Laplace Transform
Table of Contents
Given a time domain continuous signal, it can be represented in the complex frequency domain by Laplace transform. It is essentially an extension of Continuous Fourier transform.
Definition
Bilateral transform
\begin{align*} X(s) = \int_{-\infty}^{\infty} x(t) e^{-st} dt \end{align*}Unilateral transform
\begin{align*} X(s) = \int_{0_-}^{\infty} x(t) e^{-st} dt \end{align*}Inverse transform
\begin{align*} x(t) = \frac{1}{2\pi j} \int_{\sigma-j\infty}^{\sigma+j\infty} X(s) e^{st} ds \end{align*}- Region of convergence (ROC): Considering the fact that \(X(s)\) is an infinite integration, the convergence must be ensured. To this end, ROC is defined, which is comprised of the candidate values of \(s\) keeping the convergence. \(s\) is termed complex frequency.
Properties (Unilateral Transform)
\begin{align*}
x(t), t \ge 0_{-} &\leftrightarrow X(s) \\
x(at), a > 0 &\leftrightarrow \frac{1}{a}X \left( \frac{s}{a} \right) \\
x(t - t_0)\varepsilon(t - t_0), t_0>0 &\leftrightarrow X(s) e^{-st_0} \\
x(t)e^{s_0}t &\leftrightarrow X(s - s_0) \\
x^{(n)}(t) &\leftrightarrow s^nX(s) - s^{n-1}x(0_{-}) - \cdots - s^0x^{(n-1)}(0_{-}) \\
x_1(t) * x_2(t) &\leftrightarrow X_1(s) X_2(s) \\
x_1(t) x_2(t) &\leftrightarrow \frac{1}{2\pi j} \int_{C-j\infty}^{C+j\infty} X_1(\eta)X_2(s-\eta) d \eta \\
(-t)^nx(t) &\leftrightarrow \frac{d^nX(s)}{ds^{n}} \\
\frac{x(t)}{t} &\leftrightarrow \int_s^{\infty}X(\eta)d\eta
\end{align*}
Initial Value Theorem
\begin{align*}
x(0_{+}) = \lim_{s\to \infty}sX(s)
\end{align*}
Final Value Theorem
\begin{align*}
x(\infty) = \lim_{s\to 0} sX(s)
\end{align*}
Transform Pairs (Unilateral Transform)
\begin{align*}
x(t), t \ge 0_{-} &\leftrightarrow X(s) \\
\delta(t) &\leftrightarrow 1 \\
\varepsilon(t) &\leftrightarrow \frac{1}{s} \\
e^{-s_0t} &\leftrightarrow \frac{1}{s+s_0} \\
t^n, n \in \mathbb{Z}^+ &\leftrightarrow \frac{n!}{s^{n+1}} \\
\sin \omega t &\leftrightarrow \frac{\omega_0}{s^2 + \omega_0^2} \\
\cos \omega t &\leftrightarrow \frac{s}{s^2 + \omega_0^2}
\end{align*}