DFT Definition

The Discrete Fourier Transform (DFT) of a signal may be defined by

where ` ' means ``is defined as'' or ``equals by definition'', and

The sampling interval is also called the sampling period. For a tutorial on sampling continuous-time signals to obtain non-aliased discrete-time signals, see Appendix D.

When all signal samples are real, we say . If they may be complex, we write . Finally, means is any integer.

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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.