DFT of time reversed signal I was looking into proof and find something strange:


  


The last part we obtain from DFT definition.
$$X[k] = \sum^{N-1}_{n=0}x[n]W^{kn}_N, \quad\text{Where}\quad W^{kn}_N = e^{-j\frac{2\pi}{N}nk}$$ 

So, looking at $W^{-k}_NX[N-k]$ we can see that when $k = 0$ we will have $W^{0}_NX[N]$ and we know that our DFT coefficients only can be from $[0,N-1]$. 

So my question is: what does it mean when we have $X[N]$ DFT coefficient? Or did i understand something wrong?
You can evaluate $X[k]$ (and $x[n]$) outside the domain $[0,N-1]$. The sequences are periodic with period $N$, so $X[N]=X[0]$, and, more generally, $X[k]=X[k+N]$.