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I was looking into proof and find something strange:

Proof

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?

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  • $\begingroup$ There are two different visions of what time reversal means in the discrete domain. In one, $x_r[n] = x[N-1-n]$ which replaces $\mathbf x = \big[x[0], x[1],\ldots, x[N-1]\big]$ with $\mathbf x_r = \big[x[N-1], x[N-2],\ldots, x[1], x[0]\big]$ and the other which has $x_r[n] = x[-n]$ which replaces $\big[x[0], x[1],\ldots, x[N-1]\big]$ with $\big[x[0], x[N-1], x[N-2],\ldots, x[1]\big]$ $\endgroup$
    – Dilip Sarwate
    Dec 11 '17 at 3:16
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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, $X[k]=X[k+N]$.

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  • $\begingroup$ yeah, thank you, this came to my head after posting a question) $\endgroup$
    – qqffx
    Dec 10 '17 at 17:54

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