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

.
Figure 3.1:
The rectangular window.
![\includegraphics[width=3.5in]{eps/rectWindow}](http://www.dsprelated.com/josimages_new/sasp2/img306.png) |
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
aliased sinc
function:
4.1
 |
(4.5) |
(also called the
Dirichlet function
[
175,
72] or
periodic sinc function).
This (real) result is for the
zero-centered rectangular window. For the
causal case, a
linear
phase term appears:
 |
(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
![$ [0,1/T]$](http://www.dsprelated.com/josimages_new/sasp2/img321.png)
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
sinc |
(4.7) |
Specifically,
sinc |
(4.8) |
where

.
4.2
Figure 3.2:
Fourier
transform of the rectangular window.
![\includegraphics[width=\textwidth ,height=2.25in]{eps/rectWindowRawFT}](http://www.dsprelated.com/josimages_new/sasp2/img329.png) |
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.
Figure 3.3:
Magnitude (dB) of the
rectangular-window transform.
![\includegraphics[width=\twidth]{eps/rectWindowFT}](http://www.dsprelated.com/josimages_new/sasp2/img334.png) |
Figure 3.4:
Magnitude of the rectangular-window Fourier transform.
![% latex2html id marker 73365
\includegraphics[width=\twidth,height=0.3125\theight]{eps/rectWindowFTzeroX}](http://www.dsprelated.com/josimages_new/sasp2/img335.png) |
Figure 3.5:
Phase of the rectangular-window Fourier transform.
![% latex2html id marker 73369
\includegraphics[width=\twidth,height=0.3125\theight]{eps/rectWindowPhaseFT}](http://www.dsprelated.com/josimages_new/sasp2/img336.png) |
Rectangular
Window Side-Lobes
From Fig.
3.3 and Eq.

(
3.4), we see that the
main-lobe width is

radian, and the
side-lobe level is 13
dB down.
Since the
DTFT of the rectangular window approximates the
sinc
function (see (
3.4)), which has an
amplitude envelope
proportional to

(see (
3.7)), it should ``roll
off'' at approximately 6
dB per octave (since

). This is verified in the log-log
plot of Fig.
3.6.
As the
sampling rate approaches infinity, the rectangular window
transform (

) converges exactly to the
sinc
function.
Therefore, the departure of the roll-off from that of the
sinc
function can be ascribed to
aliasing in the
frequency domain,
due to
sampling in the time domain (hence the name ``

'').
Note that each side lobe has width

, as
measured between zero crossings.
4.3 The main lobe, on the other hand, is
width

. Thus, in principle, we should never confuse
side-lobe peaks with main-lobe peaks, because a peak must be at least

wide in order to be considered ``real''. However, in
complicated real-world scenarios, side-lobes can still cause
estimation errors (``
bias''). Furthermore, two
sinusoids at closely
spaced frequencies and opposite phase can partially cancel each
other's main lobes, making them appear to be narrower than

.
In summary, the DTFT of the

-sample rectangular window is
proportional to the `
aliased sinc function':
Thus, it has zero crossings at integer multiples of
 |
(4.11) |
Its main-lobe width is

and its first side-lobe is 13
dB
down from the main-lobe peak. As

gets bigger, the main-lobe
narrows, giving better
frequency resolution (as discussed in the next
section). Note that the window-length

has
no effect on
side-lobe level (ignoring aliasing). The side-lobe height is instead
a result of the abruptness of the window's transition from 1 to 0 in
the time domain. This is the same thing as the so-called
Gibbs phenomenon seen in truncated
Fourier series expansions of
periodic waveforms. The abruptness of the window discontinuity in the
time domain is also what determines the side-lobe roll-off rate
(approximately 6
dB per octave). The relation of roll-off rate to the
smoothness of the window at its endpoints is discussed in
§
B.18.
Rectangular Window Summary
The rectangular window was discussed in Chapter
5 (§
3.1). Here we summarize
the results of that discussion.
Definition (
odd):
![$\displaystyle w_R(n) \isdef \left\{\begin{array}{ll} 1, & \left\vert n\right\vert\leq\frac{M-1}{2} \\ [5pt] 0, & \hbox{otherwise} \\ \end{array} \right.$](http://www.dsprelated.com/josimages_new/sasp2/img349.png) |
(4.12) |
Transform:
 |
(4.13) |
The
DTFT of a rectangular window is shown in Fig.
3.7.
Properties:
- Zero crossings at integer multiples of
 |
(4.14) |
- Main lobe width is
.
- As
increases, the main lobe narrows (better frequency resolution).
has no effect on the height of the side lobes
(same as the ``Gibbs phenomenon'' for truncated Fourier series expansions).
- First side lobe only 13 dB down from the main-lobe peak.
- Side lobes roll off at approximately 6dB per octave.
- A phase term arises when we shift the window to make
it causal, while the window
transform is real in the zero-phase case
(i.e., centered about time 0).
Next Section: Generalized Hamming
Window FamilyPrevious Section: Spectral Interpolation