NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 transform8.6(FFT) together with a window 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 a windowed signal
, or
, which is then analyzed using the FFT. The window serves to taper the data segment gracefully to zero, thus eliminating spectral distortions due to suddenly cutting off the signal in time. Windowing is thus appropriate when
Theorem: Real symmetric FFT windows are linear 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 a linear phase term. A spectrum whose phase is a linear function offrequency (with possible discontinuities of
radians), is linear phase by definition.