Laplace Transform

calendar_today April 1, 2025 article 206 words access_time 2 min replay Modified April 1, 2025

Table of Contents

Determined using the Laplace transform integral or laplace transform table [Laplace transform table (course handout)]
Most continuous functions have a Laplace transform

$$X(s)=\int_{-\infty}^{\infty}x(t)e^{-st} \ \mathrm{d}t$$

Inverse Laplace Transform

Usually not practical using the mathematical definition of ILT
Must change the $s$-domain function so that it can be evaluated with a table [Laplace transform table (course handout)]
- Accomplished with partial fraction decomposition or clever manipulation

$$x(t) = \frac{2\pi}{j}\int_{\alpha-j\infty}^{\alpha+j\infty}X(s)e^{st} \ \mathrm{d}s$$

Every proper rational function can be expressed as a sum of first-order rational functions

$$X(s)=\frac{B(s)}{A(s)}=\frac{b_{M}s^M+b_{M-1}s^{M-1}+\dots+b_{1}s^{1}+b_{0}}{a_{M}s^M+a_{M-1}s^{M-1}+\dots+a_{1}s^{1}+a_{0}}$$

Each term in the sum is a residue, $r_k$, divided by $s$ minus a pole factor, $p_k$

$$X(s) = \sum_{k}^{N}\frac{r_{k}}{s-p_{k}}$$

Useful functions for partial fraction decomposition: residue, roots, poly $partial fraction decomposition with matlab example$

Distribute $u(t)$ and evaluate each sine independently as a general sine
The example shown below is pretty poor since it has a constant for the first term instead of a sinusoid

Two methods: algebraic manipulation and use the table, or convolution in the $s$-domain (harder)

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