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    <p>To help clarify (focusing in 2D as it easily extends to 2D transform of 3D data):</p> <h2>storage</h2> <p>An <code>Nx * Ny</code> array requires <code>Nx * (Ny / 2 + 1)</code> complex elements after a Fourier Transform.</p> <p>First, in the y-direction, the negative frequencies can be reconstructed from the complex conjugate symmetry (that comes from transforming a real sequence). The y modes <code>ky</code> then run from <code>0 to Ny/2</code> inclusive. <strong>So for y we need <code>Ny/2 + 1</code> complex values.</strong></p> <p>Next we transform in the x-direction where we cannot use the same symmetry assumption, as we are acting on complex-valued y values. Therefore we must include positive and negative frequencies, so x-modes <code>kx</code> run from <code>-Nx/2 to Nx/2</code> inclusive. However <code>kx = -Nx/2</code> and <code>kx = Nx/2</code> are equivalent so only one is stored (see <a href="http://www.fftw.org/fftw3_doc/The-1d-Discrete-Fourier-Transform-_0028DFT_0029.html#The-1d-Discrete-Fourier-Transform-_0028DFT_0029" rel="nofollow">here</a>). <strong>So for x we need <code>Nx</code> complex values.</strong></p> <h2>access</h2> <p>As tir38 points out the x index post-transform runs from 0 to Nx-1, however this doesn't mean that modes <code>kx</code> run from 0 to Nx-1. FFTW packs positive frequencies in the first half of the array, then negative frequencies in the second half (in reverse order), like:</p> <pre><code>kx = 0, 1, 2, ..., Nx/2, -Nx/2 + 1, ..., -2, -1 </code></pre> <p>There are two ways we can think about accessing these elements. First as tir38 suggests we can loop through in order and work out the mode <code>kx</code> from the index:</p> <pre><code>for (int i = 0; i &lt; Nx; i++) { // produces the list of kxs above int kx = (i &lt;= Nx/2) ? i : i - Nx; // here we index with i, but with the knowledge that the mode is kx outputRe(i, ...) = some function of kx } </code></pre> <p>or we can loop through the modes <code>kx</code> and convert to an index:</p> <pre><code>for (int kx = -Nx/2; kx &lt; Nx/2; kx++) { // work out index from mode kx int i = (kx &gt;= 0) ? i : Nx + i; // here we index with i, but with the knowledge that the mode is kx outputRe(i, ...) = some function of kx } </code></pre> <p>The two types of indexing along with the rest of the code <a href="https://gist.github.com/hemmer/5916596" rel="nofollow">can found here</a>.</p>
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