Application of the Shift Theorem to FFT Windows

In practical spectrum analysis, we most often use the *Fast
Fourier Transform*^{7.16} (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.

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

Copyright ©

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