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.