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<title>FFTW 3.3.8: 1d Real-even DFTs (DCTs)</title>

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<a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029"></a>
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Next: <a href="1d-Real_002dodd-DFTs-_0028DSTs_0029.html#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029" accesskey="n" rel="next">1d Real-odd DFTs (DSTs)</a>, Previous: <a href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" accesskey="p" rel="prev">The 1d Real-data DFT</a>, Up: <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" accesskey="u" rel="up">What FFTW Really Computes</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029-1"></a>
<h4 class="subsection">4.8.3 1d Real-even DFTs (DCTs)</h4>

<p>The Real-even symmetry DFTs in FFTW are exactly equivalent to the unnormalized
forward (and backward) DFTs as defined above, where the input array
<em>X</em> of length <em>N</em> is purely real and is also <em>even</em> symmetry.  In
this case, the output array is likewise real and even symmetry.
<a name="index-real_002deven-DFT-1"></a>
<a name="index-REDFT-1"></a>
</p>

<a name="index-REDFT00"></a>
<p>For the case of <code>REDFT00</code>, this even symmetry means that
<i>X<sub>j</sub> = X<sub>N-j</sub></i>,
where we take <em>X</em> to be periodic so that
<i>X<sub>N</sub> = X</i><sub>0</sub>.
Because of this redundancy, only the first <em>n</em> real numbers are
actually stored, where <em>N = 2(n-1)</em>.
</p>
<p>The proper definition of even symmetry for <code>REDFT10</code>,
<code>REDFT01</code>, and <code>REDFT11</code> transforms is somewhat more intricate
because of the shifts by <em>1/2</em> of the input and/or output, although
the corresponding boundary conditions are given in <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029">Real even/odd DFTs (cosine/sine transforms)</a>.  Because of the even symmetry, however,
the sine terms in the DFT all cancel and the remaining cosine terms are
written explicitly below.  This formulation often leads people to call
such a transform a <em>discrete cosine transform</em> (DCT), although it is
really just a special case of the DFT.
<a name="index-discrete-cosine-transform-2"></a>
<a name="index-DCT-2"></a>
</p>

<p>In each of the definitions below, we transform a real array <em>X</em> of
length <em>n</em> to a real array <em>Y</em> of length <em>n</em>:
</p>
<a name="REDFT00-_0028DCT_002dI_0029"></a>
<h4 class="subsubheading">REDFT00 (DCT-I)</h4>
<a name="index-REDFT00-1"></a>
<p>An <code>REDFT00</code> transform (type-I DCT) in FFTW is defined by:
<center><img src="equation-redft00.png" align="top">.</center>
Note that this transform is not defined for <em>n=1</em>.  For <em>n=2</em>,
the summation term above is dropped as you might expect.
</p>
<a name="REDFT10-_0028DCT_002dII_0029"></a>
<h4 class="subsubheading">REDFT10 (DCT-II)</h4>
<a name="index-REDFT10"></a>
<p>An <code>REDFT10</code> transform (type-II DCT, sometimes called &ldquo;the&rdquo; DCT) in FFTW is defined by:
<center><img src="equation-redft10.png" align="top">.</center>
</p>
<a name="REDFT01-_0028DCT_002dIII_0029"></a>
<h4 class="subsubheading">REDFT01 (DCT-III)</h4>
<a name="index-REDFT01"></a>
<p>An <code>REDFT01</code> transform (type-III DCT) in FFTW is defined by:
<center><img src="equation-redft01.png" align="top">.</center>
In the case of <em>n=1</em>, this reduces to
<i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
Up to a scale factor (see below), this is the inverse of <code>REDFT10</code> (&ldquo;the&rdquo; DCT), and so the <code>REDFT01</code> (DCT-III) is sometimes called the &ldquo;IDCT&rdquo;.
<a name="index-IDCT-3"></a>
</p>
<a name="REDFT11-_0028DCT_002dIV_0029"></a>
<h4 class="subsubheading">REDFT11 (DCT-IV)</h4>
<a name="index-REDFT11"></a>
<p>An <code>REDFT11</code> transform (type-IV DCT) in FFTW is defined by:
<center><img src="equation-redft11.png" align="top">.</center>
</p>
<a name="Inverses-and-Normalization"></a>
<h4 class="subsubheading">Inverses and Normalization</h4>

<p>These definitions correspond directly to the unnormalized DFTs used
elsewhere in FFTW (hence the factors of <em>2</em> in front of the
summations).  The unnormalized inverse of <code>REDFT00</code> is
<code>REDFT00</code>, of <code>REDFT10</code> is <code>REDFT01</code> and vice versa, and
of <code>REDFT11</code> is <code>REDFT11</code>.  Each unnormalized inverse results
in the original array multiplied by <em>N</em>, where <em>N</em> is the
<em>logical</em> DFT size.  For <code>REDFT00</code>, <em>N=2(n-1)</em> (note that
<em>n=1</em> is not defined); otherwise, <em>N=2n</em>.
<a name="index-normalization-10"></a>
</p>

<p>In defining the discrete cosine transform, some authors also include
additional factors of
&radic;2
(or its inverse) multiplying selected inputs and/or outputs.  This is a
mostly cosmetic change that makes the transform orthogonal, but
sacrifices the direct equivalence to a symmetric DFT.
</p>
<hr>
<div class="header">
<p>
Next: <a href="1d-Real_002dodd-DFTs-_0028DSTs_0029.html#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029" accesskey="n" rel="next">1d Real-odd DFTs (DSTs)</a>, Previous: <a href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" accesskey="p" rel="prev">The 1d Real-data DFT</a>, Up: <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" accesskey="u" rel="up">What FFTW Really Computes</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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author	Chris Cannam <cannam@all-day-breakfast.com>
date	Mon, 02 Mar 2020 14:03:47 +0000
parents	bd3cc4d1df30
children