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Rectangular Window Summary

The rectangular window was discussed in Chapter 53.1). Here we summarize the results of that discussion.


Definition ($ M$ odd):

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


Transform:

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

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

Figure 3.7: Rectangular window discrete-time Fourier transform.
\includegraphics[width=\twidth]{eps/Rect}


Properties:


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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