The DFT and its Inverse Restated

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

In this chapter, we will omit mention of an explicit sampling interval , as this is most typical in the digital signal processing literature. It is often said that the sampling frequency is . In this case, a radian frequency is in units of ``radians per sample.'' Elsewhere in this book, usually means ``radians per second.'' (Of course, there's no difference when the sampling rate is 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 .^7.1 Note that physical units of seconds and Hertz can be reintroduced by the substitution

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Subsections

• Notation and Terminology
• Modulo Indexing, Periodic Extension

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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.