An Orthonormal Sinusoidal Set

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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).

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Next: The Discrete Fourier Transform (DFT)

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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