NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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The DFT and its Inverse

Let denote an -sample complex sequence, i.e., . Then the spectrum of is defined by the Discrete Fourier Transform (DFT):

The inverse DFT (IDFT) is defined by

Note that for the first time we are not carrying along the sampling interval in our notation. This is actually the most typical treatment in the digital signal processing literature. It is often said that the sampling frequency is . However, it can be set to any desired value using the substitution

However, for the remainder of this reader, we will adopt the more common (and more mathematical) convention . In particular, we'll use the definition for this chapter only. In this case, a radian frequency is in units of ''radians per sample.'' Elsewhere in this course, always means ''radians persecond.'' (Of course, there's no difference when the sampling rateis really .) Another term we use in connection with the convention is normalized frequency: All normalized radian frequencies lie in the range , and all normalized frequencies in Hz lie in the range .

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Subsections

• Notation and Terminology
• Modulo Indexing, Periodic Extension

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