# Inverse laplace transform square wave

1. ### magnifik

360

i do not know how to solve for this due to the e^-20s shift in the denominator... is there a way to change this function to make it easier to solve?

8,390

360
4. ### magnifik

360
in analyzing JUST the function part of this...

f(t) = 1, when 0 < t < 10
f(t) = 0, when 10 < t < 20
F(s) = (-1/s)(e^(-10s) - 1)/(1 - e^(-20s))

is it correct to say that f(t) = U(t) - U(t -10) and that the period is 20??

5. ### vela

12,767
Staff Emeritus
The idea is to use the fact that

$$\frac{1}{1-x} = 1+x+x^2+x^3+x^4+\cdots$$

so that

\begin{align*} \frac{1}{1-e^{-20s}} &= 1 + e^{-20s} + (e^{-20s})^2+ (e^{-20s})^3 + \cdots \\ &= 1 + e^{-20s} + e^{-40s} + e^{-60s} + \cdots \end{align*}

Hopefully, you can see why that may result in a function periodic in the time domain.

6. ### LCKurtz

8,390
Yes. To be precise, F(s) is the Laplace Transform of the periodic extension with period 20 of your f(t).

7. ### magnifik

360
i understand that it can be represented as a series. however, i wouldn't know how to take the inverse laplace of the series

8. ### vela

12,767
Staff Emeritus
You're not taking the inverse transform of just the series. You're taking the inverse of

\begin{align*} \frac{F(s)}{1-e^{-20s}} & = F(s)[1 + e^{-20s} + e^{-40s} + e^{-60s} + \cdots] \\ & = F(s) + e^{-20s}F(s) + e^{-40s}F(s) + e^{-60s}F(s) + \cdots \end{align*}

9. ### magnifik

360
not sure how i would do this. i know that the e^-ns represents a shift in the time domain f(t-n) but wouldn't there be infinitely many time shifts using the series expansion?

10. ### vela

12,767
Staff Emeritus
Yes, you need an infinite number of shifts because a periodic signal has an infinite number of cycles.

Take your f(t) in post #4 and repeatedly shift and add. What do you get?

11. ### magnifik

360
U(t) - U(t-10) + U(t - 20) - U(t - 30) + U(t-40) - U(t - 50) + U(t - 60) + U(t - 70)...

i think i see your point.. the function will essentially be U(t) - U(t-10) repeated over and over again

12. ### LCKurtz

8,390
Vela and I are leading you in two different directions for solution. I guess you learn twice as much. Say you have a periodic function f(t) with period p and define a new function f1(t) which is 1 on (0,p) and 0 elsewhere, giving you one nonzero period of your function f(t). So you can write

f1(t) = f(t)(u(t)-u(t-p))

You know that the transform of the periodic function f(t) is

$$F(s) = \frac{L(f_1(t))}{1-e^{-ps}}$$

This tells you that when taking inverse transforms a factor of (1 - e-ps) can be suppressed and the inverse will give you one period of the original periodic function. So if you can find the inverse without that factor, just extend it periodically with period p to get the function f(t).