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I am doing a excercise on discrete fourier transforms. The excercise asks to find the resultant signal after applying DFT twice. I was able to figure it out by thinking it of in this way, multiplying the two DFT matrices directly and seeing the resultant matrix and figuring out.

But I am not able to do it (when I do the following method)

$$X[k] = \sum_{n = 0}^{N-1} x[n]e^{-j\frac{2\pi}{N}nk}$$ and then doing $$y[m] = \sum_{k = 0}^{N-1} X[k]e^{-j\frac{2\pi}{N}mk}$$ and then expanind $X[k]$ so it becomes $$y[m] = \sum_{k = 0}^{N-1} \sum_{n = 0}^{N-1} x[n]e^{-j\frac{2\pi}{N}nk}e^{-j\frac{2\pi}{N}mk}$$ and I am not able to solve this summation. How to proceed from here it doesn't seem obvious from here.

abkds
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    Try switching the order of summation. With the $n$ summation on the outside, you can pull out the $x[n]$. This will make the inner summation a finite geometric series, which has a closed form – Mike Earnest Feb 06 '15 at 18:57

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