Rectangular Window Side-Lobes

From Fig.1.11 and Eq.$\,$(1.3), we see that the main lobe width is $2\cdot 2\pi/M=4\pi/11 \approx 1.1$ 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 $1/\omega$ (see (1.5)), it should ``roll off'' at approximately 6 dB per octave (since $-20\log_{10}(2)=6.0205999\ldots$). This is verified in the log-log plot of Fig.1.12.

Figure 1.12: Roll-off of the rectangular-window Fourier transform.

As the sampling rate approaches infinity, the rectangular window transform ( $\hbox{asinc}$) 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 `` $\hbox{asinc}$'').

Note that each side lobe has width $\Omega_M \isdeftext 2\pi/M$, as measured between zero crossings.^2.7 The main lobe, on the other hand, is width $2\Omega_M$. Thus, in principle, we should never confuse side-lobe peaks with main-lobe peaks, because a peak must be at least $2\Omega_M$ 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 $2\Omega_M$.

In summary, the DTFT of the $M$-sample rectangular window is proportional to the `aliased sinc function':

$$\begin{eqnarray*} \hbox{asinc}_M(\omega) &\isdef & \frac{\sin(\omega M / 2)}{M\c... ...in] &\approx& \frac{\sin(\pi fM)}{\pi f} \isdef M\mbox{sinc}(fM) \end{eqnarray*}$$

Thus, it has zero crossings at integer multiples of

$$\Omega_M \isdef \frac{2\pi}{M}.$$

Its main lobe width is $2\Omega_M$ and its first side-lobe is 13 dB down from the main-lobe peak. As $M$ gets bigger, the main-lobe narrows, giving better frequency resolution (as discussed in the next section). Note that the window-length $M$ 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.