Laplace Transforms

Definition

$$\mathcal{L}\{x(t)\} = X(s) = \int_{0^-}^{\infty} x(t)\, e^{-st} \, dt$$

Properties

• Differentiation $\mathcal{L}\{x^{(n)}(t)\} = s^n X(s) - s^{n-1}x(0^-) - s^{n-2}\dot{x}(0^-) - \cdots - x^{(n-1)}(0^-)$
• Integration $\mathcal{L}\{\int_0^t x(\tau)\, d\tau\} = X(s)/s$
• Multiplication by t $\mathcal{L}\{t\, x(t)\} = -\frac{d}{ds} X(s)$
• Time shift $\mathcal{L}\{x(t - T)\, u(t - T)\} = e^{-sT} X(s)$ for $T \ge 0$, with $u$ the unit step
• Frequency shift $\mathcal{L}\{e^{-at} x(t)\} = X(s + a)$
• Convolution For causal signals, $(x_1 * x_2)(t) = \int_0^t x_1(t - \tau)\, x_2(\tau)\, d\tau$ and $\mathcal{L}\{x_1 * x_2\} = X_1(s) X_2(s)$

Common Pairs

For $t \ge 0$,

$$\begin{aligned} \delta(t) &\;\leftrightarrow\; 1 & 1 &\;\leftrightarrow\; \frac{1}{s} \\ t^n &\;\leftrightarrow\; \frac{n!}{s^{n+1}} & \frac{t^{n-1}}{(n-1)!} e^{-at} &\;\leftrightarrow\; \frac{1}{(s+a)^n} \\ e^{-at}\cos(\omega_0 t) &\;\leftrightarrow\; \frac{s + a}{(s + a)^2 + \omega_0^2} & e^{-at}\sin(\omega_0 t) &\;\leftrightarrow\; \frac{\omega_0}{(s + a)^2 + \omega_0^2} \end{aligned}$$

Partial Fractions

For a strictly proper rational $X(s)$ with distinct poles $p_i$,

$$X(s) = \sum_{i=1}^{n} \frac{r_i}{s - p_i}, \qquad r_i = \lim_{s \to p_i} (s - p_i) X(s)$$

If $p_i$ has multiplicity $m_i$ ($q$ distinct poles),

$$X(s) = \sum_{i=1}^{q} \sum_{j=1}^{m_i} \frac{r_{ij}}{(s - p_i)^j}$$

A complex conjugate pair $p = \sigma + j\omega$ with residue $r = a e^{j\phi}$ contributes

$$\mathcal{L}^{-1}\left\{\frac{r}{s - p} + \frac{\bar{r}}{s - \bar{p}}\right\} = r e^{pt} + \bar{r} e^{\bar{p}t} = 2a\, e^{\sigma t} \cos(\omega t + \phi)$$

Initial and Final Value Theorems

Final value theorem If all poles of $sX(s)$ are in the open left half-plane (equivalently, $X(s)$ has left half-plane poles and at most a simple pole at the origin), then

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

Initial value theorem If $X(s)$ is strictly proper, $x(0^+) = \lim_{s \to \infty} sX(s)$.