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Application of the Shift Theorem to FFT Windows

In practical spectrum analysis, we most often use the Fast Fourier Transform7.10 (FFT) together with a window function $ w(n), n=0,1,2,\ldots,N-1$. As discussed further in Chapter 8, windows are normally positive ($ w(n)>0$), symmetric about their midpoint, and look pretty much like a ``bell curve.'' A window multiplies the signal $ x$ being analyzed to form a windowed signal $ x_w(n) = w(n)x(n)$, or $ x_w = w\cdot x$, which is then analyzed using an 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 $ x$ is a short section of a longer signal (not a period or whole number of periods from a periodic signal).

Theorem: Real symmetric FFT windows are linear phase.

Proof: The midpoint of any (finite-duration) symmetric signal can be translated to the time origin to create an even signal. As established previously on page [*], the DFT of 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 of frequency (with possible discontinuities of $ \pi $ radians), is linear phase by definition.

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``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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