Fourier transform of an infinite sum

Question

Find the fourier transform of

$$\sum_{m=-\infty}^\infty f (t-mT)$$

Where $f (t) = 1 $ for $0 <t < T $ and 0 otherwise. I am not sure how to tackle it ? Also I have problems if it's correct to interchange summation with integration.

Answer 1

$$f_T(t)=\sum_{m=-\infty}^\infty f (t-mT)$$ is a periodic function with period $T$, and within each period it is equal to $f(t)$.

It has a Fourier series representation which is simple (it is just like a square wave). Assume the FS representation is $$f_T(t)=\sum_{k=-\infty}^{\infty}c_ke^{i\frac{2k\pi }{T} t}$$ Take the term-by-term Fourier transform of the series, using the linearity of Fourier transform: $$\begin{align}\mathcal{F}(f_T(t))&=\mathcal{F}\left(\sum_{k=-\infty}^{\infty}c_ke^{i\frac{2k\pi}{T} t}\right)\\ &=\sum_{k=-\infty}^{\infty}c_k\mathcal{F}\left(e^{i\frac{2k\pi}{T} t}\right)\\ &=2\pi \sum_{k=-\infty}^{\infty}c_k\delta(\omega-\frac{2k\pi}{T}) \end{align}$$