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Find if a function is the Laplace transform of a periodic function

  1. Feb 2, 2010 #1
    1. The problem statement, all variables and given/known data

    Could [tex]\[F(s) = \frac{1}{{s(1 - {e^{ - s}})}}\][/tex] be the Laplace transform of some periodic function? Why? If so, find that periodic function

    3. The attempt at a solution

    If it was the Laplace transform of some periodic function, the the Laplace transform of the first wave should be 1/s, and the period T should be 1. The function whose Laplace transform is 1/s is H(t). Then the periodic function should be the stairs function, [tex]\[f(t) = \left\{ \begin{array}{l}
    n,x \in [nT,(n + 1)T] \\
    n + 1,x \in [(n + 1)T,(n + 2)T] \\
    \end{array} \right.,n \in N + \left\{ 0 \right\}\][/tex]

    Now, the stairs function is of exponential order, since it's smaller or equal than the function t+1 for all t, and that is an exponential order function. Then it has a Laplace transform.

    So far, so good. If the stairs function transforms to F(s) there, then F(s) is the Laplace transform of some periodic function.

    But I don't think that's what the exercise asks me to do, since a latter question asks me to find the periodic function. I think there's something I have to prove through F(s) that allows me to say that it is the Laplace transform of a periodic function.

    But I don't know what is that.

    Any ideas?
     
  2. jcsd
  3. Feb 2, 2010 #2

    LCKurtz

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    I'm thinking the answer is no and you should be thinking about the "why" rather than trying to find a periodic function. Your stairs function certainly isn't periodic...
     
  4. Feb 2, 2010 #3
    Allright, it's true.

    But then why couldn't it be periodic? Is it because the inverse Laplace transform of 1/s is H(t), and that isn't a periodic function? What does F(s) has to have to check if it belongs to a periodic function or not?
     
  5. Feb 2, 2010 #4

    LCKurtz

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    Well, I didn't check your steps, but I assume you have the correct inverse with your staircase function. It isn't periodic and the FT is a 1-1 transform so doesn't that settle it?
     
  6. Feb 2, 2010 #5
    Yes, but I thought there was another argument to say that. I mean, since the question seems to be theorical.
     
  7. Feb 2, 2010 #6
    Anybody?
     
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