I'am trying to prove the following equation. $$\sum_{n=1}^{N}f(n-x_1)f(n-x_2)=\delta(x_1-x_2)$$ where $f(x):=\frac{sin(\pi x)}{Nsin(\frac{\pi}{N}x)}$ , and $x_1,x_2$ are integers.
First, I have used the Euler theorem for sine functions. $$ \frac{sin(\pi (n-x_1))}{Nsin(\frac{\pi}{N}(n-x_1))} \frac{sin(\pi (n-x_1))}{Nsin(\frac{\pi}{N}(n-x_1))} = \frac{1}{N}\frac{e^{j\pi(n-x_1)-j\pi(n-x_1)}}{e^{j\frac{\pi}{N}(n-x_1)-j\frac{\pi}{N}(n-x_1)}} \frac{1}{N}\frac{e^{j\pi(n-x_2)-j\pi(n-x_2)}}{e^{j\frac{\pi}{N}(n-x_2)-j\frac{\pi}{N}(n-x_2)}} $$ , and I am stuck here about what to do next. Any hints for next step?
