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Application of the Shift Theorem to FFT Windows

In practical spectrum analysis, we most often use the Fast Fourier Transform7.16 (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: Let $ w(n)$ denote the window samples for $ n=0,1,2,\ldots,M-1$ . Since the window is symmetric, we have $ w(n)=w(M-1-n)$ for all $ n$ . When $ M$ is odd, there is a sample at the midpoint at time $ n=(M-1)/2$ . The midpoint can be translated to the time origin to create an even signal. As established on page [*], the DFT of a real and even signal is real and even. By the shift theorem, the DFT of the original symmetric window is a real, even spectrum multiplied by a linear phase term, yielding a spectrum having a phase that is linear in frequency with possible discontinuities of $ \pm\pi$ radians. Thus, all odd-length real symmetric signals are ``linear phase'', including FFT windows.

When $ M$ is even, the window midpoint at time $ n=(M-1)/2$ lands half-way between samples, so we cannot simply translate the window to zero-centered form. However, we can still factor the window spectrum $ W(\omega_k)$ into the product of a linear phase term $ \exp[-\omega_k(M-1)/2]$ and a real spectrum (verify this as an exercise), which satisfies the definition of a linear phase signal.

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``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
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University