Theorem: For any and any integer ,
The shift theorem is often expressed in shorthand as
A zero-phase signal is thus a linear-phase signal for which the phase-slope is zero. As mentioned above (in §7.4.3), it would be more precise to say ``0-or--phase signal'' instead of ``zero-phase signal''. Another better term is ``zero-centered signal'', since every real (even) spectrum corresponds to an even (real) signal. Of course, a zero-centered symmetric signal is simply an even signal, by definition. Thus, a ``zero-phase signal'' is more precisely termed an ``even signal''.
Application of the Shift Theorem to FFT Windows
In practical spectrum analysis, we most often use the Fast Fourier Transform7.15 (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.