
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|<math>\frac{z}{\left ( z-1 \right )^{2} } \;\;\;\left | z \right | >1</math> |
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|- |
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| |
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|<math>\frac{k\left ( k-1 \right ) }{2} \varepsilon \left ( k \right ) </math> |
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|<math>\frac{z}{\left ( z-1 \right )^{3} }\;\;\;\left | z \right | >1 </math> |
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|- |
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| |
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|<math>\frac{k\left ( k-1 \right )\left ( k-2 \right ) }{3!} \varepsilon \left ( k \right )</math> |
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|<math>\frac{z}{\left ( z-1 \right )^{4} }\;\;\;\left | z \right | >1</math> |
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|- |
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| |
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|<math>\frac{k\left ( k-1 \right )\cdots \left ( k-m+1 \right ) }{m!} \varepsilon \left ( k \right )</math> |
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|<math>\frac{z}{\left ( z-1 \right )^{m+1} }\;\;\;\left | z \right | >1</math> |
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|- |
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| |
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|<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> |
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|<math>\frac{z}{\left ( z-a \right )^{m+1} }\;\;\;\left | z \right | > \left | a \right |</math> |
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|- |
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|指数序列 |
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|<math>e^{jk\omega _{0} } \varepsilon \left ( k \right )</math> |
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|<math>\frac{z}{z-e^{j\omega _{0} } }\;\;\;\left | z \right | >1</math> |
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|- |
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|正弦序列 |
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|<math>\sin \left ( \omega _{0} k \right ) \varepsilon \left ( k \right )</math> |
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|<math>\frac{2\sin \omega _{0} }{z^{2}-2z\cos \omega _{0}+1 } \;\;\;\left | z \right | >1</math> |
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|- |
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|余弦序列 |
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|<math>\cos \left ( \omega _{0} k \right ) \varepsilon \left ( k \right )</math> |
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|<math>\frac{z^{2}-z\cos \omega _{0} }{z^{2}-2z\cos \omega _{0}+1 } \;\;\;\left | z \right | >1</math> |
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|- |
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| |
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|<math>-\varepsilon \left ( -k -1\right )</math> |
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|<math>\frac{z}{z-1}\;\;\;\left | z \right | <1</math> |
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|- |
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| |
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|<math>-k\varepsilon \left ( -k -1\right )</math> |
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|<math>\frac{z}{\left ( z-1 \right )^{2} } \;\;\;\left | z \right | <1</math> |
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|} |
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2024年12月3日 (二) 22:14的最新版本
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]
3.一些卷积积分
①[math]\displaystyle{ \varepsilon \left ( t \right ) \ast \varepsilon \left ( t \right ) =t\varepsilon \left ( t \right )=r\left ( t \right ) }[/math]
②[math]\displaystyle{ 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]\displaystyle{ e^{at}\varepsilon \left ( t \right ) *e^{at}\varepsilon \left ( t \right ) =te^{at}\varepsilon \left ( t \right ) }[/math]
④[math]\displaystyle{ 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]\displaystyle{ \varepsilon \left ( k \right ) \ast \varepsilon \left ( k \right ) =\left ( k+1 \right ) \varepsilon \left ( k \right ) }[/math]
②[math]\displaystyle{ 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]\displaystyle{ a^{k}\varepsilon \left ( k \right )*\varepsilon \left ( k \right )=\frac{1-a^{k+1} }{1-a} }[/math]
④[math]\displaystyle{ 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]\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[math]\displaystyle{ \varepsilon \left ( t \right ) }[/math]
[math]\displaystyle{ \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
| 时域 | s域 | |
| 冲激信号 | [math]\displaystyle{ \delta \left ( t \right ) }[/math] | 1 |
| 阶跃信号 | [math]\displaystyle{ \varepsilon \left ( t \right ) }[/math] | [math]\displaystyle{ \frac{1}{s} }[/math] |
| 单边指数信号 | [math]\displaystyle{ e^{-at} \varepsilon \left ( t \right ) }[/math] | [math]\displaystyle{ \frac{1}{s+a} }[/math] |
| 延时冲激信号 | [math]\displaystyle{ \delta \left ( t-t_{0} \right ) }[/math] | [math]\displaystyle{ e^{-st_{0} } }[/math] |
| 冲激信号求导 | [math]\displaystyle{ \delta ^{\left ( n \right ) } \left ( t \right ) }[/math] | [math]\displaystyle{ s^{n} }[/math] |
| 幂函数 | [math]\displaystyle{ t^{n} \varepsilon \left ( t \right ) }[/math] | [math]\displaystyle{ \frac{n!}{s^{n+1} } }[/math] |
| 正弦信号 | [math]\displaystyle{ \sin t\;\varepsilon \left ( t \right ) }[/math] | [math]\displaystyle{ \frac{\omega }{s^{2+}\omega ^{2} } }[/math] |
| 余弦信号 | [math]\displaystyle{ \cos t\;\varepsilon \left ( t \right ) }[/math] | [math]\displaystyle{ \frac{s}{s^{2+}\omega ^{2} } }[/math] |
| [math]\displaystyle{ t\sin t\;\varepsilon \left ( t \right ) }[/math] | [math]\displaystyle{ \frac{2\omega s}{\left ( s^{2}+\omega ^{2} \right )^{2} } }[/math] | |
| [math]\displaystyle{ t\cos t\;\varepsilon \left ( t \right ) }[/math] | [math]\displaystyle{ \frac{s^{2}-\omega ^{2}}{\left ( s^{2}+\omega ^{2} \right )^{2} } }[/math] | |
| [math]\displaystyle{ \sum_{n=0}^{\infty} \delta \left ( t-nT \right ) }[/math] | [math]\displaystyle{ \frac{1}{1-e^{-sT} } }[/math] |
z变换
常用序列的z变换表
| 时域 | z区域 | |
| 冲激序列 | [math]\displaystyle{ \delta \left ( k \right ) }[/math] | [math]\displaystyle{ 1 }[/math] |
| 阶跃序列 | [math]\displaystyle{ \varepsilon \left ( k \right ) }[/math] | [math]\displaystyle{ \frac{z}{z-1}\;\;\;\left | z \right | \gt 1 }[/math] |
| 幂函数序列 | [math]\displaystyle{ a^{k} \varepsilon \left ( k \right ) }[/math] | [math]\displaystyle{ \frac{z}{z-a} \;\;\;\left | z \right | \gt \left | a \right | }[/math] |
| 斜升序列 | [math]\displaystyle{ k\varepsilon \left ( k \right ) }[/math] | [math]\displaystyle{ \frac{z}{\left ( z-1 \right )^{2} } \;\;\;\left | z \right | \gt 1 }[/math] |
| [math]\displaystyle{ \frac{k\left ( k-1 \right ) }{2} \varepsilon \left ( k \right ) }[/math] | [math]\displaystyle{ \frac{z}{\left ( z-1 \right )^{3} }\;\;\;\left | z \right | \gt 1 }[/math] | |
| [math]\displaystyle{ \frac{k\left ( k-1 \right )\left ( k-2 \right ) }{3!} \varepsilon \left ( k \right ) }[/math] | [math]\displaystyle{ \frac{z}{\left ( z-1 \right )^{4} }\;\;\;\left | z \right | \gt 1 }[/math] | |
| [math]\displaystyle{ \frac{k\left ( k-1 \right )\cdots \left ( k-m+1 \right ) }{m!} \varepsilon \left ( k \right ) }[/math] | [math]\displaystyle{ \frac{z}{\left ( z-1 \right )^{m+1} }\;\;\;\left | z \right | \gt 1 }[/math] | |
| [math]\displaystyle{ \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]\displaystyle{ \frac{z}{\left ( z-a \right )^{m+1} }\;\;\;\left | z \right | \gt \left | a \right | }[/math] | |
| 指数序列 | [math]\displaystyle{ e^{jk\omega _{0} } \varepsilon \left ( k \right ) }[/math] | [math]\displaystyle{ \frac{z}{z-e^{j\omega _{0} } }\;\;\;\left | z \right | \gt 1 }[/math] |
| 正弦序列 | [math]\displaystyle{ \sin \left ( \omega _{0} k \right ) \varepsilon \left ( k \right ) }[/math] | [math]\displaystyle{ \frac{2\sin \omega _{0} }{z^{2}-2z\cos \omega _{0}+1 } \;\;\;\left | z \right | \gt 1 }[/math] |
| 余弦序列 | [math]\displaystyle{ \cos \left ( \omega _{0} k \right ) \varepsilon \left ( k \right ) }[/math] | [math]\displaystyle{ \frac{z^{2}-z\cos \omega _{0} }{z^{2}-2z\cos \omega _{0}+1 } \;\;\;\left | z \right | \gt 1 }[/math] |
| [math]\displaystyle{ -\varepsilon \left ( -k -1\right ) }[/math] | [math]\displaystyle{ \frac{z}{z-1}\;\;\;\left | z \right | \lt 1 }[/math] | |
| [math]\displaystyle{ -k\varepsilon \left ( -k -1\right ) }[/math] | [math]\displaystyle{ \frac{z}{\left ( z-1 \right )^{2} } \;\;\;\left | z \right | \lt 1 }[/math] |