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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, viewed as a linear function of
radian-frequency

, is

. In general, the

*time
delay in samples* equals

*minus the slope of the linear 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 a

*positively 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.

**Previous:** Shift Theorem**Next:** Linear Phase Signals

**About the Author: ** 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 (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.