#### Application of the Shift Theorem to FFT Windows

In practical spectrum analysis, we most often use the*Fast Fourier Transform*

^{7.15}(FFT) together with a

*window function*. As discussed further in Chapter 8, 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 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 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 denote the window samples for . Since the window is symmetric, we have for all . When is odd, there is a sample at the midpoint at time . 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 radians. Thus, all odd-length real symmetric signals are ``linear phase'', including FFT windows. When is even, the window midpoint at time lands half-way between samples, so we cannot simply translate the window to zero-centered form. However, we can still factor the window spectrum into the product of a linear phase term and a real spectrum (verify this as an exercise), which satisfies the definition of a linear phase signal.

**Next Section:**

Normalized DFT Power Theorem

**Previous Section:**

Zero Phase Signals