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
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(4.11) |