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When determining the Fourier Series representation of an impulse train,

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

I have noticed that most proofs determine the coefficients using the bounds of $[-T/2, T/2]$ instead of $[0, T]$:

$a_k = \frac{1}{T}\int_{-T/2}^{T/2}\delta(t)e^{-jk2\pi t/T}dt$

Why? Is there an issue with having the impulses located directly on the endpoints of the integral bounds?

Vivek Kumar
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  • The impulse associated with the $\delta(t)$ in the integrand is located at $t=0$, not at $t=-\frac{T}{2}$ or at $t=\frac{T}{2}$. Therefore using $[0, T]$ instead of $\left[-\frac{T}{2}, \frac{T}{2}\right]$ would cause a problem. Also $\frac{1}{T} \int\limits_{-a}^a \delta(t), e^{-\frac{j k 2 \pi t}{T}},dt=\frac{1}{T}$ for all $a>0$. – Steven Clark Jul 10 '23 at 01:19

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