Rectangular Window Summary

The rectangular window was discussed in Chapter 5 (§3.1). Here we summarize the results of that discussion.

Definition ($M$ odd):

$$w_R(n) \isdef \left\{\begin{array}{ll} 1, & \left\vert n\right\vert\leq\frac{M-1}{2} \\ [5pt] 0, & \hbox{otherwise} \\ \end{array} \right.$$ (4.12)

Transform:

$$W_R(\omega) = M\cdot \hbox{asinc}_M(\omega) \isdef \frac{\sin\left(M\frac{\omega}{2}\right)}{\sin\left(\frac{\omega}{2}\right)}$$ (4.13)

The DTFT of a rectangular window is shown in Fig.3.7.

Figure 3.7: Rectangular window discrete-time Fourier transform.

Properties:

• Zero crossings at integer multiples of

  $$\Omega_M \isdef \frac{2\pi}{M} = \hbox{frequency sampling interval for a length $M$\ DFT.}$$ (4.14)

  
  

• Main lobe width is $2 \Omega_M = \frac{4\pi}{M}$ .
• As $M$ increases, the main lobe narrows (better frequency resolution).
• $M$ 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).