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The Context:

Find $X(ω)$ which is the frequency domain representations of $x(t)$.

$$ x(t) = \sum_{k=-\infty}^\infty \delta(t-4k) $$

This my professor's solution:

solution

As we can see, the way he proceeded is as follows: 1) Since x(t) is periodic, he first calculated the Fourier Series coefficient of x(t). 2) He used the following formula which allows to convert from Fourier Series to Fourier Transform:

relation

a_k represents the Fourier series coefficient.

My Question:

Why can't this same result be obtained using the Fourier Transform Analysis equation or the Fourier Transform Tables?

For example, using the Fourier Transform Analysis Equation, we obtain the following.

alternative solution

As we can see, the result obtained does not seem to be the same as my professor's. Or is it? Are the two results equivalent if we were to somehow simplify my answer algebraically? If they're not equivalent, how would I know which method to use, mine or my professor's, to solve this type of problem? Do I use my professor's method because x(t) is periodic?

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  • $\begingroup$ Because that's a complicated theorem that the Fourier transform of the Dirac comb is itself, it is also the Poisson summation formula. It is equivalent to the Fourier series inversion theorem, part of the much more general Fourier_inversion_theorem which can be fully understood only with the help of the distributions $\endgroup$ – reuns Apr 19 '16 at 13:07
  • $\begingroup$ In one word : for proving that the Dirac comb is its own Fourier transform, you have to rely on some sort of weak version of the Fourier_inversion_theorem. $\endgroup$ – reuns Apr 19 '16 at 13:09
  • $\begingroup$ @user1952009 Thanks for that! But, is the answer obtained with the Fourier Analysis equation equivalent to the answer obtained using the Fourier Series <-> Fourier Transform relationship equation? How do I know which method to use? $\endgroup$ – Georan Apr 19 '16 at 13:33
  • $\begingroup$ I don't understand what you are asking. Can you see what are the theorems used by your teacher in each step of his derivation ? $\endgroup$ – reuns Apr 19 '16 at 13:35
  • $\begingroup$ @user1952009 The way my teacher calculates the frequency domain representation of x(t) is as follows: 1) Since x(t) is periodic, he calculates the fourier series coefficient a_k. 2) He finds the fourier transform using the formula shown in my post. The result he obtains is dirac comb. On the other hand, I propose to simply use the fourier analysis equation to find the fourier transform. In this case, the result obtained is the one shown at the bottom of my post. Is my answer equivalent to the professor's if we were to somehow simplify it algebraically? $\endgroup$ – Georan Apr 19 '16 at 13:44

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