# Laplace/frequency domain

1. Feb 6, 2005

### edlewis

This is not a Bode plot question, but similar.

I have data on the laplace transform of a tracer response at different times, "t". I want to plot the response against "s".

Since the Laplace transform of t is 1/s^2, do I convert to "s" by s=(1/t)^.5?

2. Feb 7, 2005

### Atrus

At the beginning, sorry for both my English and lousy typing skills, but my knowledge of TeX is... well, nonexistent and I do my best with English :)

If I understood correctly, you have a function f(t) which is your time-domain response, and you want to plot it's s-domain representation in Laplace variable s? Or is it vice versa?

In any case, making a plain substitute of the time variable (t) or the frequency variable (s) won't give a correct answer.

Theoretically, Laplace transform of a function f is
Integral(0 -> +Inf) (f(t)e^(-s*t)dt)

In practice, you would use pre-made tables of standard terms like e^(a*t), t, sin(a*t) and such. You then transform your function into the sum of terms given in the table and then substitute time-domain terms into s-domain.

E.g.

f(t) = sin(2*t)*cos(3*t)

f(t) = 0.5 * sin(5*t) - 0.5*sin(t)

as Laplace(sin(a*t)) = a/(s^2+a^2)

F(s) = Laplace(f(t)) = 0.5*5/(s^2+25)-0.5/(s^2+1)

You can collect the terms to make it into a single function which you can then plot against s.

If your function can not be disassembled into terms found in tables of Laplace transform, well, then you'll have to calculate above integral transform by hand.

Hope I helped

P.S. most of calculus or engineering math handbooks have Laplace transform tables in their appendices.

Last edited: Feb 7, 2005