What is the DFT of DFT of discrete signal
let

$$\begin{align}
 X[k] &amp;= \mathcal{DFT} \Big\{ x[n] \Big\} \\
      &amp;\triangleq \sum\limits_{n=0}^{N-1} x[n] \, e^{-j2\pi nk/N}
\end{align} $$

and 

$$ y[n] \triangleq X[n]  $$

(note the substitution of $n$ in for $k$.)  then

$$ Y[k] = \mathcal{DFT} \Big\{ y[n] \Big\} $$

then, if the DFT is defined the most common way (as above):

$$ Y[n] = N \cdot x[-n] $$

where periodicity is implied:  $x[n+N]=x[n]$ for all $n$.