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

The complex sinusoids corresponding to the frequencies are

These sinusoids are generated by the th

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

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University