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Linear Phase Terms

The reason $ e^{-j \omega_k \Delta}$ is called a linear phase term is that its phase is a linear function of frequency:

$\displaystyle \angle e^{-j \omega_k \Delta} = - \Delta \cdot \omega_k

Thus, the slope of the phase, viewed as a linear function of radian-frequency $ \omega_k$, is $ -\Delta$. 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

$\displaystyle X(k) = G(k) e^{j\Theta(k)},

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

$\displaystyle e^{-j \omega_k \Delta} X(k) \isdef
e^{-j \omega_k \Delta} G(k) e^{j\Theta(k)}
= G(k) e^{j[\Theta(k)-\omega_k\Delta]}

where $ \omega_k \isdeftext 2\pi
k/N$. 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.

Definition: A signal is said to be a linear phase signal if its phase is of the form

$\displaystyle \Theta(\omega_k)= \pm \Delta \cdot \omega_k\pm \pi I(\omega_k),

where $ \pm\Delta$ is any real constant, and $ I(\omega_k)$ is an indicator function which takes on the values 0 or $ 1$ over the points $ \omega_k$, $ k=0,1,2,\ldots,N-1$.

A zero-phase signal is thus a linear phase signal for which the phase-slope $ \Delta$ is zero.

As mentioned above (in §7.4.3), it would be more precise to say ``piecewise constant phase'' instead of ``zero phase''. Similarly, ``linear phase'' is better described as ``linear phase interrupt by discontinuities.''

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[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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