# Spectral Phase

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

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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### Spectral Phase

As for the phase of the spectrum, what do we expect? We have chosen thesinusoid phase to be zero. The window is symmetric about its middle. Therefore, we expect a linear phase term with slope -(M-1)/2 samples. Also, the window transform has sidelobes which cause a phase of pi radians to switch in and out. Thus, we expect to see samples of a straight line with slope -15 across the main lobe of the window transform, together with a switching offset by pi in every other sidelobe away from the main lobe, starting with the immediately adjacent sidelobes.

In the plot, you can see the negatively sloped line across the main lobe of the window transform, but the sidelobes are hard to follow.

```plot(fn,phs,'*'); hold on; plot(fni,phsi); grid;
title('Spectral Phase');
xlabel('Normalized Frequency (cycles per sample))');