Show that $S[n]= \frac1M \sum e^{\frac{2\pi l n}M}$ for $l = 0$ to $M - 1$

Question

I am learning about image processing. Currently, I am reading about upsampling and down sampling. Here is some background what I am reading:

I was wondering if someone show me a proof of how equation (70) is true. It simply says "It can be shown that".

Thanks!

Answer 1

HINT:

First of all, Eq. $(70)$ is wrong, there's an imaginary unit $j$ missing in the exponent. If you add that missing $j$, note that for $n=kM$, $k\in\mathbb Z$, you have $e^{j2\pi ln/M}=1$, so the result of the scaled sum equals $1$. In order to see that the sum equals zero for all other values of $n$, use the formula of the geometric series.

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