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Rectangular Window Transform (Cont'd)

Above, we found the rectangular window transform to be the aliased sinc function:

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

\epsfig{file=eps/rectWindowRawFT.eps,width=6in,height=3in}
This (real) result is for the zero-centered rectangular window. For the causal case, a linear phase term appears:

$\displaystyle W^c_R(\omega) = e^{-j\frac{M-1}{2}\omega}M\hbox{asinc}_M(\omega)
$

As the sampling rate goes to infinity, the aliased sinc function approaches the regular sinc function

   sinc$\displaystyle (x) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\frac{\sin(\pi x)}{\pi x}
$

More generally, we may plot both the magnitude and phase of the window transform versus frequency:


\begin{psfrags}\psfrag{freq}{ \footnotesize $ \omega / \Omega_M $\ }\begin{center}
\epsfig{file=eps/rectWindowFTzeroX.eps,width=6in,height=3.5in} \\
\end{center} % was epsfbox
\end{psfrags}


\begin{psfrags}\psfrag{freq}{ \footnotesize $ \omega / \Omega_M $\ }\begin{center}
\epsfig{file=eps/rectWindowPhaseFT.eps,width=6in,height=3.5in} \\
\end{center} % was epsfbox
\end{psfrags}

In audio work, we more typically plot the window transform magnitude on a decibel (dB) scale:


\begin{psfrags}\psfrag{freq}{$\omega T$\ (radians per sample)}\begin{center}
\epsfig{file=eps/rectWindowFT.eps,width=6in} \\
\end{center}
\end{psfrags}

Since the DTFT of the rectangular window approximates the sinc function, it should ``roll off'' at approximately 6 dB per octave, as verified in the log-log plot below:


\begin{psfrags}\psfrag{freq}{$\omega T$\ (radians per sample)}\begin{center}
\epsfig{file=eps/rectWindowLLFT.eps,width=6in} \\
\end{center}
\end{psfrags}

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.


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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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