/usr/share/doc/pyxplot/html/sect0028.html is in pyxplot-doc 0.8.4-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 | <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
<head>
<meta name="generator" content="plasTeX" />
<meta content="text/html; charset=utf-8" http-equiv="content-type" />
<title>PyXPlot Users' Guide: Fourier Transforms</title>
<link href="sect0029.html" title="Window Functions" rel="next" />
<link href="ex-interpolation.html" title="Datafile Interpolation" rel="prev" />
<link href="ch-numerics.html" title="Working with Data" rel="up" />
<link rel="stylesheet" href="styles/styles.css" />
</head>
<body>
<div class="navigation">
<table cellspacing="2" cellpadding="0" width="100%">
<tr>
<td><a href="ex-interpolation.html" title="Datafile Interpolation"><img alt="Previous: Datafile Interpolation" border="0" src="icons/previous.gif" width="32" height="32" /></a></td>
<td><a href="ch-numerics.html" title="Working with Data"><img alt="Up: Working with Data" border="0" src="icons/up.gif" width="32" height="32" /></a></td>
<td><a href="sect0029.html" title="Window Functions"><img alt="Next: Window Functions" border="0" src="icons/next.gif" width="32" height="32" /></a></td>
<td class="navtitle" align="center">PyXPlot Users' Guide</td>
<td><a href="index.html" title="Table of Contents"><img border="0" alt="" src="icons/contents.gif" width="32" height="32" /></a></td>
<td><a href="sect0255.html" title="Index"><img border="0" alt="" src="icons/index.gif" width="32" height="32" /></a></td>
<td><img border="0" alt="" src="icons/blank.gif" width="32" height="32" /></td>
</tr>
</table>
</div>
<div class="breadcrumbs">
<span>
<span>
<a href="index.html">PyXPlot Users' Guide</a> <b>:</b>
</span>
</span><span>
<span>
<a href="sect0001.html">Introduction to PyXPlot</a> <b>:</b>
</span>
</span><span>
<span>
<a href="ch-numerics.html">Working with Data</a> <b>:</b>
</span>
</span><span>
<span>
<b class="current">Fourier Transforms</b>
</span>
</span>
<hr />
</div>
<div><h1 id="a0000000029">5.8 Fourier Transforms</h1>
<p>The <tt class="tt">fft</tt> and <tt class="tt">ifft</tt> commands<a name="a0000000529" id="a0000000529"></a><a name="a0000000530" id="a0000000530"></a> take Fourier transforms and inverse Fourier transforms respectively of data supplied either from a file or from a function. In each case, a regular grid of abscissa values must be specified on which to take the discrete Fourier transform, which can extend over an arbitrary number of dimensions. The following example demonstrates the syntax of these commands as applied to a two-dimensional top-hat function: </p><pre>
step(x,y) = tophat(x,0.2) * tophat(y,0.4)
fft [ 0: 1:0.01][ 0: 1:0.01] f() of step()
ifft [-50:49:1 ][-50:49:1 ] g() of f()
</pre><p>In the <tt class="tt">fft</tt> command above, <img src="images/img-0187.png" alt="$N_ x=100$" style="vertical-align:-2px;
width:74px;
height:14px" class="math gen" /> equally-spaced samples of the function <tt class="tt">step</tt><img src="images/img-0188.png" alt="$(x,y)$" style="vertical-align:-4px;
width:40px;
height:18px" class="math gen" /> are taken between limits of <img src="images/img-0189.png" alt="$x_\mathrm {min}=0$" style="vertical-align:-2px;
width:68px;
height:14px" class="math gen" /> and <img src="images/img-0190.png" alt="$x_\mathrm {max}=1$" style="vertical-align:-2px;
width:69px;
height:14px" class="math gen" /> for each of <img src="images/img-0191.png" alt="$N_ y=100$" style="vertical-align:-5px;
width:74px;
height:17px" class="math gen" /> equally-spaced values of <img src="images/img-0020.png" alt="$y$" style="vertical-align:-4px;
width:9px;
height:12px" class="math gen" /> on an identical raster, giving a total of <img src="images/img-0192.png" alt="$10^4$" style="vertical-align:0px;
width:24px;
height:16px" class="math gen" /> samples. These are converted into a rectangular grid of <img src="images/img-0192.png" alt="$10^4$" style="vertical-align:0px;
width:24px;
height:16px" class="math gen" /> samples of the Fourier transform <tt class="tt">f</tt><img src="images/img-0193.png" alt="$(\omega _ x,\omega _ y)$" style="vertical-align:-5px;
width:59px;
height:19px" class="math gen" /> at </p><table id="a0000000531" cellpadding="7" width="100%" cellspacing="0" class="eqnarray">
<tr id="a0000000532">
<td style="width:40%"> </td>
<td style="vertical-align:middle; text-align:right"><img src="images/img-0194.png" alt="$\displaystyle \omega _ x = \frac{j}{\Delta x} $" style="vertical-align:-12px; width:72px; height:37px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="images/img-0195.png" alt="$\displaystyle \textrm{for} $" style="vertical-align:0px; width:22px; height:12px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:left"><img src="images/img-0196.png" alt="$\displaystyle -\frac{N_ x}{2}\leq j <\frac{N_ x}{2} \; \left(\textrm{equivalently, for} -\frac{N_ x}{2\Delta x}\leq \omega _ x <\frac{N_ x}{2\Delta x} \right), \nonumber $" style="vertical-align:-17px; width:461px; height:44px" class="math gen" /></td>
<td style="width:40%"> </td>
<td style="width:20%" class="eqnnum"> </td>
</tr><tr id="a0000000533">
<td style="width:40%"> </td>
<td style="vertical-align:middle; text-align:right"><img src="images/img-0197.png" alt="$\displaystyle \omega _ y = \frac{k}{\Delta y} $" style="vertical-align:-17px; width:70px; height:42px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="images/img-0195.png" alt="$\displaystyle \textrm{for} $" style="vertical-align:0px; width:22px; height:12px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:left"><img src="images/img-0198.png" alt="$\displaystyle -\frac{N_ y}{2}\leq k <\frac{N_ y}{2} \; \left(\textrm{equivalently, for} -\frac{N_ y}{2\Delta y}\leq \omega _ y <\frac{N_ y}{2\Delta y} \right). \nonumber $" style="vertical-align:-17px; width:459px; height:44px" class="math gen" /></td>
<td style="width:40%"> </td>
<td style="width:20%" class="eqnnum"> </td>
</tr>
</table><p>where <img src="images/img-0199.png" alt="$\Delta x=x_\mathrm {max}-x_\mathrm {min}$" style="vertical-align:-2px;
width:142px;
height:14px" class="math gen" /> and <img src="images/img-0200.png" alt="$\Delta y$" style="vertical-align:-4px;
width:23px;
height:16px" class="math gen" /> is analogously defined. These samples are interpolated stepwise, such that an evaluation of the function <tt class="tt">f</tt><img src="images/img-0193.png" alt="$(\omega _ x,\omega _ y)$" style="vertical-align:-5px;
width:59px;
height:19px" class="math gen" /> for general inputs yields the nearest sample, or zero outside the rectangular grid where samples are available. In general, even the Fourier transforms of real functions are complex, and their evaluation when complex arithmetic is not enabled (see Section <a href="sec-complex_numbers.html">4.5</a>) is likely to fail. For this reason, a warning is issued if complex arithmetic is disabled when a Fourier transform function is evaluated. </p><p>In the example above, we go on to convert this set of samples back into the function with which we started by instructing the <tt class="tt">ifft</tt> command<a name="a0000000534" id="a0000000534"></a> to take <img src="images/img-0187.png" alt="$N_ x=100$" style="vertical-align:-2px;
width:74px;
height:14px" class="math gen" /> equally-spaced samples along the <img src="images/img-0201.png" alt="$\omega _ x$" style="vertical-align:-2px;
width:19px;
height:10px" class="math gen" />-axis between <img src="images/img-0202.png" alt="$\omega _{x,\mathrm{min}}=-{N_ x}/{2\Delta x}$" style="vertical-align:-5px;
width:152px;
height:18px" class="math gen" /> and <img src="images/img-0203.png" alt="$\omega _{x,\mathrm{max}}=(N_ x-1)/{2\Delta x}$" style="vertical-align:-5px;
width:186px;
height:19px" class="math gen" />, with similar sampling along the <img src="images/img-0204.png" alt="$\omega _ y$" style="vertical-align:-5px;
width:18px;
height:13px" class="math gen" />-axis. </p><p>Taking the simpler example of a one-dimensional Fourier transform for clarity, as might be calculated by the instructions </p><pre>
step(x) = tophat(x,0.2)
fft [ 0: 1:0.01] f() of step()
</pre><p> the <tt class="tt">fft</tt> and <tt class="tt">ifft</tt> commands<a name="a0000000535" id="a0000000535"></a><a name="a0000000536" id="a0000000536"></a> compute, respectively, discrete sets of samples <img src="images/img-0205.png" alt="$F_ m$" style="vertical-align:-2px;
width:24px;
height:14px" class="math gen" /> and <img src="images/img-0206.png" alt="$I_ n$" style="vertical-align:-2px;
width:16px;
height:14px" class="math gen" /> of the functions <img src="images/img-0207.png" alt="$F(\omega _ x)$" style="vertical-align:-4px;
width:47px;
height:18px" class="math gen" /> and <img src="images/img-0208.png" alt="$I(x)$" style="vertical-align:-4px;
width:32px;
height:18px" class="math gen" />, which are given by </p><table id="a0000000537" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="images/img-0209.png" alt="\[ F_ j = \sum _{m=0}^{N-1} I_ m e^{-2\pi ijm/N} \, \delta x,\; \textrm{for}\; -\frac{N}{2}\leq j <\frac{N}{2} , \]" style="width:363px;
height:54px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> and </p><table id="a0000000538" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="images/img-0210.png" alt="\[ I_ j = \sum _{m=0}^{N-1} F_ m e^{ 2\pi ijm/N} \, \delta \omega _ x,\; \textrm{for}\; -\frac{N}{2}\leq j <\frac{N}{2} , \]" style="width:362px;
height:54px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> where: </p><table cellspacing="0" class="tabular">
<tr>
<td style="text-align:right"><p> <img src="images/img-0208.png" alt="$I(x)$" style="vertical-align:-4px;
width:32px;
height:18px" class="math gen" /> </p></td>
<td style="text-align:center"><p> = </p></td>
<td style="text-align:left"><p> Function being Fourier transformed. </p></td>
</tr><tr>
<td style="text-align:right"><p><img src="images/img-0207.png" alt="$F(\omega _ x)$" style="vertical-align:-4px;
width:47px;
height:18px" class="math gen" /> </p></td>
<td style="text-align:center"><p> = </p></td>
<td style="text-align:left"><p> Fourier transform of <img src="images/img-0211.png" alt="$I()$" style="vertical-align:-4px;
width:22px;
height:18px" class="math gen" />. </p></td>
</tr><tr>
<td style="text-align:right"><p><img src="images/img-0151.png" alt="$N$" style="vertical-align:0px;
width:16px;
height:12px" class="math gen" /> </p></td>
<td style="text-align:center"><p> = </p></td>
<td style="text-align:left"><p> The number of values sampled along the abscissa axis. </p></td>
</tr><tr>
<td style="text-align:right"><p><img src="images/img-0212.png" alt="$\delta x$" style="vertical-align:0px;
width:18px;
height:12px" class="math gen" /> </p></td>
<td style="text-align:center"><p> = </p></td>
<td style="text-align:left"><p> Spacing of values sampled along the abscissa axis. </p></td>
</tr><tr>
<td style="text-align:right"><p><img src="images/img-0213.png" alt="$\delta \omega _ x$" style="vertical-align:-2px;
width:27px;
height:14px" class="math gen" /> </p></td>
<td style="text-align:center"><p> = </p></td>
<td style="text-align:left"><p> Spacing of abscissa values sampled along the <img src="images/img-0201.png" alt="$\omega _ x$" style="vertical-align:-2px;
width:19px;
height:10px" class="math gen" /> axis. </p></td>
</tr><tr>
<td style="text-align:right"><p><img src="images/img-0214.png" alt="$i$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" /> </p></td>
<td style="text-align:center"><p> = </p></td>
<td style="text-align:left"><p> <img src="images/img-0007.png" alt="$\sqrt {-1}$" style="vertical-align:-3px;
width:38px;
height:19px" class="math gen" />. </p></td>
</tr>
</table><p>It may be shown in the limit that <img src="images/img-0212.png" alt="$\delta x$" style="vertical-align:0px;
width:18px;
height:12px" class="math gen" /> becomes small – i.e. when the number of samples taken becomes very large – that these sums approximate the integrals </p><table id="a0000000539" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="images/img-0215.png" alt="\begin{equation} F(\omega _ x) = \int I(x) e^{-2\pi ix\omega _ x} \, \mathrm{d}x , \end{equation}" style="width:409px;
height:41px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"><span>(<span>5.1</span>)</span></td>
</tr>
</table><p> and </p><table id="ex:fourier" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="images/img-0216.png" alt="\begin{equation} I(x) = \int F(\omega _ x) e^{ 2\pi ix\omega _ x} \, \mathrm{d}\omega _ x . \end{equation}" style="width:408px;
height:41px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"><span>(<span>5.2</span>)</span></td>
</tr>
</table><p>Fourier transforms may also be taken of data stored in datafiles using syntax of the form </p><pre>
fft [-10:10:0.1] f() of 'datafile.dat'
</pre><p>In such cases, the data read from the datafile for an <img src="images/img-0151.png" alt="$N$" style="vertical-align:0px;
width:16px;
height:12px" class="math gen" />-dimensional FFT must be arranged in <img src="images/img-0217.png" alt="$N+1$" style="vertical-align:-1px;
width:46px;
height:13px" class="math gen" /> columns<a href="#a0000000540" class="footnote"><sup class="footnotemark">1</sup></a>, with the first <img src="images/img-0151.png" alt="$N$" style="vertical-align:0px;
width:16px;
height:12px" class="math gen" /> containing the abscissa values for each of the <img src="images/img-0151.png" alt="$N$" style="vertical-align:0px;
width:16px;
height:12px" class="math gen" /> dimensions, and the final column containing the data to be Fourier transformed. The abscissa values must strictly match those in the raster specified in the <tt class="tt">fft</tt> or <tt class="tt">ifft</tt> command, and must be arranged strictly in row-major order. </p><p> <span class="upshape"><span class="mdseries"><span class="rm">The Fourier transform of a top-hat function.</span></span></span></p><div>
<table cellspacing="0" class="tabular">
<tr>
<td style="border-top-style:solid; border-left:1px solid black; border-right:1px solid black; border-top-color:black; border-top-width:1px; text-align:left"><p> It is straightforward to show that the Fourier transform of a top-hat function of unit width is the function <img src="images/img-0218.png" alt="${\rm sinc}(x^\prime =\pi x)=\sin (x^\prime )/x^\prime $" style="vertical-align:-5px;
width:204px;
height:20px" class="math gen" />. If </p><table id="a0000000541" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="images/img-0219.png" alt="\[ f(x)=\left\{ \begin{array}{l}1\; |x|\leq \nicefrac {1}{2}\\ 0\; |x|>\nicefrac {1}{2}\end{array}\right. , \]" style="width:177px;
height:44px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> the Fourier transform <img src="images/img-0220.png" alt="$F(\omega )$" style="vertical-align:-4px;
width:39px;
height:18px" class="math gen" /> of <img src="images/img-0078.png" alt="$f(x)$" style="vertical-align:-4px;
width:32px;
height:18px" class="math gen" /> is </p><table id="a0000000542" cellpadding="7" width="100%" cellspacing="0" class="eqnarray">
<tr id="a0000000543">
<td style="width:40%"> </td>
<td style="vertical-align:middle; text-align:right"><img src="images/img-0221.png" alt="$\displaystyle F(\omega ) $" style="vertical-align:-4px; width:39px; height:18px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="images/img-0058.png" alt="$\displaystyle = $" style="vertical-align:2px; width:12px; height:4px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:left"><img src="images/img-0222.png" alt="$\displaystyle \int _0^\infty f(x) \exp \left(-2\pi ix\omega \right) \, \mathrm{d}x = \int _{-\nicefrac {1}{2}}^{\nicefrac {1}{2}} \exp \left(-2\pi ix\omega \right) \, \mathrm{d}x $" style="vertical-align:-19px; width:408px; height:50px" class="math gen" /></td>
<td style="width:40%"> </td>
<td style="width:20%" class="eqnnum"> </td>
</tr><tr id="a0000000544">
<td style="width:40%"> </td>
<td style="vertical-align:middle; text-align:right"><img src="images/img-0175.png" alt="$\displaystyle $" style="vertical-align:0px; width:1px; height:1px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="images/img-0058.png" alt="$\displaystyle = $" style="vertical-align:2px; width:12px; height:4px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:left"><img src="images/img-0223.png" alt="$\displaystyle \frac{1}{2\pi \omega }\left[ \exp \left(\pi i\omega \right) - \exp \left(-\pi i\omega \right) \right] = \frac{\sin (\pi \omega )}{\pi \omega } = {\rm sinc}(\pi \omega ). $" style="vertical-align:-13px; width:415px; height:39px" class="math gen" /></td>
<td style="width:40%"> </td>
<td style="width:20%" class="eqnnum"> </td>
</tr>
</table></td>
</tr><tr>
<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><p>In this example, we demonstrate this numerically by taking the Fourier transform of such a step function, and comparing the result against the function <tt class="tt">sinc(x)</tt> which is pre-defined within PyXPlot: </p></td>
</tr><tr>
<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><p><tt class="tt">set numerics complex</tt><br /><tt class="tt">step(x) = tophat(x,0.5)</tt><br /><tt class="tt">fft [-1:1:0.01] f() of step()</tt><br /><tt class="tt">plot [-10:10] Re(f(x)), sinc(pi*x)</tt> </p></td>
</tr><tr>
<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><p>Note that the function <tt class="tt">Re(x)</tt> is needed in the <tt class="tt">plot</tt> statement here, since although the Fourier transform of a symmetric function is in theory real, in practice any numerical Fourier transform will yield a small imaginary component at the level of the accuracy of the numerical method used. Although the calculated numerical Fourier transform is defined throughout the range <img src="images/img-0224.png" alt="$-50\leq x<50$" style="vertical-align:-3px;
width:108px;
height:16px" class="math gen" />, discretised with steps of size <img src="images/img-0225.png" alt="$\Updelta x=0.5$" style="vertical-align:0px;
width:69px;
height:14px" class="math gen" />, we only plot the central region in order to show clearly the stepping of the function: </p></td>
</tr><tr>
<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><center> <img src="images/img-0227.png" alt="\includegraphics{examples/eps/ex_fft}" style="width:516px; height:319px" />
</center></td>
</tr><tr>
<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><p>In the following steps, we take the square of the function <img src="images/img-0233.png" alt="${\rm sinc}(\pi x)$" style="vertical-align:-4px;
width:64px;
height:18px" class="math gen" /> just calculated, and then plot the numerical inverse Fourier transform of the result: </p></td>
</tr><tr>
<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><p><tt class="tt">g(x) = f(x)**2</tt><br /><tt class="tt">ifft [-50:49.5:0.5] h(x) of g(x)</tt><br /><tt class="tt">plot [-2:2] Re(h(x))</tt> </p></td>
</tr><tr>
<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><center> <img src="images/img-0235.png" alt="\includegraphics{examples/eps/ex_fft2}" style="width:520px; height:324px" />
</center></td>
</tr><tr>
<td style="border-bottom-style:solid; border-bottom-width:1px; border-left:1px solid black; border-right:1px solid black; text-align:left; border-bottom-color:black"><p>As can be seen, the result is a triangle function. This is the result which would be expected from the convolution theorem, which states that when the Fourier transforms of two functions are multiplied together and then inverse transformed, the result is the convolution of the two original functions. The convolution of a top-hat function with itself is, indeed, a triangle function. </p></td>
</tr>
</table>
</div></div>
<div class="contents section-contents"><!--<strong>Subsections</strong>-->
<ul>
<li><a href="sect0029.html">5.8.1 Window Functions</a>
</li>
</ul>
</div>
<div id="footnotes">
<p><b>Footnotes</b></p>
<ol>
<li id="a0000000540">The <tt class="tt">using</tt>, <tt class="tt">every</tt>, <tt class="tt">index</tt> and <tt class="tt">select</tt> modifiers can be used to arrange it into this form.</li>
</ol>
</div>
<div class="navigation">
<table cellspacing="2" cellpadding="0" width="100%">
<tr>
<td><a href="ex-interpolation.html" title="Datafile Interpolation"><img alt="Previous: Datafile Interpolation" border="0" src="icons/previous.gif" width="32" height="32" /></a></td>
<td><a href="ch-numerics.html" title="Working with Data"><img alt="Up: Working with Data" border="0" src="icons/up.gif" width="32" height="32" /></a></td>
<td><a href="sect0029.html" title="Window Functions"><img alt="Next: Window Functions" border="0" src="icons/next.gif" width="32" height="32" /></a></td>
<td class="navtitle" align="center">PyXPlot Users' Guide</td>
<td><a href="index.html" title="Table of Contents"><img border="0" alt="" src="icons/contents.gif" width="32" height="32" /></a></td>
<td><a href="sect0255.html" title="Index"><img border="0" alt="" src="icons/index.gif" width="32" height="32" /></a></td>
<td><img border="0" alt="" src="icons/blank.gif" width="32" height="32" /></td>
</tr>
</table>
</div>
<script language="javascript" src="icons/imgadjust.js" type="text/javascript"></script>
</body>
</html>
|