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<li class="toctree-l1"><a class="reference internal" href="notes/amp_examples.html">Automatic Mixed Precision examples</a></li>
<li class="toctree-l1"><a class="reference internal" href="notes/autograd.html">Autograd mechanics</a></li>
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<div class="section" id="torch-fft">
<span id="torch-fft-module"></span><h1>torch.fft<a class="headerlink" href="#torch-fft" title="Permalink to this headline">¶</a></h1>
<p>Discrete Fourier transforms and related functions.</p>
<p>To use these functions the torch.fft module must be imported since its name
conflicts with the <a class="reference internal" href="generated/torch.fft.html#torch.fft" title="torch.fft"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.fft()</span></code></a> function.</p>
<div class="section" id="functions">
<h2>Functions<a class="headerlink" href="#functions" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="torch.fft.fft">
<code class="sig-prename descclassname">torch.fft.</code><code class="sig-name descname">fft</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">n=None</em>, <em class="sig-param">dim=-1</em>, <em class="sig-param">norm=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.fft.fft" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the one dimensional discrete Fourier transform of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The Fourier domain representation of any real signal satisfies the
Hermitian property: <cite>X[i] = conj(X[-i])</cite>. This function always returns both
the positive and negative frequency terms even though, for real inputs, the
negative frequencies are redundant. <a class="reference internal" href="#torch.fft.rfft" title="torch.fft.rfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">rfft()</span></code></a> returns the
more compact one-sided representation where only the positive frequencies
are returned.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor</p></li>
<li><p><strong>n</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><em>optional</em>) – Signal length. If given, the input will either be zero-padded
or trimmed to this length before computing the FFT.</p></li>
<li><p><strong>dim</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><em>optional</em>) – The dimension along which to take the one dimensional FFT.</p></li>
<li><p><strong>norm</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.9)"><em>str</em></a><em>, </em><em>optional</em>) – <p>Normalization mode. For the forward transform
(<a class="reference internal" href="#torch.fft.fft" title="torch.fft.fft"><code class="xref py py-func docutils literal notranslate"><span class="pre">fft()</span></code></a>), these correspond to:</p>
<ul>
<li><p><code class="docutils literal notranslate"><span class="pre">"forward"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"backward"</span></code> - no normalization</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"ortho"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/sqrt(n)</span></code> (making the FFT orthonormal)</p></li>
</ul>
<p>Calling the backward transform (<a class="reference internal" href="#torch.fft.ifft" title="torch.fft.ifft"><code class="xref py py-func docutils literal notranslate"><span class="pre">ifft()</span></code></a>) with the same
normalization mode will apply an overall normalization of <code class="docutils literal notranslate"><span class="pre">1/n</span></code> between
the two transforms. This is required to make <a class="reference internal" href="#torch.fft.ifft" title="torch.fft.ifft"><code class="xref py py-func docutils literal notranslate"><span class="pre">ifft()</span></code></a>
the exact inverse.</p>
<p>Default is <code class="docutils literal notranslate"><span class="pre">"backward"</span></code> (no normalization).</p>
</p></li>
</ul>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">torch.fft</span>
<span class="gp">>>> </span><span class="n">t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">t</span>
<span class="go">tensor([0, 1, 2, 3])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="go">tensor([ 6.+0.j, -2.+2.j, -2.+0.j, -2.-2.j])</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">t</span> <span class="o">=</span> <span class="n">tensor</span><span class="p">([</span><span class="mf">0.</span><span class="o">+</span><span class="mf">1.</span><span class="n">j</span><span class="p">,</span> <span class="mf">2.</span><span class="o">+</span><span class="mf">3.</span><span class="n">j</span><span class="p">,</span> <span class="mf">4.</span><span class="o">+</span><span class="mf">5.</span><span class="n">j</span><span class="p">,</span> <span class="mf">6.</span><span class="o">+</span><span class="mf">7.</span><span class="n">j</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="go">tensor([12.+16.j, -8.+0.j, -4.-4.j, 0.-8.j])</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.fft.ifft">
<code class="sig-prename descclassname">torch.fft.</code><code class="sig-name descname">ifft</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">n=None</em>, <em class="sig-param">dim=-1</em>, <em class="sig-param">norm=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.fft.ifft" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the one dimensional inverse discrete Fourier transform of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor</p></li>
<li><p><strong>n</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><em>optional</em>) – Signal length. If given, the input will either be zero-padded
or trimmed to this length before computing the IFFT.</p></li>
<li><p><strong>dim</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><em>optional</em>) – The dimension along which to take the one dimensional IFFT.</p></li>
<li><p><strong>norm</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.9)"><em>str</em></a><em>, </em><em>optional</em>) – <p>Normalization mode. For the backward transform
(<a class="reference internal" href="#torch.fft.ifft" title="torch.fft.ifft"><code class="xref py py-func docutils literal notranslate"><span class="pre">ifft()</span></code></a>), these correspond to:</p>
<ul>
<li><p><code class="docutils literal notranslate"><span class="pre">"forward"</span></code> - no normalization</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"backward"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"ortho"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/sqrt(n)</span></code> (making the IFFT orthonormal)</p></li>
</ul>
<p>Calling the forward transform (<a class="reference internal" href="#torch.fft.fft" title="torch.fft.fft"><code class="xref py py-func docutils literal notranslate"><span class="pre">fft()</span></code></a>) with the same
normalization mode will apply an overall normalization of <code class="docutils literal notranslate"><span class="pre">1/n</span></code> between
the two transforms. This is required to make <a class="reference internal" href="#torch.fft.ifft" title="torch.fft.ifft"><code class="xref py py-func docutils literal notranslate"><span class="pre">ifft()</span></code></a>
the exact inverse.</p>
<p>Default is <code class="docutils literal notranslate"><span class="pre">"backward"</span></code> (normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code>).</p>
</p></li>
</ul>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">torch.fft</span>
<span class="gp">>>> </span><span class="n">t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span> <span class="mf">6.</span><span class="o">+</span><span class="mf">0.</span><span class="n">j</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="o">+</span><span class="mf">2.</span><span class="n">j</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="o">+</span><span class="mf">0.</span><span class="n">j</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="o">-</span><span class="mf">2.</span><span class="n">j</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">ifft</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="go">tensor([0.+0.j, 1.+0.j, 2.+0.j, 3.+0.j])</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.fft.fftn">
<code class="sig-prename descclassname">torch.fft.</code><code class="sig-name descname">fftn</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">s=None</em>, <em class="sig-param">dim=None</em>, <em class="sig-param">norm=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.fft.fftn" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the N dimensional discrete Fourier transform of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The Fourier domain representation of any real signal satisfies the
Hermitian property: <code class="docutils literal notranslate"><span class="pre">X[i_1,</span> <span class="pre">...,</span> <span class="pre">i_n]</span> <span class="pre">=</span> <span class="pre">conj(X[-i_1,</span> <span class="pre">...,</span> <span class="pre">-i_n])</span></code>. This
function always returns all positive and negative frequency terms even
though, for real inputs, half of these values are redundant.
<a class="reference internal" href="#torch.fft.rfftn" title="torch.fft.rfftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">rfftn()</span></code></a> returns the more compact one-sided representation
where only the positive frequencies of the last dimension are returned.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor</p></li>
<li><p><strong>s</strong> (<em>Tuple</em><em>[</em><a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>]</em><em>, </em><em>optional</em>) – Signal size in the transformed dimensions.
If given, each dimension <code class="docutils literal notranslate"><span class="pre">dim[i]</span></code> will either be zero-padded or
trimmed to the length <code class="docutils literal notranslate"><span class="pre">s[i]</span></code> before computing the FFT.
If a length <code class="docutils literal notranslate"><span class="pre">-1</span></code> is specified, no padding is done in that dimension.
Default: <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">=</span> <span class="pre">[input.size(d)</span> <span class="pre">for</span> <span class="pre">d</span> <span class="pre">in</span> <span class="pre">dim]</span></code></p></li>
<li><p><strong>dim</strong> (<em>Tuple</em><em>[</em><a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>]</em><em>, </em><em>optional</em>) – Dimensions to be transformed.
Default: all dimensions, or the last <code class="docutils literal notranslate"><span class="pre">len(s)</span></code> dimensions if <code class="xref py py-attr docutils literal notranslate"><span class="pre">s</span></code> is given.</p></li>
<li><p><strong>norm</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.9)"><em>str</em></a><em>, </em><em>optional</em>) – <p>Normalization mode. For the forward transform
(<a class="reference internal" href="#torch.fft.fftn" title="torch.fft.fftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">fftn()</span></code></a>), these correspond to:</p>
<ul>
<li><p><code class="docutils literal notranslate"><span class="pre">"forward"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"backward"</span></code> - no normalization</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"ortho"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/sqrt(n)</span></code> (making the FFT orthonormal)</p></li>
</ul>
<p>Where <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">=</span> <span class="pre">prod(s)</span></code> is the logical FFT size.
Calling the backward transform (<a class="reference internal" href="#torch.fft.ifftn" title="torch.fft.ifftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">ifftn()</span></code></a>) with the same
normalization mode will apply an overall normalization of <code class="docutils literal notranslate"><span class="pre">1/n</span></code>
between the two transforms. This is required to make
<a class="reference internal" href="#torch.fft.ifftn" title="torch.fft.ifftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">ifftn()</span></code></a> the exact inverse.</p>
<p>Default is <code class="docutils literal notranslate"><span class="pre">"backward"</span></code> (no normalization).</p>
</p></li>
</ul>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">torch.fft</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">complex64</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">fftn</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fftn</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
</pre></div>
</div>
<p>The discrete Fourier transform is separable, so <a class="reference internal" href="#torch.fft.fftn" title="torch.fft.fftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">fftn()</span></code></a>
here is equivalent to two one-dimensional <a class="reference internal" href="#torch.fft.fft" title="torch.fft.fft"><code class="xref py py-func docutils literal notranslate"><span class="pre">fft()</span></code></a> calls:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">two_ffts</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">),</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">fftn</span><span class="p">,</span> <span class="n">two_ffts</span><span class="p">)</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.fft.ifftn">
<code class="sig-prename descclassname">torch.fft.</code><code class="sig-name descname">ifftn</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">s=None</em>, <em class="sig-param">dim=None</em>, <em class="sig-param">norm=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.fft.ifftn" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the N dimensional inverse discrete Fourier transform of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor</p></li>
<li><p><strong>s</strong> (<em>Tuple</em><em>[</em><a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>]</em><em>, </em><em>optional</em>) – Signal size in the transformed dimensions.
If given, each dimension <code class="docutils literal notranslate"><span class="pre">dim[i]</span></code> will either be zero-padded or
trimmed to the length <code class="docutils literal notranslate"><span class="pre">s[i]</span></code> before computing the IFFT.
If a length <code class="docutils literal notranslate"><span class="pre">-1</span></code> is specified, no padding is done in that dimension.
Default: <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">=</span> <span class="pre">[input.size(d)</span> <span class="pre">for</span> <span class="pre">d</span> <span class="pre">in</span> <span class="pre">dim]</span></code></p></li>
<li><p><strong>dim</strong> (<em>Tuple</em><em>[</em><a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>]</em><em>, </em><em>optional</em>) – Dimensions to be transformed.
Default: all dimensions, or the last <code class="docutils literal notranslate"><span class="pre">len(s)</span></code> dimensions if <code class="xref py py-attr docutils literal notranslate"><span class="pre">s</span></code> is given.</p></li>
<li><p><strong>norm</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.9)"><em>str</em></a><em>, </em><em>optional</em>) – <p>Normalization mode. For the backward transform
(<a class="reference internal" href="#torch.fft.ifftn" title="torch.fft.ifftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">ifftn()</span></code></a>), these correspond to:</p>
<ul>
<li><p><code class="docutils literal notranslate"><span class="pre">"forward"</span></code> - no normalization</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"backward"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"ortho"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/sqrt(n)</span></code> (making the IFFT orthonormal)</p></li>
</ul>
<p>Where <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">=</span> <span class="pre">prod(s)</span></code> is the logical IFFT size.
Calling the forward transform (<a class="reference internal" href="#torch.fft.fftn" title="torch.fft.fftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">fftn()</span></code></a>) with the same
normalization mode will apply an overall normalization of <code class="docutils literal notranslate"><span class="pre">1/n</span></code> between
the two transforms. This is required to make <a class="reference internal" href="#torch.fft.ifftn" title="torch.fft.ifftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">ifftn()</span></code></a>
the exact inverse.</p>
<p>Default is <code class="docutils literal notranslate"><span class="pre">"backward"</span></code> (normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code>).</p>
</p></li>
</ul>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">torch.fft</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">complex64</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">ifftn</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">ifftn</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
</pre></div>
</div>
<p>The discrete Fourier transform is separable, so <a class="reference internal" href="#torch.fft.ifftn" title="torch.fft.ifftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">ifftn()</span></code></a>
here is equivalent to two one-dimensional <a class="reference internal" href="#torch.fft.ifft" title="torch.fft.ifft"><code class="xref py py-func docutils literal notranslate"><span class="pre">ifft()</span></code></a> calls:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">two_iffts</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">ifft</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">ifft</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">),</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">ifftn</span><span class="p">,</span> <span class="n">two_iffts</span><span class="p">)</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.fft.rfft">
<code class="sig-prename descclassname">torch.fft.</code><code class="sig-name descname">rfft</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">n=None</em>, <em class="sig-param">dim=-1</em>, <em class="sig-param">norm=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.fft.rfft" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the one dimensional Fourier transform of real-valued <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<p>The FFT of a real signal is Hermitian-symmetric, <code class="docutils literal notranslate"><span class="pre">X[i]</span> <span class="pre">=</span> <span class="pre">conj(X[-i])</span></code> so
the output contains only the positive frequencies below the Nyquist frequency.
To compute the full output, use <a class="reference internal" href="#torch.fft.fft" title="torch.fft.fft"><code class="xref py py-func docutils literal notranslate"><span class="pre">fft()</span></code></a></p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the real input tensor</p></li>
<li><p><strong>n</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><em>optional</em>) – Signal length. If given, the input will either be zero-padded
or trimmed to this length before computing the real FFT.</p></li>
<li><p><strong>dim</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><em>optional</em>) – The dimension along which to take the one dimensional real FFT.</p></li>
<li><p><strong>norm</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.9)"><em>str</em></a><em>, </em><em>optional</em>) – <p>Normalization mode. For the forward transform
(<a class="reference internal" href="#torch.fft.rfft" title="torch.fft.rfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">rfft()</span></code></a>), these correspond to:</p>
<ul>
<li><p><code class="docutils literal notranslate"><span class="pre">"forward"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"backward"</span></code> - no normalization</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"ortho"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/sqrt(n)</span></code> (making the FFT orthonormal)</p></li>
</ul>
<p>Calling the backward transform (<a class="reference internal" href="#torch.fft.irfft" title="torch.fft.irfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">irfft()</span></code></a>) with the same
normalization mode will apply an overall normalization of <code class="docutils literal notranslate"><span class="pre">1/n</span></code> between
the two transforms. This is required to make <a class="reference internal" href="#torch.fft.irfft" title="torch.fft.irfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">irfft()</span></code></a>
the exact inverse.</p>
<p>Default is <code class="docutils literal notranslate"><span class="pre">"backward"</span></code> (no normalization).</p>
</p></li>
</ul>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">torch.fft</span>
<span class="gp">>>> </span><span class="n">t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">t</span>
<span class="go">tensor([0, 1, 2, 3])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfft</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="go">tensor([ 6.+0.j, -2.+2.j, -2.+0.j])</span>
</pre></div>
</div>
<p>Compare against the full output from <a class="reference internal" href="#torch.fft.fft" title="torch.fft.fft"><code class="xref py py-func docutils literal notranslate"><span class="pre">fft()</span></code></a>:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="go">tensor([ 6.+0.j, -2.+2.j, -2.+0.j, -2.-2.j])</span>
</pre></div>
</div>
<p>Notice that the symmetric element <code class="docutils literal notranslate"><span class="pre">T[-1]</span> <span class="pre">==</span> <span class="pre">T[1].conj()</span></code> is omitted.
At the Nyquist frequency <code class="docutils literal notranslate"><span class="pre">T[-2]</span> <span class="pre">==</span> <span class="pre">T[2]</span></code> is it’s own symmetric pair,
and therefore must always be real-valued.</p>
</dd></dl>
<dl class="function">
<dt id="torch.fft.irfft">
<code class="sig-prename descclassname">torch.fft.</code><code class="sig-name descname">irfft</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">n=None</em>, <em class="sig-param">dim=-1</em>, <em class="sig-param">norm=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.fft.irfft" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the inverse of <a class="reference internal" href="#torch.fft.rfft" title="torch.fft.rfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">rfft()</span></code></a>.</p>
<p><code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is interpreted as a one-sided Hermitian signal in the Fourier
domain, as produced by <a class="reference internal" href="#torch.fft.rfft" title="torch.fft.rfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">rfft()</span></code></a>. By the Hermitian property, the
output will be real-valued.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Some input frequencies must be real-valued to satisfy the Hermitian
property. In these cases the imaginary component will be ignored.
For example, any imaginary component in the zero-frequency term cannot
be represented in a real output and so will always be ignored.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The correct interpretation of the Hermitian input depends on the length of
the original data, as given by <code class="xref py py-attr docutils literal notranslate"><span class="pre">n</span></code>. This is because each input shape
could correspond to either an odd or even length signal. By default, the
signal is assumed to be even length and odd signals will not round-trip
properly. So, it is recommended to always pass the signal length <code class="xref py py-attr docutils literal notranslate"><span class="pre">n</span></code>.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor representing a half-Hermitian signal</p></li>
<li><p><strong>n</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><em>optional</em>) – Output signal length. This determines the length of the
output signal. If given, the input will either be zero-padded or trimmed to this
length before computing the real IFFT.
Defaults to even output: <code class="docutils literal notranslate"><span class="pre">n=2*(input.size(dim)</span> <span class="pre">-</span> <span class="pre">1)</span></code>.</p></li>
<li><p><strong>dim</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><em>optional</em>) – The dimension along which to take the one dimensional real IFFT.</p></li>
<li><p><strong>norm</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.9)"><em>str</em></a><em>, </em><em>optional</em>) – <p>Normalization mode. For the backward transform
(<a class="reference internal" href="#torch.fft.irfft" title="torch.fft.irfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">irfft()</span></code></a>), these correspond to:</p>
<ul>
<li><p><code class="docutils literal notranslate"><span class="pre">"forward"</span></code> - no normalization</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"backward"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"ortho"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/sqrt(n)</span></code> (making the real IFFT orthonormal)</p></li>
</ul>
<p>Calling the forward transform (<a class="reference internal" href="#torch.fft.rfft" title="torch.fft.rfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">rfft()</span></code></a>) with the same
normalization mode will apply an overall normalization of <code class="docutils literal notranslate"><span class="pre">1/n</span></code> between
the two transforms. This is required to make <a class="reference internal" href="#torch.fft.irfft" title="torch.fft.irfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">irfft()</span></code></a>
the exact inverse.</p>
<p>Default is <code class="docutils literal notranslate"><span class="pre">"backward"</span></code> (normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code>).</p>
</p></li>
</ul>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">torch.fft</span>
<span class="gp">>>> </span><span class="n">t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">t</span>
<span class="go">tensor([0, 1, 2, 3, 4])</span>
<span class="gp">>>> </span><span class="n">T</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfft</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">T</span>
<span class="go">tensor([10.0000+0.0000j, -2.5000+3.4410j, -2.5000+0.8123j])</span>
</pre></div>
</div>
<p>Without specifying the output length to <a class="reference internal" href="#torch.fft.irfft" title="torch.fft.irfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">irfft()</span></code></a>, the output
will not round-trip properly because the input is odd-length:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">irfft</span><span class="p">(</span><span class="n">T</span><span class="p">)</span>
<span class="go">tensor([0.6250, 1.4045, 3.1250, 4.8455])</span>
</pre></div>
</div>
<p>So, it is recommended to always pass the signal length <code class="xref py py-attr docutils literal notranslate"><span class="pre">n</span></code>:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">irfft</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">t</span><span class="o">.</span><span class="n">numel</span><span class="p">())</span>
<span class="go">tensor([0.0000, 1.0000, 2.0000, 3.0000, 4.0000])</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.fft.rfftn">
<code class="sig-prename descclassname">torch.fft.</code><code class="sig-name descname">rfftn</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">s=None</em>, <em class="sig-param">dim=None</em>, <em class="sig-param">norm=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.fft.rfftn" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the N-dimensional discrete Fourier transform of real <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<p>The FFT of a real signal is Hermitian-symmetric,
<code class="docutils literal notranslate"><span class="pre">X[i_1,</span> <span class="pre">...,</span> <span class="pre">i_n]</span> <span class="pre">=</span> <span class="pre">conj(X[-i_1,</span> <span class="pre">...,</span> <span class="pre">-i_n])</span></code> so the full
<a class="reference internal" href="#torch.fft.fftn" title="torch.fft.fftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">fftn()</span></code></a> output contains redundant information.
<a class="reference internal" href="#torch.fft.rfftn" title="torch.fft.rfftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">rfftn()</span></code></a> instead omits the negative frequencies in the
last dimension.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor</p></li>
<li><p><strong>s</strong> (<em>Tuple</em><em>[</em><a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>]</em><em>, </em><em>optional</em>) – Signal size in the transformed dimensions.
If given, each dimension <code class="docutils literal notranslate"><span class="pre">dim[i]</span></code> will either be zero-padded or
trimmed to the length <code class="docutils literal notranslate"><span class="pre">s[i]</span></code> before computing the real FFT.
If a length <code class="docutils literal notranslate"><span class="pre">-1</span></code> is specified, no padding is done in that dimension.
Default: <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">=</span> <span class="pre">[input.size(d)</span> <span class="pre">for</span> <span class="pre">d</span> <span class="pre">in</span> <span class="pre">dim]</span></code></p></li>
<li><p><strong>dim</strong> (<em>Tuple</em><em>[</em><a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>]</em><em>, </em><em>optional</em>) – Dimensions to be transformed.
Default: all dimensions, or the last <code class="docutils literal notranslate"><span class="pre">len(s)</span></code> dimensions if <code class="xref py py-attr docutils literal notranslate"><span class="pre">s</span></code> is given.</p></li>
<li><p><strong>norm</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.9)"><em>str</em></a><em>, </em><em>optional</em>) – <p>Normalization mode. For the forward transform
(<a class="reference internal" href="#torch.fft.rfftn" title="torch.fft.rfftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">rfftn()</span></code></a>), these correspond to:</p>
<ul>
<li><p><code class="docutils literal notranslate"><span class="pre">"forward"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"backward"</span></code> - no normalization</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"ortho"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/sqrt(n)</span></code> (making the real FFT orthonormal)</p></li>
</ul>
<p>Where <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">=</span> <span class="pre">prod(s)</span></code> is the logical FFT size.
Calling the backward transform (<a class="reference internal" href="#torch.fft.irfftn" title="torch.fft.irfftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">irfftn()</span></code></a>) with the same
normalization mode will apply an overall normalization of <code class="docutils literal notranslate"><span class="pre">1/n</span></code> between
the two transforms. This is required to make <a class="reference internal" href="#torch.fft.irfftn" title="torch.fft.irfftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">irfftn()</span></code></a>
the exact inverse.</p>
<p>Default is <code class="docutils literal notranslate"><span class="pre">"backward"</span></code> (no normalization).</p>
</p></li>
</ul>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">torch.fft</span>
<span class="gp">>>> </span><span class="n">t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">rfftn</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfftn</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">rfftn</span><span class="o">.</span><span class="n">size</span><span class="p">()</span>
<span class="go">torch.Size([10, 6])</span>
</pre></div>
</div>
<p>Compared against the full output from <a class="reference internal" href="#torch.fft.fftn" title="torch.fft.fftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">fftn()</span></code></a>, we have all
elements up to the Nyquist frequency.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">fftn</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fftn</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">fftn</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">6</span><span class="p">],</span> <span class="n">rfftn</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
<p>The discrete Fourier transform is separable, so <a class="reference internal" href="#torch.fft.rfftn" title="torch.fft.rfftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">rfftn()</span></code></a>
here is equivalent to a combination of <a class="reference internal" href="#torch.fft.fft" title="torch.fft.fft"><code class="xref py py-func docutils literal notranslate"><span class="pre">fft()</span></code></a> and
<a class="reference internal" href="#torch.fft.rfft" title="torch.fft.rfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">rfft()</span></code></a>:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">two_ffts</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfft</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">rfftn</span><span class="p">,</span> <span class="n">two_ffts</span><span class="p">)</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.fft.irfftn">
<code class="sig-prename descclassname">torch.fft.</code><code class="sig-name descname">irfftn</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">s=None</em>, <em class="sig-param">dim=None</em>, <em class="sig-param">norm=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.fft.irfftn" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the inverse of <a class="reference internal" href="#torch.fft.rfftn" title="torch.fft.rfftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">rfftn()</span></code></a>.</p>
<p><code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is interpreted as a one-sided Hermitian signal in the Fourier
domain, as produced by <a class="reference internal" href="#torch.fft.rfftn" title="torch.fft.rfftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">rfftn()</span></code></a>. By the Hermitian property, the
output will be real-valued.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Some input frequencies must be real-valued to satisfy the Hermitian
property. In these cases the imaginary component will be ignored.
For example, any imaginary component in the zero-frequency term cannot
be represented in a real output and so will always be ignored.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The correct interpretation of the Hermitian input depends on the length of
the original data, as given by <code class="xref py py-attr docutils literal notranslate"><span class="pre">s</span></code>. This is because each input shape
could correspond to either an odd or even length signal. By default, the
signal is assumed to be even length and odd signals will not round-trip
properly. So, it is recommended to always pass the signal shape <code class="xref py py-attr docutils literal notranslate"><span class="pre">s</span></code>.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor</p></li>
<li><p><strong>s</strong> (<em>Tuple</em><em>[</em><a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>]</em><em>, </em><em>optional</em>) – Signal size in the transformed dimensions.
If given, each dimension <code class="docutils literal notranslate"><span class="pre">dim[i]</span></code> will either be zero-padded or
trimmed to the length <code class="docutils literal notranslate"><span class="pre">s[i]</span></code> before computing the real FFT.
If a length <code class="docutils literal notranslate"><span class="pre">-1</span></code> is specified, no padding is done in that dimension.
Defaults to even output in the last dimension:
<code class="docutils literal notranslate"><span class="pre">s[-1]</span> <span class="pre">=</span> <span class="pre">2*(input.size(dim[-1])</span> <span class="pre">-</span> <span class="pre">1)</span></code>.</p></li>
<li><p><strong>dim</strong> (<em>Tuple</em><em>[</em><a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>]</em><em>, </em><em>optional</em>) – Dimensions to be transformed.
The last dimension must be the half-Hermitian compressed dimension.
Default: all dimensions, or the last <code class="docutils literal notranslate"><span class="pre">len(s)</span></code> dimensions if <code class="xref py py-attr docutils literal notranslate"><span class="pre">s</span></code> is given.</p></li>
<li><p><strong>norm</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.9)"><em>str</em></a><em>, </em><em>optional</em>) – <p>Normalization mode. For the backward transform
(<a class="reference internal" href="#torch.fft.irfftn" title="torch.fft.irfftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">irfftn()</span></code></a>), these correspond to:</p>
<ul>
<li><p><code class="docutils literal notranslate"><span class="pre">"forward"</span></code> - no normalization</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"backward"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"ortho"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/sqrt(n)</span></code> (making the real IFFT orthonormal)</p></li>
</ul>
<p>Where <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">=</span> <span class="pre">prod(s)</span></code> is the logical IFFT size.
Calling the forward transform (<a class="reference internal" href="#torch.fft.rfftn" title="torch.fft.rfftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">rfftn()</span></code></a>) with the same
normalization mode will apply an overall normalization of <code class="docutils literal notranslate"><span class="pre">1/n</span></code> between
the two transforms. This is required to make <a class="reference internal" href="#torch.fft.irfftn" title="torch.fft.irfftn"><code class="xref py py-func docutils literal notranslate"><span class="pre">irfftn()</span></code></a>
the exact inverse.</p>
<p>Default is <code class="docutils literal notranslate"><span class="pre">"backward"</span></code> (normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code>).</p>
</p></li>
</ul>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">torch.fft</span>
<span class="gp">>>> </span><span class="n">t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">T</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfftn</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
</pre></div>
</div>
<p>Without specifying the output length to <a class="reference internal" href="#torch.fft.irfft" title="torch.fft.irfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">irfft()</span></code></a>, the output
will not round-trip properly because the input is odd-length in the last
dimension:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">irfftn</span><span class="p">(</span><span class="n">T</span><span class="p">)</span><span class="o">.</span><span class="n">size</span><span class="p">()</span>
<span class="go">torch.Size([10, 10])</span>
</pre></div>
</div>
<p>So, it is recommended to always pass the signal shape <code class="xref py py-attr docutils literal notranslate"><span class="pre">s</span></code>.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">roundtrip</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">irfftn</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">t</span><span class="o">.</span><span class="n">size</span><span class="p">())</span>
<span class="gp">>>> </span><span class="n">roundtrip</span><span class="o">.</span><span class="n">size</span><span class="p">()</span>
<span class="go">torch.Size([10, 9])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">roundtrip</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.fft.hfft">
<code class="sig-prename descclassname">torch.fft.</code><code class="sig-name descname">hfft</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">n=None</em>, <em class="sig-param">dim=-1</em>, <em class="sig-param">norm=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.fft.hfft" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the one dimensional discrete Fourier transform of a Hermitian
symmetric <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> signal.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p><a class="reference internal" href="#torch.fft.hfft" title="torch.fft.hfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">hfft()</span></code></a>/<a class="reference internal" href="#torch.fft.ihfft" title="torch.fft.ihfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">ihfft()</span></code></a> are analogous to
<a class="reference internal" href="#torch.fft.rfft" title="torch.fft.rfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">rfft()</span></code></a>/<a class="reference internal" href="#torch.fft.irfft" title="torch.fft.irfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">irfft()</span></code></a>. The real FFT expects
a real signal in the time-domain and gives a Hermitian symmetry in the
frequency-domain. The Hermitian FFT is the opposite; Hermitian symmetric in
the time-domain and real-valued in the frequency-domain. For this reason,
special care needs to be taken with the length argument <code class="xref py py-attr docutils literal notranslate"><span class="pre">n</span></code>, in the
same way as with <a class="reference internal" href="#torch.fft.irfft" title="torch.fft.irfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">irfft()</span></code></a>.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Because the signal is Hermitian in the time-domain, the result will be
real in the frequency domain. Note that some input frequencies must be
real-valued to satisfy the Hermitian property. In these cases the imaginary
component will be ignored. For example, any imaginary component in
<code class="docutils literal notranslate"><span class="pre">input[0]</span></code> would result in one or more complex frequency terms which
cannot be represented in a real output and so will always be ignored.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The correct interpretation of the Hermitian input depends on the length of
the original data, as given by <code class="xref py py-attr docutils literal notranslate"><span class="pre">n</span></code>. This is because each input shape
could correspond to either an odd or even length signal. By default, the
signal is assumed to be even length and odd signals will not round-trip
properly. So, it is recommended to always pass the signal length <code class="xref py py-attr docutils literal notranslate"><span class="pre">n</span></code>.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor representing a half-Hermitian signal</p></li>
<li><p><strong>n</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><em>optional</em>) – Output signal length. This determines the length of the
real output. If given, the input will either be zero-padded or trimmed to this
length before computing the Hermitian FFT.
Defaults to even output: <code class="docutils literal notranslate"><span class="pre">n=2*(input.size(dim)</span> <span class="pre">-</span> <span class="pre">1)</span></code>.</p></li>
<li><p><strong>dim</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><em>optional</em>) – The dimension along which to take the one dimensional Hermitian FFT.</p></li>
<li><p><strong>norm</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.9)"><em>str</em></a><em>, </em><em>optional</em>) – <p>Normalization mode. For the forward transform
(<a class="reference internal" href="#torch.fft.hfft" title="torch.fft.hfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">hfft()</span></code></a>), these correspond to:</p>
<ul>
<li><p><code class="docutils literal notranslate"><span class="pre">"forward"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"backward"</span></code> - no normalization</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"ortho"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/sqrt(n)</span></code> (making the Hermitian FFT orthonormal)</p></li>
</ul>
<p>Calling the backward transform (<a class="reference internal" href="#torch.fft.ihfft" title="torch.fft.ihfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">ihfft()</span></code></a>) with the same
normalization mode will apply an overall normalization of <code class="docutils literal notranslate"><span class="pre">1/n</span></code> between
the two transforms. This is required to make <a class="reference internal" href="#torch.fft.ihfft" title="torch.fft.ihfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">ihfft()</span></code></a>
the exact inverse.</p>
<p>Default is <code class="docutils literal notranslate"><span class="pre">"backward"</span></code> (no normalization).</p>
</p></li>
</ul>
</dd>
</dl>
<p class="rubric">Example</p>
<p>Taking a real-valued frequency signal and bringing it into the time domain
gives Hermitian symmetric output:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">torch.fft</span>
<span class="gp">>>> </span><span class="n">t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">t</span>
<span class="go">tensor([0, 1, 2, 3, 4])</span>
<span class="gp">>>> </span><span class="n">T</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">ifft</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">T</span>
<span class="go">tensor([ 2.0000+-0.0000j, -0.5000-0.6882j, -0.5000-0.1625j, -0.5000+0.1625j,</span>
<span class="go"> -0.5000+0.6882j])</span>
</pre></div>
</div>
<p>Note that <code class="docutils literal notranslate"><span class="pre">T[1]</span> <span class="pre">==</span> <span class="pre">T[-1].conj()</span></code> and <code class="docutils literal notranslate"><span class="pre">T[2]</span> <span class="pre">==</span> <span class="pre">T[-2].conj()</span></code> is
redundant. We can thus compute the forward transform without considering
negative frequencies:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">hfft</span><span class="p">(</span><span class="n">T</span><span class="p">[:</span><span class="mi">3</span><span class="p">],</span> <span class="n">n</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="go">tensor([0., 1., 2., 3., 4.])</span>
</pre></div>
</div>
<p>Like with <a class="reference internal" href="#torch.fft.irfft" title="torch.fft.irfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">irfft()</span></code></a>, the output length must be given in order
to recover an even length output:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">hfft</span><span class="p">(</span><span class="n">T</span><span class="p">[:</span><span class="mi">3</span><span class="p">])</span>
<span class="go">tensor([0.5000, 1.1236, 2.5000, 3.8764])</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.fft.ihfft">
<code class="sig-prename descclassname">torch.fft.</code><code class="sig-name descname">ihfft</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">n=None</em>, <em class="sig-param">dim=-1</em>, <em class="sig-param">norm=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.fft.ihfft" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the inverse of <a class="reference internal" href="#torch.fft.hfft" title="torch.fft.hfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">hfft()</span></code></a>.</p>
<p><code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> must be a real-valued signal, interpreted in the Fourier domain.
The IFFT of a real signal is Hermitian-symmetric, <code class="docutils literal notranslate"><span class="pre">X[i]</span> <span class="pre">=</span> <span class="pre">conj(X[-i])</span></code>.
<a class="reference internal" href="#torch.fft.ihfft" title="torch.fft.ihfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">ihfft()</span></code></a> represents this in the one-sided form where only the
positive frequencies below the Nyquist frequency are included. To compute the
full output, use <a class="reference internal" href="#torch.fft.ifft" title="torch.fft.ifft"><code class="xref py py-func docutils literal notranslate"><span class="pre">ifft()</span></code></a>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the real input tensor</p></li>
<li><p><strong>n</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><em>optional</em>) – Signal length. If given, the input will either be zero-padded
or trimmed to this length before computing the Hermitian IFFT.</p></li>
<li><p><strong>dim</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><em>optional</em>) – The dimension along which to take the one dimensional Hermitian IFFT.</p></li>
<li><p><strong>norm</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.9)"><em>str</em></a><em>, </em><em>optional</em>) – <p>Normalization mode. For the backward transform
(<a class="reference internal" href="#torch.fft.ihfft" title="torch.fft.ihfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">ihfft()</span></code></a>), these correspond to:</p>
<ul>
<li><p><code class="docutils literal notranslate"><span class="pre">"forward"</span></code> - no normalization</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"backward"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"ortho"</span></code> - normalize by <code class="docutils literal notranslate"><span class="pre">1/sqrt(n)</span></code> (making the IFFT orthonormal)</p></li>
</ul>
<p>Calling the forward transform (<a class="reference internal" href="#torch.fft.hfft" title="torch.fft.hfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">hfft()</span></code></a>) with the same
normalization mode will apply an overall normalization of <code class="docutils literal notranslate"><span class="pre">1/n</span></code> between
the two transforms. This is required to make <a class="reference internal" href="#torch.fft.ihfft" title="torch.fft.ihfft"><code class="xref py py-func docutils literal notranslate"><span class="pre">ihfft()</span></code></a>
the exact inverse.</p>
<p>Default is <code class="docutils literal notranslate"><span class="pre">"backward"</span></code> (normalize by <code class="docutils literal notranslate"><span class="pre">1/n</span></code>).</p>
</p></li>
</ul>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">torch.fft</span>
<span class="gp">>>> </span><span class="n">t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">t</span>
<span class="go">tensor([0, 1, 2, 3, 4])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">ihfft</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="go">tensor([ 2.0000+-0.0000j, -0.5000-0.6882j, -0.5000-0.1625j])</span>
</pre></div>
</div>
<p>Compare against the full output from <a class="reference internal" href="#torch.fft.ifft" title="torch.fft.ifft"><code class="xref py py-func docutils literal notranslate"><span class="pre">ifft()</span></code></a>:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">ifft</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="go">tensor([ 2.0000+-0.0000j, -0.5000-0.6882j, -0.5000-0.1625j, -0.5000+0.1625j,</span>
<span class="go"> -0.5000+0.6882j])</span>
</pre></div>
</div>
</dd></dl>
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