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Replace these with versions built using an older toolset (so as to avoid ABI compatibilities when linking on Ubuntu 14.04 for packaging purposes)
author Chris Cannam
date Fri, 07 Feb 2020 11:51:13 +0000
parents 37bf6b4a2645
children
rev   line source
Chris@10 1 <html lang="en">
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Chris@10 3 <title>The 1d Discrete Fourier Transform (DFT) - FFTW 3.3.3</title>
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Chris@10 48 <div class="node">
Chris@10 49 <a name="The-1d-Discrete-Fourier-Transform-(DFT)"></a>
Chris@10 50 <a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029"></a>
Chris@10 51 <p>
Chris@10 52 Next:&nbsp;<a rel="next" accesskey="n" href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT">The 1d Real-data DFT</a>,
Chris@10 53 Previous:&nbsp;<a rel="previous" accesskey="p" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>,
Chris@10 54 Up:&nbsp;<a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>
Chris@10 55 <hr>
Chris@10 56 </div>
Chris@10 57
Chris@10 58 <h4 class="subsection">4.8.1 The 1d Discrete Fourier Transform (DFT)</h4>
Chris@10 59
Chris@10 60 <p><a name="index-discrete-Fourier-transform-292"></a><a name="index-DFT-293"></a>The forward (<code>FFTW_FORWARD</code>) discrete Fourier transform (DFT) of a
Chris@10 61 1d complex array X of size n computes an array Y,
Chris@10 62 where:
Chris@10 63 <center><img src="equation-dft.png" align="top">.</center>The backward (<code>FFTW_BACKWARD</code>) DFT computes:
Chris@10 64 <center><img src="equation-idft.png" align="top">.</center>
Chris@10 65
Chris@10 66 <p><a name="index-normalization-294"></a>FFTW computes an unnormalized transform, in that there is no coefficient
Chris@10 67 in front of the summation in the DFT. In other words, applying the
Chris@10 68 forward and then the backward transform will multiply the input by
Chris@10 69 n.
Chris@10 70
Chris@10 71 <p><a name="index-frequency-295"></a>From above, an <code>FFTW_FORWARD</code> transform corresponds to a sign of
Chris@10 72 -1 in the exponent of the DFT. Note also that we use the
Chris@10 73 standard &ldquo;in-order&rdquo; output ordering&mdash;the k-th output
Chris@10 74 corresponds to the frequency k/n (or k/T, where T
Chris@10 75 is your total sampling period). For those who like to think in terms of
Chris@10 76 positive and negative frequencies, this means that the positive
Chris@10 77 frequencies are stored in the first half of the output and the negative
Chris@10 78 frequencies are stored in backwards order in the second half of the
Chris@10 79 output. (The frequency -k/n is the same as the frequency
Chris@10 80 (n-k)/n.)
Chris@10 81
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