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Rectangular Window

The rectangular window was discussed in Chapter 44.5). Here we summarize the results of that discussion.


Definition ($ M$ odd):

$\displaystyle 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.
$


Transform:

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

The DTFT of a rectangular window is shown in Fig.3.1.

Figure 3.1: Rectangular window discrete-time Fourier transform.
\includegraphics[width=\twidth]{eps/Rect}


Properties:

  • Zero crossings at integer multiples of

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

  • 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).


Previous: Spectrum Analysis Windows
Next: Generalized Hamming Window Family

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About the Author: 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.


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