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The Rectangular Window

The rectangular window may be defined as:

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

\epsfig{file=eps/rectWindow.eps,width=5in}

To see what happens in the frequency domain, we need to look at the DTFT of the window:


\begin{eqnarray*}
W_R(\omega )
& = & \hbox{\sc DTFT}_\omega(w_R) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\sum_{n=-\infty}^\infty
w_R(n)e^{-j\omega n} \\
& = & \sum_{n=-\frac{M-1}{2}}^{\frac{M-1}{2}} e^{-j \omega n}
= \frac{e^{j \omega \frac{M-1}{2}} - e^{-j \omega \frac{M+1}{2}} }{1 - e^{-j \omega }}
\end{eqnarray*}

where we used the closed form of a geometric series:

$\displaystyle \sum_{n=L}^U r^n = \frac{ r^L - r^{U+1}}{1-r}
$

We can factor out linear phase terms from the numerator and denominator of the above expression to get

\begin{eqnarray*}
W_R(\omega)
&=& \frac{e^{-j \omega \frac{1}{2}}}{e^{-j\omega \frac{1}{2}}}
\left[ \frac{ e^{j \omega \frac{M}{2}}-e^{-j\omega\frac{M}{2}}}
{e^{j \omega\frac{1}{2}}-e^{-j\omega\frac{1}{2}}} \right]
\nonumber \\ &=&
\frac{\sin\left(M\frac{\omega}{2}\right)}{\sin\left(\frac{\omega}{2}\right)}
\mathrel{\stackrel{\mathrm{\Delta}}{=}}M\cdot \hbox{asinc}_M(\omega)
\protect
\end{eqnarray*}

where $ \hbox{asinc}_M(\omega)$ denotes the aliased sinc function.

$\displaystyle \hbox{asinc}_M(\omega) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\frac{\sin(M\omega/2)}{M\cdot \sin(\omega/2)}
$

(also called the Dirichlet function)


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``Music 421 Overview'', by Julius O. Smith III, (From Lecture Overheads, Music 421).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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