Orthogonality of Sinusoids

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

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

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

The complex sinusoids corresponding to the frequencies are

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

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Subsections

• Nth Roots of Unity
• DFT Sinusoids

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Order a Hardcopy of Mathematics of the DFT

Previous: Geometric Series
Next: Nth Roots of Unity

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written by 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.