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<div><h2 id="a0000000035">5.8.1 Window functions</h2>
<p> <a name="a0000000580" id="a0000000580"></a> </p><p>A range of commonly-used window functions may automatically be applied to data as it is read into the <tt class="tt">fft</tt> and <tt class="tt">ifft</tt> commands; these are listed together with their algebraic forms in Table <a href="sect0034.html#tab:windowfuncs">5.1</a> and shown in Figure <a href="sect0034.html#fig:windowfuncs">5.1</a>. In each case, the window functions are given for sample number <img src="images/img-0025.png" alt="$n$" style="vertical-align:0px;
width:11px;
height:8px" class="math gen" />, which ranges between <img src="images/img-0155.png" alt="$0$" style="vertical-align:0px;
width:9px;
height:12px" class="math gen" /> and <img src="images/img-0241.png" alt="$N_ x$" style="vertical-align:-2px;
width:23px;
height:14px" class="math gen" />. The window functions may be invoked using the following syntax: </p><pre>
fft [...] <out>() of <in>() window <window_name>
</pre><p>Where multi-dimensional FFTs are performed, window functions are applied to each dimension in turn. Other arbitrary window functions may be implemented by pre-multiplying data before entry to the <tt class="tt">fft</tt> and <tt class="tt">ifft</tt> commands. </p><div id="tab:windowfuncs" class="table"><center> <table cellspacing="0" class="tabular">
<tr>
<td style="border-top-style:solid; text-align:left; border-top-color:black; border-top-width:1px; border-left:1px solid black"><p> <b class="bf">Window Name</b> </p></td>
<td style="border-top-style:solid; text-align:left; border-top-color:black; border-top-width:1px; border-right:1px solid black"><p> <b class="bf">Algebraic Definition</b> </p></td>
</tr><tr>
<td style="border-top-style:solid; text-align:left; border-top-color:black; border-top-width:1px; border-left:1px solid black"><p>Bartlett </p></td>
<td style="border-top-style:solid; text-align:left; border-top-color:black; border-top-width:1px; border-right:1px solid black"><p> <img src="images/img-0242.png" alt="$\displaystyle w(n) = \left( \frac{2}{N_ x-1} \right) \left( \frac{N_ x-1}{2} - \left| n - \frac{N_ x-1}{2} \right| \right)$" style="vertical-align:-17px;
width:359px;
height:45px" class="math gen" /> <img src="images/img-0243.png" alt="\rule{0pt}{20pt}" style="vertical-align:0px; width:1px; height:1px" class="box gen" /></p></td>
</tr><tr>
<td style="text-align:left; border-left:1px solid black"><p>BartlettHann </p></td>
<td style="text-align:left; border-right:1px solid black"><p> <img src="images/img-0244.png" alt="$\displaystyle w(n) = a_0 - a_1\left|\frac{n}{N_ x-1}-\frac{1}{2}\right| - a_2\cos \left(\frac{2\pi n}{N_ x-1}\right),\; \textrm{for}$" style="vertical-align:-17px;
width:423px;
height:45px" class="math gen" /> <img src="images/img-0245.png" alt="\rule{0pt}{\wfgap }" style="vertical-align:0px; width:1px; height:1px" class="box gen" /></p></td>
</tr><tr>
<td style="text-align:left; border-left:1px solid black"> </td>
<td style="text-align:left; border-right:1px solid black"><p> <img src="images/img-0246.png" alt="$a_0=0.62,\; a_1=0.48,\; a_2=0.38.$" style="vertical-align:-4px;
width:252px;
height:16px" class="math gen" /> <img src="images/img-0243.png" alt="\rule{0pt}{20pt}" style="vertical-align:0px; width:1px; height:1px" class="box gen" /></p></td>
</tr><tr>
<td style="text-align:left; border-left:1px solid black"><p>Cosine </p></td>
<td style="text-align:left; border-right:1px solid black"><p> <img src="images/img-0247.png" alt="$\displaystyle w(n) = \cos \left(\frac{\pi n}{N_ x-1} - \frac{\pi }{2} \right)$" style="vertical-align:-17px;
width:209px;
height:44px" class="math gen" /> <img src="images/img-0245.png" alt="\rule{0pt}{\wfgap }" style="vertical-align:0px; width:1px; height:1px" class="box gen" /></p></td>
</tr><tr>
<td style="text-align:left; border-left:1px solid black"><p>Gauss </p></td>
<td style="text-align:left; border-right:1px solid black"><p> <img src="images/img-0248.png" alt="$\displaystyle w(n) = \exp \left\{ -\frac{1}{2}\left[ \frac{n-(N_ x-1)/2}{\sigma (N_ x-1)/2} \right]^2 \right\} ,\; \textrm{for}\; \sigma =0.5$" style="vertical-align:-23px;
width:404px;
height:55px" class="math gen" /> <img src="images/img-0245.png" alt="\rule{0pt}{\wfgap }" style="vertical-align:0px; width:1px; height:1px" class="box gen" /></p></td>
</tr><tr>
<td style="text-align:left; border-left:1px solid black"><p>Hamming </p></td>
<td style="text-align:left; border-right:1px solid black"><p> <img src="images/img-0249.png" alt="$\displaystyle w(n) = 0.54 - 0.46\cos \left(\frac{2\pi n}{N_ x-1}\right)$" style="vertical-align:-17px;
width:262px;
height:44px" class="math gen" /> <img src="images/img-0245.png" alt="\rule{0pt}{\wfgap }" style="vertical-align:0px; width:1px; height:1px" class="box gen" /></p></td>
</tr><tr>
<td style="text-align:left; border-left:1px solid black"><p>Hann </p></td>
<td style="text-align:left; border-right:1px solid black"><p> <img src="images/img-0250.png" alt="$\displaystyle w(n) = 0.5 \left[ 1 - \cos \left(\frac{2\pi n}{N_ x-1}\right) \right]$" style="vertical-align:-17px;
width:246px;
height:44px" class="math gen" /> <img src="images/img-0245.png" alt="\rule{0pt}{\wfgap }" style="vertical-align:0px; width:1px; height:1px" class="box gen" /></p></td>
</tr><tr>
<td style="text-align:left; border-left:1px solid black"><p>Lanczos </p></td>
<td style="text-align:left; border-right:1px solid black"><p> <img src="images/img-0251.png" alt="$\displaystyle w(n) = \mathrm{sinc}\left( \frac{2n}{N_ x-1} - 1 \right)$" style="vertical-align:-17px;
width:209px;
height:44px" class="math gen" /> <img src="images/img-0245.png" alt="\rule{0pt}{\wfgap }" style="vertical-align:0px; width:1px; height:1px" class="box gen" /></p></td>
</tr><tr>
<td style="text-align:left; border-left:1px solid black"><p>Rectangular </p></td>
<td style="text-align:left; border-right:1px solid black"><p> <img src="images/img-0252.png" alt="$\displaystyle w(n) = 1$" style="vertical-align:-4px;
width:70px;
height:18px" class="math gen" /> <img src="images/img-0245.png" alt="\rule{0pt}{\wfgap }" style="vertical-align:0px; width:1px; height:1px" class="box gen" /></p></td>
</tr><tr>
<td style="border-bottom-color:black; border-bottom-width:1px; text-align:left; border-bottom-style:solid; border-left:1px solid black"><p>Triangular </p></td>
<td style="border-bottom-color:black; border-bottom-width:1px; text-align:left; border-bottom-style:solid; border-right:1px solid black"><p> <img src="images/img-0253.png" alt="$\displaystyle w(n) = \left( \frac{2}{N_ x} \right) \left( \frac{N_ x}{2} - \left| n - \frac{N_ x-1}{2} \right| \right)$" style="vertical-align:-17px;
width:297px;
height:45px" class="math gen" /> <img src="images/img-0245.png" alt="\rule{0pt}{\wfgap }" style="vertical-align:0px; width:1px; height:1px" class="box gen" /></p></td>
</tr>
</table> </center><div class="caption"><b>Table 5.1</b>: <span>Window functions available in the <tt class="tt">fft</tt> and <tt class="tt">ifft</tt> commands.</span></div></div><div id="fig:windowfuncs" class="figure"><center> <img src="images/img-0255.png" alt="\includegraphics{examples/eps/ex_windowfuncs}" style="width:498px; height:772px" />
</center><div class="caption"><b>Figure 5.1</b>: <span>Window functions available in the <tt class="tt">fft</tt> and <tt class="tt">ifft</tt> commands.</span></div></div></div>
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