The Rectangular Window

The (zero-centered) *rectangular window* may be defined by

(4.2) |

where is the window length in samples (assumed odd for now). A plot of the rectangular window appears in Fig.3.1 for length . It is sometimes convenient to define windows so that their dc gain is 1, in which case we would multiply the definition above by .

To see what happens in the frequency domain, we need to look at the DTFT of the window:

where the last line was derived using the closed form of a geometric series:

(4.3) |

We can factor out linear phase terms from the numerator and denominator of the above expression to get

where denotes the

(also called the

(4.6) |

The term ``aliased sinc function'' refers to the fact that it may be
simply obtained by *sampling* the length-
*continuous-time* rectangular window, which has Fourier transform
sinc
(given amplitude
in the time domain). Sampling at intervals of
seconds in
the time domain corresponds to aliasing in the frequency domain over
the interval
Hz, and by direct derivation, we have found the
result. It is interesting to consider what happens as the window
duration increases continuously in the time domain: the magnitude
spectrum can only change in discrete jumps as new samples are
included, even though it is continuously parametrized in
.

As the sampling rate goes to infinity, the aliased sinc function
therefore approaches the *sinc function*

Specifically,

where .

Figure 3.2 illustrates
for
. Note that this is the complete
window transform, not just its real part. We obtain real window
transforms like this only for zero-centered, symmetric windows. Note
that the phase of rectangular-window transform
is
*zero* for
, which is the width of the
*main lobe*. This is why zero-centered windows are often called
*zero-phase windows*; while the phase
actually alternates between 0
and
radians, the
values
occur only within side-lobes which are routinely neglected (in fact,
the window is normally designed to ensure that all side-lobes can be
neglected).

More generally, we may plot both the *magnitude* and *phase*
of the window versus frequency, as shown in
Figures 3.4 and 3.5 below. In
audio work, we more typically plot the window transform magnitude on a
*decibel (dB) scale*, as shown in Fig.3.3 below. It
is common to normalize the peak of the dB magnitude to 0 dB, as we
have done here.

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