NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 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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