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Resolution Bandwidth (resolving sinusoids)


Our ability to resolve two closely spaced sinusoids is determined by the main-lobe-width and sidelobe-level of our window's Fourier transform.

Let $ B_w$ denote the main lobe width in Hz, with the main lobe width defined as the width between zero crossings:

\epsfig{file=eps/WR2.eps,width=5in,height=3.5in}

For the Rectangular Window (length $ M$ ), we have

$\displaystyle W_R(\omega) = \hbox{asinc}_M(\omega)
\mathrel{\stackrel{\Delta}{=}}\frac{ \sin \left( M\omega T/2 \right)}{\sin(\omega T/2)}
= \frac{ \sin \left( M\pi f T \right)}{\sin(\pi f T)}
$

Main lobe width is ``two sidelobes wide''

$\displaystyle \Rightarrow \quad \zbox{B_w = 2\frac{\Omega_M}{2\pi} = 2\frac{f_s}{M}}\quad \hbox{(\mbox{Hz})}
$


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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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