Rectangular Window Side-Lobes

From Fig.1.11 and Eq.(1.3), 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 (1.3)), which has an amplitude envelope proportional to (see (1.5)), it should ``roll off'' at approximately 6 dB per octave (since ). This is verified in the log-log plot of Fig.1.12.

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.^2.7 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 §2.5.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.