Can $\sum_{a = -\infty}^{\infty} e^{i\omega aT_0}$ be represented by dirac delta functions?

Question

The usual definition of dirac delta function says that $\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ip(x-\alpha)}\ dp $.

The appearance similarity makes me think that it may be possible to write $\sum_{a = -\infty}^{\infty} e^{i\omega aT_0}$ as sum of dirac delta functions, where $T_0$ is some rational constant number. Can this be done?

Answer 1

The function $f(\omega) = \frac{T_0}{2\pi}\sum_{k=-\infty}^\infty e^{i k \omega T_0}$ is a Fourier series of $\sum_{k=-\infty}^\infty \delta\left(\omega - \frac{2\pi k}{T_0}\right)$. This function is most commonly known under the names the Dirac Comb and the Shah Function.

Answer 2

The discrete-time Fourier transform (DTFT) is defined as $X(\theta)=\sum_n f[n]e^{-i\theta n} $ and it is the equivalent Fourier transform for discrete time series. The resulting $X(\theta)$ is in continuous time and is $2\pi$-periodic.

In your case, $f[n]=1$ and you're asking for $X(\theta)|_{\theta=-\omega T_0}$. The DTFT of $1$ should be a delta function, but it should be periodic as well:

$X(\theta)=2\pi \sum_k \delta(\theta+2\pi k)$.