Laplace Transform

What Are Laplace Transformations?

This is another frequency mapping technique that can be easier to use than the Fourier Transforms as these use a table and map certain identities to known transforms. Laplace transforms are used to describe transfer functions since laplace domain makes transfer function maths a lot easier. When a signal is affected by a transfer function, the input is simply multiplied by the transfer function to produce the output which is a much simpler operation to complete than convolution in the time domain.

Quick Facts About Laplace Transforms

Laplace Transforms:

Are linear.
Uses "s" as its variable, a complex numnber.
Require initial conditions.
Can be done forwards and backwards (or in inverse) without losing information

Time Domain: f(t)	Laplace Domain: F(s)
Impulse \( \delta(t) \)	\( 1 \)
Step \( 1(t) \)	\( \frac{1}{s} \)
Ramp \( t \)	\( \frac{1}{s^{2}} \)
\( t^{n} \)	\( \frac{n!}{s^{n+1}} \)
\( e^{-at} \)	\( \frac{1}{s+a} \)
\( te^{-at} \)	\( \frac{1}{(s+a)^2} \)
\( \sin(\omega t) \)	\( \frac{\omega}{s^2+\omega^2} \)
\( \cos(\omega t) \)	\( \frac{s}{s^2+\omega^2} \)
\( e^{-at}\sin(\omega t) \)	\( \frac{\omega}{(s+a)^2+\omega^2} \)
\( e^{-at}\cos(\omega t) \)	\( \frac{s+a}{(s+a)^2+\omega^2} \)

Where \( t \geq 0 \) for all time domain functions.

Common Signals In Laplace Domain

Function	Time Domain: f(t)	Laplace Domain: F(s)
Step	\( x(t) \begin{cases} &0\ \ \ \ t\leq 0\\ &K\ \ \ t>0\\ \end{cases} \)	\( \frac{K}{s} \)
Ramp	\( r(t) \begin{cases} &0\ \ \ \ \ \ t\leq 0\\ &Kt\ \ \ t>0\\ \end{cases} \)	\( \frac{K}{s^2} \)
Impulse	\( \delta(t) \)	\( 1 \)
Pulse/Rectangle	\( rect(t) = \begin{cases} &1\ \ \ t_0<t<t_1\\ &0\ \ \ else\\ \end{cases} \)	\( \frac{1}{s}-\frac{e^{-t_{1}s}}{s} \)