4 Example 2.10 OGATA 4ed (obtaining the Laplace transform of a graph)

4 Example 2.10 OGATA 4ed (obtaining the Laplace transform of a graph)

We do have a different book's form, but giving the same result. Function over time is as follows as it has period T.

$\begin{displaymath}f(t)=\sum_{n=0}^{\infty}(1(t-n\cdot T)-2\cdot 1(t-\frac{T}{2}-n\cdot T)+ 1(t-T-n\cdot T))\end{displaymath}$

So the Laplace transform of the function we would:

$\begin{displaymath}F(s)=\frac{\sum_{n=0}^{\infty}(e^{-n\cdot T\cdot s}-2\cdot e^... ...e^{-T\cdot s})\cdot \sum_{n=0}^{\infty}e^{-n\cdot T\cdot s}}{s}\end{displaymath}$

As seen in the ejercidio A2.9:

$\begin{displaymath}\sum_{n=0}^{\infty}e^{-n\cdot T\cdot s}=\frac{1}{(1-e^{-T\cdot s})}\end{displaymath}$

Ie the equation above would be as follows:

$\begin{displaymath}F(s)=(1-2\cdot e^{-\frac{T\cdot s}{2}}+e^{-T\cdot s})\cdot \frac{1}{s\cdot (1-e^{-T\cdot s})}\end{displaymath}$