**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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## Linear Phase Terms

The reason is called a

linear phase termis that its phase is a linear function of frequency:

Thus, the slope of the phase versus radian frequency is . In general, thetime delay in samplesequalsminus the slope of thelinear phase term.

If we express the original spectrum in polar form as

where and are the magnitude and phase of , respectively (both real), we can see that a linear phase term only modifies the spectral phase :

where . A positive time delay (waveform shift to the right) adds anegatively slopedlinear phase to the original spectral phase. A negative time delay (waveform shift to the left) adds apositively slopedlinear phase to the original spectral phase. If we seem to be belaboring this relationship, it is because it is one of the most useful in practice.

Definition:A signal is said to belinear phase signalif its phase is of the form

where is anindicator functionwhich takes on the values or .