**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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## Application of the Shift Theorem to FFT Windows

In practical spectrum analysis, we most often use the

fast Fourier transform^{8.6}(FFT) together with awindow function. Windows are normally positive (), symmetric about their midpoint, and look pretty much like a “bell curve.” A window multiplies the signal being analyzed to form awindowed signal, or , which is then analyzed using the FFT. The window serves totaperthe data segment gracefully to zero, thus eliminating spectral distortions due to suddenly cutting off the signal in time. Windowing is thus appropriate when is a short section of a longer signal.

Theorem:Real symmetric FFT windows arelinear phase.

Proof:The midpoint of a symmetric signal can be translated to the time origin to create an even signal. As established previously, the DFTof a real even signal is real and even. By the shift theorem, the DFT of the original symmetric signal is a real even spectrum multiplied by alinear phase term. A spectrum whose phase is a linear function offrequency (with possible discontinuities of radians), islinear phaseby definition.