Laplace Transform Table

TABLE 9.1 PROPERTIES OF THE LAPLACE TRANSFORM

Section	Property	Signal	Laplace Transform	ROC
		$x(t)$	$X(s)$	$R$
		$x_1(t)$	$X_1(s)$	$R_1$
		$x_2(t)$	$X_2(s)$	$R_2$
9.5.1	Linearity	$ax_1(t) + bx_2(t)$	$aX_1(s) + bX_2(s)$	At least $R_1 \cap R_2$
9.5.2	Time shifting	$x(t - t_0)$	$e^{-st_0}X(s)$	$R$
9.5.3	Shifting in the s-Domain	$e^{s_0t}x(t)$	$X(s - s_0)$	Shifted version of $R$ (i.e., $s$ is in the ROC if $s - s_0$ is in $R$)
9.5.4	Time scaling	$x(at)$	$\frac{1}{	a
9.5.5	Conjugation	$x^*(t)$	$X^*(s^*)$	$R$
9.5.6	Convolution	$x_1(t) * x_2(t)$	$X_1(s)X_2(s)$	At least $R_1 \cap R_2$
9.5.7	Differentiation in the Time Domain	$\frac{d}{dt}x(t)$	$sX(s)$	At least $R$
9.5.8	Differentiation in the s-Domain	$-tx(t)$	$\frac{d}{ds}X(s)$	$R$
9.5.9	Integration in the Time Domain	$\int_{-\infty}^t x(\tau)d\tau$	$\frac{1}{s}X(s)$	At least $R \cap \{\text{Re}[s] > 0\}$

9.5.10 Initial- and Final-Value Theorems

Initial-Value Theorem (初值定理)

If $x(t) = 0$ for $t < 0$ and $x(t)$ contains no impulses or higher-order singularities at $t = 0$, then

$$x(0^+) = \lim_{s \to \infty} sX(s)$$

Final-Value Theorem (终值定理)

If $x(t) = 0$ for $t < 0$ and $x(t)$ has a finite limit as $t \to \infty$, then

$$\lim_{t \to \infty} x(t) = \lim_{s \to 0} sX(s)$$

Table 9.2 常用拉普拉斯变换对

序号	信号 $f(t)$	Laplace变换 $F(s)$	收敛域 ROC
1	$\delta(t)$	$1$	All $s$
2	$u(t)$	$\frac{1}{s}$	$\text{Re}\{s\} > 0$
3	$-u(-t)$	$\frac{1}{s}$	$\text{Re}\{s\} < 0$
4	$\frac{t^{n-1}}{(n-1)!}u(t)$	$\frac{1}{s^n}$	$\text{Re}\{s\} > 0$
5	$-\frac{t^{n-1}}{(n-1)!}u(-t)$	$\frac{1}{s^n}$	$\text{Re}\{s\} < 0$
6	$e^{-\alpha t}u(t)$	$\frac{1}{s+\alpha}$	$\text{Re}\{s\} > -\alpha$
7	$-e^{-\alpha t}u(-t)$	$\frac{1}{s+\alpha}$	$\text{Re}\{s\} < -\alpha$
8	$\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t)$	$\frac{1}{(s+\alpha)^n}$	$\text{Re}\{s\} > -\alpha$
9	$-\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t)$	$\frac{1}{(s+\alpha)^n}$	$\text{Re}\{s\} < -\alpha$
10	$\delta(t-T)$	$e^{-sT}$	All $s$
11	$[\cos \omega_0 t]u(t)$	$\frac{s}{s^2+\omega_0^2}$	$\text{Re}\{s\} > 0$
12	$[\sin \omega_0 t]u(t)$	$\frac{\omega_0}{s^2+\omega_0^2}$	$\text{Re}\{s\} > 0$
13	$[e^{-\alpha t} \cos \omega_0 t]u(t)$	$\frac{s+\alpha}{(s+\alpha)^2+\omega_0^2}$	$\text{Re}\{s\} > -\alpha$
14	$[e^{-\alpha t} \sin \omega_0 t]u(t)$	$\frac{\omega_0}{(s+\alpha)^2+\omega_0^2}$	$\text{Re}\{s\} > -\alpha$
15	$\frac{d^n \delta(t)}{dt^n}$	$s^n$	All $s$
16	$u_{-n}(t) = \underbrace{u(t) * \cdots * u(t)}_{n \text{ times}}$	$\frac{1}{s^n}$	$\text{Re}\{s\} > 0$

单边Laplace变换对

性质	信号	单边Laplace变换
基本定义	$x(t)$	$\mathcal{X}(s)$
	$x_1(t)$	$\mathcal{X}_1(s)$
	$x_2(t)$	$\mathcal{X}_2(s)$
线性性	$ax_1(t) + bx_2(t)$	$a\mathcal{X}_1(s) + b\mathcal{X}_2(s)$
$s$域移位	$e^{s_0 t}x(t)$	$\mathcal{X}(s - s_0)$
时间尺度变换	$x(at)$, $a > 0$	$\frac{1}{a}\mathcal{X}\left(\frac{s}{a}\right)$
共轭	$x^*(t)$	$\mathcal{X}^*(s^*)$
卷积	$x_1(t) * x_2(t)$	$\mathcal{X}_1(s)\mathcal{X}_2(s)$
（假设$x_1(t)$和$x_2(t)$在$t < 0$时恒为零）
时域微分	$\frac{d}{dt}x(t)$	$s\mathcal{X}(s) - x(0^-)$
$s$域微分	$-tx(t)$	$\frac{d}{ds}\mathcal{X}(s)$
时域积分	$\int_0^t x(\tau)d\tau$	$\frac{1}{s}\mathcal{X}(s)$

$$\mathcal{L} [f^{(n)}(t)] = s^n \mathcal{X}(s) - \left[f^{(n-1)}(0) + \cdots + s^{n-1} f(0)\right]$$

$$\mathcal{X}^{(n)}(p) = \mathcal{L}\left[(-t)^n f(t)\right]$$