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Choosing Window Length to Resolve Sinusoids

A conservative requirement for resolving 2 sinusoids (in noisy conditions) with a spacing of $ \Delta f$ Hz is to choose a window length $ M$ long enough so that their main lobes are clearly discernible. For example, we may require that their main lobes meet at the first zero crossings.


\begin{psfrags}\psfrag{freq}{$\omega T$\ (radians per sample)}\begin{center}
\epsfig{file=eps/sinesAnn.eps,width=5in} \\
\end{center} % was epsfbox
\end{psfrags}

To obtain the separation shown above, we must have $ B_w \le \Delta f$ , where $ B_w$ is the main lobe width in Hz, and $ \Delta f$ is the sinusoidal frequency separation in Hz.

For the rectangular window, $ B_w$ can be expressed as

$\displaystyle B_w = 2 \frac{f_s}{M}
$

Hence we need:

\begin{eqnarray*}
B_w&=& 2 \frac{f_s}{M} \le \Delta f \\
&\Rightarrow& M \ge 2 \frac{f_s}{\Delta f}
\end{eqnarray*}

or

$\displaystyle \zbox{M \ge 2 \frac{f_s}{\vert f_2-f_1\vert}}
$


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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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