# Math Help - Help Laplace transform of Dirac comb

1. ## Help Laplace transform of Dirac comb

Please I need some help to solve this problem the Laplace transform of :

2. ## Re: Help Laplace transform of Dirac comb

the summation over what?

3. ## Re: Help Laplace transform of Dirac comb

from 0 to +inf

4. ## Re: Help Laplace transform of Dirac comb

Originally Posted by ameva
from 0 to +inf
there is no summation index in the expression.... is it a?

Yes it's a.

6. ## Re: Help Laplace transform of Dirac comb

Originally Posted by ameva
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

$\large \mathscr{L} \{ \displaystyle{\sum_{k=0}^\infty} \delta(t - kT) \} = \dfrac 1 {1 - e^{-s T}}$

7. ## Re: Help Laplace transform of Dirac comb

that is what I looking for! thanks!

8. ## Re: Help Laplace transform of Dirac comb

just a question, did you change "a" by "kT"?

9. ## Re: Help Laplace transform of Dirac comb

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

i.e.

$\dfrac 1 {1 - e^{-s}}$

Thanks!

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

I've just put the summation in a calculator program and it returned me this. Is it correct? Why is that?

12. ## Re: Help Laplace transform of Dirac comb

PD: On the paper I get the same as you

13. ## Re: Help Laplace transform of Dirac comb

Originally Posted by ameva
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.