DFT dirac simplification
it's not a Dirac delta, it's a Kroenecker (discrete-time) delta.

$$ \begin{align} \frac{1}{N} \sum\limits_{k=0}^{N-1} \left(-W_N^n \right)^k \ &amp; = \ \frac{1}{N} \sum\limits_{k=0}^{N-1} \left(-e^{j 2 \pi n/N} \right)^k \\
&amp; = \ \frac{1}{N} \sum\limits_{k=0}^{N-1} \left(e^{-j\pi}e^{j 2 \pi n/N} \right)^k \\
&amp; = \ \frac{1}{N} \sum\limits_{k=0}^{N-1} \left(e^{j 2 \pi (n-N/2)/N}\right)^k  \\
&amp; = \ \frac{1}{N} \frac{1 - e^{j 2 \pi (n-N/2)}}{1 - e^{j 2 \pi (n-N/2)/N}} \\
&amp; = \ \begin{cases} \frac{1}{N} N \text{  for } n = N/2 + mM \text{  where } m \in \text{integers} \\ \frac{1}{N} 0 \text{  otherwise} \\ \end{cases}
\end{align} $$

so it's not exactly $ \delta[n-N/2] $ but more like discrete kronecker comb.

$$ \frac{1}{N} \sum\limits_{k=0}^{N-1} \left(-e^{j 2 \pi n/N} \right)^k  =  \sum\limits_{m=-\infty}^{+\infty} \delta[n-N/2+mN]$$