常用拉普拉斯变换对

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拉普拉斯变换

线性时不变 (LTI) 系统由微分方程描述。拉普拉斯变换是一种数学工具，它将时域中的微分方程转换为频域（或s域）中的代数方程。

如果$\mathrm{\mathit{x\left ( t \right )}}$是时间函数，则该函数的拉普拉斯变换定义为−

$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{=}X\left ( s \right )\mathrm{=}\int_{-\infty }^{\infty }x\left ( t \right )e^{-st}\:dt\; \; \cdot \cdot \cdot\left ( \mathrm{1} \right ) }}$$

其中，s 是一个复变量，由下式给出：

$$\mathrm{\mathit{s\mathrm{=}\sigma \mathrm{+ }j\omega }}$$

拉普拉斯逆变换

拉普拉斯逆变换定义为−

$$\mathrm{\mathit{L^{\mathrm{-1}}\left [X\left ( s \right ) \right ]\mathrm{=}x\left ( t \right )\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi j}\int_{\sigma -j\infty }^{\sigma \mathrm{+ }j\infty }X\left ( s \right )e^{st}\:ds\; \; \cdot \cdot \cdot\left ( \mathrm{2} \right ) }}$$

公式 (1) 和 (2) 构成了拉普拉斯变换对，可以表示为：

$$\mathrm{\mathit{x\left ( t \right )\overset{LT}{\leftrightarrow} X\left ( s \right ) }}$$

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常用拉普拉斯变换对

下表提供了许多拉普拉斯变换。该表还指定了收敛域 (ROC) −

函数 $\mathrm{\mathit{\left\{ x\left ( t \right )\mathrm{=}L^{-1}\left [ x\left ( t \right ) \right ]\right\}}}$	拉普拉斯变换 $\mathrm{\mathit{\left\{ L\left [ x\left ( t \right ) \right ]\mathrm{=}X\left ( s \right )\right\}}}$	收敛域 (ROC)
$\mathrm{\mathit{\delta \left ( t \right )}}$	1	$\mathit{\mathrm{所有}\; s}$
$\mathrm{\mathit{\delta \left ( t-a \right )}}$	$\mathrm{\mathit {e^{-as}}}$	$\mathit{\mathrm{所有}\; s}$
$\mathrm{\mathit{u \left ( t \right )}}$	$\mathrm{\mathit {\frac{\mathrm{1}}{s}}}$	$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{u \left ( t-a \right )}}$	$\mathrm{\mathit {\frac{e^{-as}}{s}}}$	$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{u \left ( -t \right )}}$	$\mathrm{\mathit -{\frac{\mathrm{1}}{s}}}$	$\mathit{\mathrm{Re}\left ( s \right )< \mathrm{0}}$
$\mathrm{\mathit{tu \left ( t \right )}}$	$\mathrm{\mathit {\frac{\mathrm{1}}{s^{\mathrm{2}}}}}$	$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{t^{\mathrm{2}}u \left ( t \right )}}$	$\mathrm{\mathit {\frac{\mathrm{2!}}{s^{\mathrm{3}}}}}$	$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{t^{n}u \left ( t \right )}}$	$\mathrm{\mathit {\frac{n!}{s^{\left ( n\mathrm{+ }\mathrm{1} \right )}}}}$	$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{e^{-at}u \left ( t \right )}}$	$\mathrm{\mathit {\frac{\mathrm{1}}{\left ( s\mathrm{+ }a \right )}}}$	$\mathit{\mathrm{Re}\left ( s \right )> -a}$
$\mathrm{\mathit{e^{at}u \left ( t \right )}}$	$\mathrm{\mathit {\frac{\mathrm{1}}{\left ( s-a \right )}}}$	$\mathit{\mathrm{Re}\left ( s \right )> a}$
$\mathrm{\mathit{te^{-at}u \left ( t \right )}}$	$\mathrm{\mathit {\frac{\mathrm{1}}{\left ( s\mathrm{+ }a \right )^{\mathrm{2}}}}}$	$\mathit{\mathrm{Re}\left ( s \right )> -a}$
$\mathrm{\mathit{t^{n}e^{-at}u \left ( t \right )}}$	$\mathrm{\mathit {\frac{n!}{\left ( s\mathrm{+ }a \right )^{n\mathrm{+ }\mathrm{1}}}}}$	$\mathit{\mathrm{Re}\left ( s \right )> -a}$
$\mathrm{\mathit{\mathrm{sin}\: \omega t\: u \left ( t \right )}}$	$\mathrm{\mathit {\frac{\omega }{\left ( s^{\mathrm{2}}\mathrm{+ }\omega ^{\mathrm{2}} \right )}}}$	$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{\mathrm{cos}\: \omega t\: u \left ( t \right )}}$	$\mathrm{\mathit {\frac{s }{\left ( s^{\mathrm{2}}\mathrm{+ }\omega ^{\mathrm{2}} \right )}}}$	$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{e^{-at}\: \mathrm{sin}\: \omega t\: u \left ( t \right )}}$	$\mathrm{\mathit {\frac{\omega }{\left ( s\mathrm{+ }a \right )^{\mathrm{2}}\mathrm{+ }\omega ^{\mathrm{2}}}}}$	$\mathit{\mathrm{Re}\left ( s \right )> -a}$
$\mathrm{\mathit{e^{-at}\: \mathrm{cos}\: \omega t\: u \left ( t \right )}}$	$\mathrm{\mathit {\frac{\left ( s\mathrm{+ }a \right )}{\left ( s\mathrm{+ }a \right )^{\mathrm{2}}\mathrm{+ }\omega ^{\mathrm{2}}}}}$	$\mathit{\mathrm{Re}\left ( s \right )> -a}$
$\mathrm{\mathit{\mathrm{sin}\left ( \omega t\mathrm{+ }\theta \right )}}$	$$\mathrm{\mathit {\frac{s\: \mathrm{sin}\, \theta \mathrm{+ }\omega \: \mathrm{cos}\, \theta }{\left ( s^{\mathrm{2}}\mathrm{+ }\omega^{\mathrm{2}} \right )}}}$$	$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{\mathrm{cos}\left ( \omega t\mathrm{+ }\theta \right )}}$	$\mathrm{\mathit {\frac{s\: \mathrm{cos}\, \theta \mathrm{+ }\omega \: \mathrm{sin}\, \theta }{\left ( s^{\mathrm{2}}\mathrm{+ }\omega^{\mathrm{2}} \right )}}}$	$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit t\: {\mathrm{sin}\:\mathit{\omega t\: u\left ( t \right )}}}$	$\mathrm{\mathit {\frac{\mathrm{2}\omega s}{\left ( s^{\mathrm{2}}\mathrm{+ }\omega^{\mathrm{2}} \right )\mathrm{^{2}}}}}$	$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit t\: {\mathrm{cos}\:\mathit{\omega t\: u\left ( t \right )}}}$	$\mathrm{\mathit {\frac{ s^{\mathrm{2}}-\omega ^{\mathrm{2}}}{\left ( s^{\mathrm{2}}\mathrm{+ }\omega^{\mathrm{2}} \right )\mathrm{^{2}}}}}$	$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit {\mathrm{sinh}\:\mathit{\omega t\: u\left ( t \right )}}}$	$\mathrm{\mathit {\frac{ \omega }{\left ( s^{\mathrm{2}}-\omega^{\mathrm{2}} \right )}}}$	$\mathit{\mathrm{Re}\left ( s \right )> \omega }$
$\mathrm{\mathit {\mathrm{cosh}\:\mathit{\omega t\: u\left ( t \right )}}}$	$\mathrm{\mathit {\frac{ s }{\left ( s^{\mathrm{2}}-\omega^{\mathrm{2}} \right )}}}$	$\mathit{\mathrm{Re}\left ( s \right )> \omega }$
$\mathrm{\mathit {e^{-at}\: \mathrm{sinh}\:\mathit{\omega t\: u\left ( t \right )}}}$	$\mathrm{\mathit {\frac{ \omega }{\left ( s\mathrm{+ }a\right )^{\mathrm{2}}-\omega^{\mathrm{2}}}}}$	$\mathit{\mathrm{Re}\left ( s \right )> \left ( \omega -a \right ) }$
$\mathrm{\mathit {e^{-at}\: \mathrm{cosh}\:\mathit{\omega t\: u\left ( t \right )}}}$	$\mathrm{\mathit {\frac{ s\mathrm{+ }a }{\left ( s\mathrm{+ }a\right )^{\mathrm{2}}-\omega^{\mathrm{2}}}}}$	$\mathit{\mathrm{Re}\left ( s \right )> \left ( \omega -a \right ) }$