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Norm of the DFT Sinusoids

For $ k=l$, we follow the previous derivation to the next-to-last step to get

$\displaystyle \left<s_k,s_k\right> = \sum_{n=0}^{N-1}e^{j2\pi (k-k) n /N} = N $

which proves

$\displaystyle \zbox {\left\Vert\,s_k\,\right\Vert = \sqrt{N}.} $

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Orthogonality of the DFT Sinusoids