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Math Help - Help Laplace transform of Dirac comb

  1. #1
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    Help Laplace transform of Dirac comb

    Please I need some help to solve this problem the Laplace transform of :Help Laplace transform of Dirac comb-sin-titulo.jpg
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  2. #2
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    Re: Help Laplace transform of Dirac comb

    the summation over what?
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  3. #3
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    Re: Help Laplace transform of Dirac comb

    from 0 to +inf
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  4. #4
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    Re: Help Laplace transform of Dirac comb

    Quote Originally Posted by ameva View Post
    from 0 to +inf
    there is no summation index in the expression.... is it a?
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  5. #5
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    Re: Help Laplace transform of Dirac comb

    Yes it's a.
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  6. #6
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    Re: Help Laplace transform of Dirac comb

    Quote Originally Posted by ameva View Post
    Yes it's a.
    take a single element of the series

    $f(t)=\delta(t-k T)$

    I used a sampling time $T$ in the expression. You can set this so 1 if you like.

    The Laplace transform of $\delta(t)$ is $1$

    The Laplace transform of $g(t-kT)$ is $e^{-skT}G(s)$

    so

    $\large \mathscr{L} \{ \delta(t-kT) \} = e^{-s k T} * 1 = e^{-s k T}$

    The Laplace transform is linear so the Laplace transform of the sum is the sum of the transforms, i.e.

    $\large \mathscr{L} \{ \displaystyle{\sum_{k=0}^\infty} f_k(t) \} = \displaystyle{\sum_{k=0}^\infty} F_k(s)$ where $F_k(s) = \mathscr{L}\{ f_k(t) \}$

    thus

    $\large \mathscr{L} \{ \displaystyle{\sum_{k=0}^\infty} \delta(t - kT) \} = \displaystyle{\sum_{k=0}^\infty} e^{-s k T}$

    This last term is a geometric series in $k$

    $\large \displaystyle{\sum_{k=0}^\infty} e^{-s k T} = \dfrac 1 {1 - e^{-s T}}$

    so

    $\large \mathscr{L} \{ \displaystyle{\sum_{k=0}^\infty} \delta(t - kT) \} = \dfrac 1 {1 - e^{-s T}}$
    Thanks from ameva
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  7. #7
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    Re: Help Laplace transform of Dirac comb

    that is what I looking for! thanks!
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  8. #8
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    Re: Help Laplace transform of Dirac comb

    just a question, did you change "a" by "kT"?
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  9. #9
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    Re: Help Laplace transform of Dirac comb

    Quote Originally Posted by ameva View Post
    just a question, did you change "a" by "kT"?
    yes. In the final answer just set $T=1$

    i.e.

    $\dfrac 1 {1 - e^{-s}}$
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  10. #10
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    Re: Help Laplace transform of Dirac comb

    Thanks!
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  11. #11
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    Re: Help Laplace transform of Dirac comb

    I'm not pretty sure about this: Help Laplace transform of Dirac comb-sin-titulo44.jpg

    I've just put the summation in a calculator program and it returned me this. Is it correct? Why is that?
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  12. #12
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    Re: Help Laplace transform of Dirac comb

    PD: On the paper I get the same as you
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  13. #13
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    Re: Help Laplace transform of Dirac comb

    Quote Originally Posted by ameva View Post
    PD: On the paper I get the same as you
    divide that second solution top and bottom by $\large e^s$. They are the same answer.
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