Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


An Orthonormal Sinusoidal Set

We can normalize the DFT sinusoids to obtain an orthonormal set:

$\displaystyle \tilde{s}_k(n) \isdef \frac{s_k(n)}{\sqrt{N}} = \frac{e^{j2\pi k n /N}}{\sqrt{N}}
$

The orthonormal sinusoidal basis signals satisfy

$\displaystyle \left<\tilde{s}_k,\tilde{s}_l\right> = \left\{\begin{array}{ll}
1, & k=l \\ [5pt]
0, & k\neq l. \\
\end{array} \right.
$

We call these the normalized DFT sinusoids. In §6.10 below, we will project signals onto them to obtain the normalized DFT (NDFT).


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``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 © 2014-10-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA