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Mathematics of the DFT
    Fourier Theorems for the DFT
       Fourier Theorems
          Shift Theorem
             Linear Phase Terms

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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.


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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 Associate Professor (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.


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