**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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

spectrumof is defined by theDiscrete Fourier Transform(DFT):

Theinverse 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 definitionfor 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 isnormalized frequency: All normalized radian frequencies lie in the range , and all normalized frequencies in Hz lie in the range .

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