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Orthogonality of Sinusoids

A key property of sinusoids is that they are orthogonal at different frequencies. That is,

$\displaystyle \omega_1 \neq \omega_2 \implies
A_1\sin(\omega_1 t + \phi_1) \perp
A_2\sin(\omega_2 t + \phi_2).

This is true whether they are complex or real, and whatever amplitude and phase they may have. All that matters is that the frequencies be different. Note, however, that the durations must be infinity (in general).

For length $ N$ sampled sinusoidal signal segments, such as used by the DFT, exact orthogonality holds only for the harmonics of the sampling-rate-divided-by-$ N$ , i.e., only for the frequencies (in Hz)

$\displaystyle f_k = k \frac{f_s}{N}, \quad k=0,1,2,3,\ldots,N-1.

These are the only frequencies that have a whole number of periods in $ N$ samples (depicted in Fig.6.2 for $ N=8$ ).6.1

The complex sinusoids corresponding to the frequencies $ f_k$ are

$\displaystyle s_k(n) \isdef e^{j\omega_k nT},
\quad \omega_k \isdef k \frac{2\pi}{N}f_s,
\quad k = 0,1,2,\ldots,N-1.

These sinusoids are generated by the $ N$ th roots of unity in the complex plane.

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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 © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University