# Linear Phase Terms

## GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Linear Phase Terms

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 term is that its phase is a linear function of frequency:

Thus, the slope of the phase versus radian frequency is . In general, the time delay in samples equals minus 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 a negatively sloped linear phase to the original spectral phase. A negative time delay (waveform shift to the left) adds apositively sloped linear 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 be linear phase signal if its phase is of the form

where is an indicator function which takes on the values or .

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