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
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
Zero-Phase Zero Padding