信号与系统：修订间差异
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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>
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 ==
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|冲激序列
|<math>\delta \left ( k \right ) </math>
|<math>1</math>
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|阶跃序列
|<math>\varepsilon \left ( k \right ) </math>
|<math>\frac{z}{z-1}\;\;\;\left | z \right | >1</math>
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|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>
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|斜升序列
|<math>k\varepsilon \left ( k \right ) </math>
|<math>\frac{z}{\left ( z-1 \right )^{2} } \;\;\;\left | z \right | >1</math>
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|<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>
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|<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>
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|<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>
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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>
|<math>\frac{z}{\left ( z-a \right )^{m+1} }\;\;\;\left | z \right | > \left | a \right |</math>
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|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>
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|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>
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|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>
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|<math>-\varepsilon \left ( -k -1\right )</math>
|<math>\frac{z}{z-1}\;\;\;\left | z \right | <1</math>
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|<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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