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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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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA