Application of the Shift Theorem to FFT Windows

In practical spectrum analysis, we most often use the fast Fourier transform^7.8 (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 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 $x$ is a short section of a longer 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, 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.