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<section id="discrete-fourier-transform">
<span id="module-numpy.fft"></span><span id="routines-fft"></span><h1>Discrete Fourier Transform<a class="headerlink" href="#discrete-fourier-transform" title="Link to this heading">#</a></h1>
<p>The SciPy module <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/fft.html#module-scipy.fft" title="(in SciPy v1.15.2)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.fft</span></code></a> is a more comprehensive superset
of <a class="reference internal" href="#module-numpy.fft" title="numpy.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.fft</span></code></a>, which includes only a basic set of routines.</p>
<section id="standard-ffts">
<h2>Standard FFTs<a class="headerlink" href="#standard-ffts" title="Link to this heading">#</a></h2>
<div class="pst-scrollable-table-container"><table class="autosummary longtable table autosummary">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.fft.html#numpy.fft.fft" title="numpy.fft.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fft</span></code></a>(a[, n, axis, norm, out])</p></td>
<td><p>Compute the one-dimensional discrete Fourier Transform.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.ifft.html#numpy.fft.ifft" title="numpy.fft.ifft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ifft</span></code></a>(a[, n, axis, norm, out])</p></td>
<td><p>Compute the one-dimensional inverse discrete Fourier Transform.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.fft2.html#numpy.fft.fft2" title="numpy.fft.fft2"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fft2</span></code></a>(a[, s, axes, norm, out])</p></td>
<td><p>Compute the 2-dimensional discrete Fourier Transform.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.ifft2.html#numpy.fft.ifft2" title="numpy.fft.ifft2"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ifft2</span></code></a>(a[, s, axes, norm, out])</p></td>
<td><p>Compute the 2-dimensional inverse discrete Fourier Transform.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.fftn.html#numpy.fft.fftn" title="numpy.fft.fftn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fftn</span></code></a>(a[, s, axes, norm, out])</p></td>
<td><p>Compute the N-dimensional discrete Fourier Transform.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.ifftn.html#numpy.fft.ifftn" title="numpy.fft.ifftn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ifftn</span></code></a>(a[, s, axes, norm, out])</p></td>
<td><p>Compute the N-dimensional inverse discrete Fourier Transform.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="real-ffts">
<h2>Real FFTs<a class="headerlink" href="#real-ffts" title="Link to this heading">#</a></h2>
<div class="pst-scrollable-table-container"><table class="autosummary longtable table autosummary">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.rfft.html#numpy.fft.rfft" title="numpy.fft.rfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfft</span></code></a>(a[, n, axis, norm, out])</p></td>
<td><p>Compute the one-dimensional discrete Fourier Transform for real input.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.irfft.html#numpy.fft.irfft" title="numpy.fft.irfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">irfft</span></code></a>(a[, n, axis, norm, out])</p></td>
<td><p>Computes the inverse of <a class="reference internal" href="generated/numpy.fft.rfft.html#numpy.fft.rfft" title="numpy.fft.rfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfft</span></code></a>.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.rfft2.html#numpy.fft.rfft2" title="numpy.fft.rfft2"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfft2</span></code></a>(a[, s, axes, norm, out])</p></td>
<td><p>Compute the 2-dimensional FFT of a real array.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.irfft2.html#numpy.fft.irfft2" title="numpy.fft.irfft2"><code class="xref py py-obj docutils literal notranslate"><span class="pre">irfft2</span></code></a>(a[, s, axes, norm, out])</p></td>
<td><p>Computes the inverse of <a class="reference internal" href="generated/numpy.fft.rfft2.html#numpy.fft.rfft2" title="numpy.fft.rfft2"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfft2</span></code></a>.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.rfftn.html#numpy.fft.rfftn" title="numpy.fft.rfftn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfftn</span></code></a>(a[, s, axes, norm, out])</p></td>
<td><p>Compute the N-dimensional discrete Fourier Transform for real input.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.irfftn.html#numpy.fft.irfftn" title="numpy.fft.irfftn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">irfftn</span></code></a>(a[, s, axes, norm, out])</p></td>
<td><p>Computes the inverse of <a class="reference internal" href="generated/numpy.fft.rfftn.html#numpy.fft.rfftn" title="numpy.fft.rfftn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfftn</span></code></a>.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="hermitian-ffts">
<h2>Hermitian FFTs<a class="headerlink" href="#hermitian-ffts" title="Link to this heading">#</a></h2>
<div class="pst-scrollable-table-container"><table class="autosummary longtable table autosummary">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.hfft.html#numpy.fft.hfft" title="numpy.fft.hfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">hfft</span></code></a>(a[, n, axis, norm, out])</p></td>
<td><p>Compute the FFT of a signal that has Hermitian symmetry, i.e., a real spectrum.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.ihfft.html#numpy.fft.ihfft" title="numpy.fft.ihfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ihfft</span></code></a>(a[, n, axis, norm, out])</p></td>
<td><p>Compute the inverse FFT of a signal that has Hermitian symmetry.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="helper-routines">
<h2>Helper routines<a class="headerlink" href="#helper-routines" title="Link to this heading">#</a></h2>
<div class="pst-scrollable-table-container"><table class="autosummary longtable table autosummary">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.fftfreq.html#numpy.fft.fftfreq" title="numpy.fft.fftfreq"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fftfreq</span></code></a>(n[, d, device])</p></td>
<td><p>Return the Discrete Fourier Transform sample frequencies.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.rfftfreq.html#numpy.fft.rfftfreq" title="numpy.fft.rfftfreq"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfftfreq</span></code></a>(n[, d, device])</p></td>
<td><p>Return the Discrete Fourier Transform sample frequencies (for usage with rfft, irfft).</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.fftshift.html#numpy.fft.fftshift" title="numpy.fft.fftshift"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fftshift</span></code></a>(x[, axes])</p></td>
<td><p>Shift the zero-frequency component to the center of the spectrum.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.ifftshift.html#numpy.fft.ifftshift" title="numpy.fft.ifftshift"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ifftshift</span></code></a>(x[, axes])</p></td>
<td><p>The inverse of <a class="reference internal" href="generated/numpy.fft.fftshift.html#numpy.fft.fftshift" title="numpy.fft.fftshift"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fftshift</span></code></a>.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="background-information">
<h2>Background information<a class="headerlink" href="#background-information" title="Link to this heading">#</a></h2>
<p>Fourier analysis is fundamentally a method for expressing a function as a
sum of periodic components, and for recovering the function from those
components. When both the function and its Fourier transform are
replaced with discretized counterparts, it is called the discrete Fourier
transform (DFT). The DFT has become a mainstay of numerical computing in
part because of a very fast algorithm for computing it, called the Fast
Fourier Transform (FFT), which was known to Gauss (1805) and was brought
to light in its current form by Cooley and Tukey <a class="reference internal" href="#rfb1dc64dd6a5-ct" id="id1">[CT]</a>. Press et al. <a class="reference internal" href="#rfb1dc64dd6a5-nr" id="id2">[NR]</a>
provide an accessible introduction to Fourier analysis and its
applications.</p>
<p>Because the discrete Fourier transform separates its input into
components that contribute at discrete frequencies, it has a great number
of applications in digital signal processing, e.g., for filtering, and in
this context the discretized input to the transform is customarily
referred to as a <em>signal</em>, which exists in the <em>time domain</em>. The output
is called a <em>spectrum</em> or <em>transform</em> and exists in the <em>frequency
domain</em>.</p>
</section>
<section id="implementation-details">
<h2>Implementation details<a class="headerlink" href="#implementation-details" title="Link to this heading">#</a></h2>
<p>There are many ways to define the DFT, varying in the sign of the
exponent, normalization, etc. In this implementation, the DFT is defined
as</p>
<div class="math notranslate nohighlight">
\[A_k = \sum_{m=0}^{n-1} a_m \exp\left\{-2\pi i{mk \over n}\right\}
\qquad k = 0,\ldots,n-1.\]</div>
<p>The DFT is in general defined for complex inputs and outputs, and a
single-frequency component at linear frequency <span class="math notranslate nohighlight">\(f\)</span> is
represented by a complex exponential
<span class="math notranslate nohighlight">\(a_m = \exp\{2\pi i\,f m\Delta t\}\)</span>, where <span class="math notranslate nohighlight">\(\Delta t\)</span>
is the sampling interval.</p>
<p>The values in the result follow so-called “standard” order: If <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">=</span>
<span class="pre">fft(a,</span> <span class="pre">n)</span></code>, then <code class="docutils literal notranslate"><span class="pre">A[0]</span></code> contains the zero-frequency term (the sum of
the signal), which is always purely real for real inputs. Then <code class="docutils literal notranslate"><span class="pre">A[1:n/2]</span></code>
contains the positive-frequency terms, and <code class="docutils literal notranslate"><span class="pre">A[n/2+1:]</span></code> contains the
negative-frequency terms, in order of decreasingly negative frequency.
For an even number of input points, <code class="docutils literal notranslate"><span class="pre">A[n/2]</span></code> represents both positive and
negative Nyquist frequency, and is also purely real for real input. For
an odd number of input points, <code class="docutils literal notranslate"><span class="pre">A[(n-1)/2]</span></code> contains the largest positive
frequency, while <code class="docutils literal notranslate"><span class="pre">A[(n+1)/2]</span></code> contains the largest negative frequency.
The routine <code class="docutils literal notranslate"><span class="pre">np.fft.fftfreq(n)</span></code> returns an array giving the frequencies
of corresponding elements in the output. The routine
<code class="docutils literal notranslate"><span class="pre">np.fft.fftshift(A)</span></code> shifts transforms and their frequencies to put the
zero-frequency components in the middle, and <code class="docutils literal notranslate"><span class="pre">np.fft.ifftshift(A)</span></code> undoes
that shift.</p>
<p>When the input <em class="xref py py-obj">a</em> is a time-domain signal and <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">=</span> <span class="pre">fft(a)</span></code>, <code class="docutils literal notranslate"><span class="pre">np.abs(A)</span></code>
is its amplitude spectrum and <code class="docutils literal notranslate"><span class="pre">np.abs(A)**2</span></code> is its power spectrum.
The phase spectrum is obtained by <code class="docutils literal notranslate"><span class="pre">np.angle(A)</span></code>.</p>
<p>The inverse DFT is defined as</p>
<div class="math notranslate nohighlight">
\[a_m = \frac{1}{n}\sum_{k=0}^{n-1}A_k\exp\left\{2\pi i{mk\over n}\right\}
\qquad m = 0,\ldots,n-1.\]</div>
<p>It differs from the forward transform by the sign of the exponential
argument and the default normalization by <span class="math notranslate nohighlight">\(1/n\)</span>.</p>
</section>
<section id="type-promotion">
<h2>Type Promotion<a class="headerlink" href="#type-promotion" title="Link to this heading">#</a></h2>
<p><a class="reference internal" href="#module-numpy.fft" title="numpy.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.fft</span></code></a> promotes <code class="docutils literal notranslate"><span class="pre">float32</span></code> and <code class="docutils literal notranslate"><span class="pre">complex64</span></code> arrays to <code class="docutils literal notranslate"><span class="pre">float64</span></code> and
<code class="docutils literal notranslate"><span class="pre">complex128</span></code> arrays respectively. For an FFT implementation that does not
promote input arrays, see <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/fftpack.html#module-scipy.fftpack" title="(in SciPy v1.15.2)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.fftpack</span></code></a>.</p>
</section>
<section id="normalization">
<h2>Normalization<a class="headerlink" href="#normalization" title="Link to this heading">#</a></h2>
<p>The argument <code class="docutils literal notranslate"><span class="pre">norm</span></code> indicates which direction of the pair of direct/inverse
transforms is scaled and with what normalization factor.
The default normalization (<code class="docutils literal notranslate"><span class="pre">"backward"</span></code>) has the direct (forward) transforms
unscaled and the inverse (backward) transforms scaled by <span class="math notranslate nohighlight">\(1/n\)</span>. It is
possible to obtain unitary transforms by setting the keyword argument <code class="docutils literal notranslate"><span class="pre">norm</span></code>
to <code class="docutils literal notranslate"><span class="pre">"ortho"</span></code> so that both direct and inverse transforms are scaled by
<span class="math notranslate nohighlight">\(1/\sqrt{n}\)</span>. Finally, setting the keyword argument <code class="docutils literal notranslate"><span class="pre">norm</span></code> to
<code class="docutils literal notranslate"><span class="pre">"forward"</span></code> has the direct transforms scaled by <span class="math notranslate nohighlight">\(1/n\)</span> and the inverse
transforms unscaled (i.e. exactly opposite to the default <code class="docutils literal notranslate"><span class="pre">"backward"</span></code>).
<em class="xref py py-obj">None</em> is an alias of the default option <code class="docutils literal notranslate"><span class="pre">"backward"</span></code> for backward
compatibility.</p>
</section>
<section id="real-and-hermitian-transforms">
<h2>Real and Hermitian transforms<a class="headerlink" href="#real-and-hermitian-transforms" title="Link to this heading">#</a></h2>
<p>When the input is purely real, its transform is Hermitian, i.e., the
component at frequency <span class="math notranslate nohighlight">\(f_k\)</span> is the complex conjugate of the
component at frequency <span class="math notranslate nohighlight">\(-f_k\)</span>, which means that for real
inputs there is no information in the negative frequency components that
is not already available from the positive frequency components.
The family of <a class="reference internal" href="generated/numpy.fft.rfft.html#numpy.fft.rfft" title="numpy.fft.rfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfft</span></code></a> functions is
designed to operate on real inputs, and exploits this symmetry by
computing only the positive frequency components, up to and including the
Nyquist frequency. Thus, <code class="docutils literal notranslate"><span class="pre">n</span></code> input points produce <code class="docutils literal notranslate"><span class="pre">n/2+1</span></code> complex
output points. The inverses of this family assumes the same symmetry of
its input, and for an output of <code class="docutils literal notranslate"><span class="pre">n</span></code> points uses <code class="docutils literal notranslate"><span class="pre">n/2+1</span></code> input points.</p>
<p>Correspondingly, when the spectrum is purely real, the signal is
Hermitian. The <a class="reference internal" href="generated/numpy.fft.hfft.html#numpy.fft.hfft" title="numpy.fft.hfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">hfft</span></code></a> family of functions exploits this symmetry by
using <code class="docutils literal notranslate"><span class="pre">n/2+1</span></code> complex points in the input (time) domain for <code class="docutils literal notranslate"><span class="pre">n</span></code> real
points in the frequency domain.</p>
<p>In higher dimensions, FFTs are used, e.g., for image analysis and
filtering. The computational efficiency of the FFT means that it can
also be a faster way to compute large convolutions, using the property
that a convolution in the time domain is equivalent to a point-by-point
multiplication in the frequency domain.</p>
</section>
<section id="higher-dimensions">
<h2>Higher dimensions<a class="headerlink" href="#higher-dimensions" title="Link to this heading">#</a></h2>
<p>In two dimensions, the DFT is defined as</p>
<div class="math notranslate nohighlight">
\[A_{kl} = \sum_{m=0}^{M-1} \sum_{n=0}^{N-1}
a_{mn}\exp\left\{-2\pi i \left({mk\over M}+{nl\over N}\right)\right\}
\qquad k = 0, \ldots, M-1;\quad l = 0, \ldots, N-1,\]</div>
<p>which extends in the obvious way to higher dimensions, and the inverses
in higher dimensions also extend in the same way.</p>
</section>
<section id="references">
<h2>References<a class="headerlink" href="#references" title="Link to this heading">#</a></h2>
<div role="list" class="citation-list">
<div class="citation" id="rfb1dc64dd6a5-ct" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">CT</a><span class="fn-bracket">]</span></span>
<p>Cooley, James W., and John W. Tukey, 1965, “An algorithm for the
machine calculation of complex Fourier series,” <em>Math. Comput.</em>
19: 297-301.</p>
</div>
<div class="citation" id="rfb1dc64dd6a5-nr" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id2">NR</a><span class="fn-bracket">]</span></span>
<p>Press, W., Teukolsky, S., Vetterline, W.T., and Flannery, B.P.,
2007, <em>Numerical Recipes: The Art of Scientific Computing</em>, ch.
12-13. Cambridge Univ. Press, Cambridge, UK.</p>
</div>
</div>
</section>
<section id="examples">
<h2>Examples<a class="headerlink" href="#examples" title="Link to this heading">#</a></h2>
<p>For examples, see the various functions.</p>
</section>
</section>
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