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<h3 class="section">D.3 <code>fftw</code></h3>

<p>Some functions in the <code>fftw</code> fast Fourier transform library are
<a name="index-Fourier-transforms-712"></a><a name="index-Hartley-transforms-713"></a>callable by virtual code programs of the form
<code>library('fftw',f)</code>, where <code>f</code> can be one of the following
character strings.

     <dl>
<dt><code>u_fw_dft</code><dd>(uni-dimensional forward Discrete Fourier transform)
<br><dt><code>u_bw_dft</code><dd>(uni-dimensional backward Discrete Fourier transform)
<br><dt><code>b_fw_dft</code><dd>(bi-dimensional forward Discrete Fourier transform)
<br><dt><code>b_bw_dft</code><dd>(bi-dimensional backward Discrete Fourier transform)
<br><dt><code>u_dht</code><dd>(uni-dimensional Discrete Hartley transform)
<br><dt><code>b_dht</code><dd>(bi-dimensional Discrete Hartley transform)
</dl>

   <p>These stand for the discrete Fourier transform, in one dimension and two
dimensions, either backward or forward, and the discrete Hartley
transform in one dimension and two dimensions. The <code>fftw</code> library
documentation (<a href="http://www.fftw.org">http://www.fftw.org</a>) can give more information
about the meaning of these transformations.

   <p>The interface is somewhat simplified compared to the API for the <code>fftw</code>
C library because there are no considerations of memory management or
planning, nor any provision for dimensions higher than two.

   <p>Furthermore, from the virtual side of the interface, these
functions operate on lists rather than arrays. The one dimensional
Fourier transforms take a list of complex numbers to a list of complex
numbers (see <a href="complex.html#complex">complex</a>), and the one dimensional Hartley
transforms take a list of reals to a list of reals (see
<a href="math.html#math">math</a>). The two dimensional transforms are analogous but they
take a matrix represented as a list of lists. Error messages
pertaining to invalid input documented at the beginning of
this section (<a href="External-Libraries.html#External-Libraries">External Libraries</a>) are relevant.

   <p>Finally, unlike the native API for <code>fftw</code>, these transformations are
scaled so that the backward transformation is the inverse of the
forward, and the Hartley transformations are their own inverses
(subject to roundoff error).

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