Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Sidelobe Roll-Off Rate

In general, if the first $ n$ derivatives of a continuous function $ w(t)$ exist (i.e., they are finite and uniquely defined), then its Fourier Transform magnitude is asymptotically proportional to

$\displaystyle \vert W(\omega)\vert \to \frac{\hbox{constant}}{\omega^{n+1}}
\quad(\hbox{as }\omega\to\infty)
$

Proof: Look up ``roll-off rate'' in text index.

Examples: For discrete-time windows, the roll-off rate slows down at high frequencies due to aliasing.

In summary, the DTFT of the $ M$ -sample rectangular window is proportional to the `aliased sinc function':

\begin{eqnarray*}
\hbox{asinc}_M(\omega T) &\mathrel{\stackrel{\Delta}{=}}& \frac{\sin(\omega M T / 2)}{M\sin(\omega T/2)} \\ [0.2in]
&\approx& \frac{\sin(\pi fMT)}{M\pi fT} \mathrel{\stackrel{\mathrm{\Delta}}{=}}\mbox{sinc}(fMT)
\end{eqnarray*}

Some important points (rect window transform):


Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Download Intro421.pdf
Download Intro421_2up.pdf
Download Intro421_4up.pdf
[Comment on this page via email]

``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
CCRMA  [Automatic-links disclaimer]