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<li><a class="reference internal" href="#">1. Filtering a <code class="docutils literal"><span class="pre">TimeSeries</span></code> to detect gravitational waves</a></li>
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<div class="section" id="filtering-a-timeseries-to-detect-gravitational-waves">
<span id="gwpy-example-signal-gw150914"></span><h1>1. Filtering a <a class="reference internal" href="../../api/gwpy.timeseries.TimeSeries.html#gwpy.timeseries.TimeSeries" title="gwpy.timeseries.TimeSeries"><code class="xref py py-obj docutils literal"><span class="pre">TimeSeries</span></code></a> to detect gravitational waves<a class="headerlink" href="#filtering-a-timeseries-to-detect-gravitational-waves" title="Permalink to this headline">¶</a></h1>
<p>The raw ‘strain’ output of the LIGO detectors is recorded as a <a class="reference internal" href="../../api/gwpy.timeseries.TimeSeries.html#gwpy.timeseries.TimeSeries" title="gwpy.timeseries.TimeSeries"><code class="xref py py-obj docutils literal"><span class="pre">TimeSeries</span></code></a>
with contributions from a large number of known and unknown noise sources,
as well as possible gravitational wave signals.</p>
<p>In order to uncover a real signal we need to filter out noises that otherwise
hide the signal in the data. We can do this by using the <code class="xref py py-mod docutils literal"><span class="pre">gwpy.signal</span></code>
module to design a digital filter to cut out low and high frequency noise, as
well as notch out fixed frequencies polluted by known artefacts.</p>
<p>First we download the raw strain data from the LOSC public archive:</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">gwpy.timeseries</span> <span class="kn">import</span> <span class="n">TimeSeries</span>
<span class="n">data</span> <span class="o">=</span> <span class="n">TimeSeries</span><span class="o">.</span><span class="n">fetch_open_data</span><span class="p">(</span><span class="s1">'H1'</span><span class="p">,</span> <span class="mi">1126259446</span><span class="p">,</span> <span class="mi">1126259478</span><span class="p">)</span>
</pre></div>
</div>
<p>Next we can design a zero-pole-gain filter to remove the extranious noise.</p>
<p>First we import the <code class="xref py py-obj docutils literal"><span class="pre">gwpy.signal.filter_design</span></code> module and create a
<code class="xref py py-meth docutils literal"><span class="pre">bandpass()</span></code> filter to remove both low and
high frequency content</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">gwpy.signal</span> <span class="kn">import</span> <span class="n">filter_design</span>
<span class="n">bp</span> <span class="o">=</span> <span class="n">filter_design</span><span class="o">.</span><span class="n">bandpass</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span> <span class="mi">250</span><span class="p">,</span> <span class="n">data</span><span class="o">.</span><span class="n">sample_rate</span><span class="p">)</span>
</pre></div>
</div>
<p>Now we want to combine the bandpass with a series of
<code class="xref py py-meth docutils literal"><span class="pre">notch()</span></code> filters, so we create those
for the first three harmonics of the 60 Hz AC mains power:</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">notches</span> <span class="o">=</span> <span class="p">[</span><span class="n">filter_design</span><span class="o">.</span><span class="n">notch</span><span class="p">(</span><span class="n">line</span><span class="p">,</span> <span class="n">data</span><span class="o">.</span><span class="n">sample_rate</span><span class="p">)</span> <span class="k">for</span>
<span class="n">line</span> <span class="ow">in</span> <span class="p">(</span><span class="mi">60</span><span class="p">,</span> <span class="mi">120</span><span class="p">,</span> <span class="mi">180</span><span class="p">)]</span>
</pre></div>
</div>
<p>and concatenate each of our filters together to create a single ZPK:</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">zpk</span> <span class="o">=</span> <span class="n">filter_design</span><span class="o">.</span><span class="n">concatenate_zpks</span><span class="p">(</span><span class="n">bp</span><span class="p">,</span> <span class="o">*</span><span class="n">notches</span><span class="p">)</span>
</pre></div>
</div>
<p>Now, we can apply our combined filter to the data, using <code class="xref py py-obj docutils literal"><span class="pre">filtfilt=True</span></code>
to filter both backwards and forwards to preserve the correct phase
at all frequencies</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">b</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">filter</span><span class="p">(</span><span class="n">zpk</span><span class="p">,</span> <span class="n">filtfilt</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
</pre></div>
</div>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">The <code class="xref py py-mod docutils literal"><span class="pre">filter_design</span></code> methods return digital filters
by default, so we apply them using <a class="reference internal" href="../../api/gwpy.timeseries.TimeSeries.html#gwpy.timeseries.TimeSeries.filter" title="gwpy.timeseries.TimeSeries.filter"><code class="xref py py-obj docutils literal"><span class="pre">TimeSeries.filter</span></code></a>. If we had
analogue filters (perhaps by passing <code class="xref py py-obj docutils literal"><span class="pre">analog=True</span></code> to the filter design
method), the easiest application would be <a class="reference internal" href="../../api/gwpy.timeseries.TimeSeries.html#gwpy.timeseries.TimeSeries.zpk" title="gwpy.timeseries.TimeSeries.zpk"><code class="xref py py-obj docutils literal"><span class="pre">TimeSeries.zpk</span></code></a></p>
</div>
<p>Finally, we can <a class="reference internal" href="../../api/gwpy.timeseries.TimeSeries.html#gwpy.timeseries.TimeSeries.plot" title="gwpy.timeseries.TimeSeries.plot"><code class="xref py py-meth docutils literal"><span class="pre">plot()</span></code></a> the original and filtered data,
adding some code to prettify the figure:</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">gwpy.plotter</span> <span class="kn">import</span> <span class="n">TimeSeriesPlot</span>
<span class="n">plot</span> <span class="o">=</span> <span class="n">TimeSeriesPlot</span><span class="p">(</span>
<span class="n">data</span><span class="o">.</span><span class="n">crop</span><span class="p">(</span><span class="o">*</span><span class="n">data</span><span class="o">.</span><span class="n">span</span><span class="o">.</span><span class="n">contract</span><span class="p">(</span><span class="mi">1</span><span class="p">)),</span>
<span class="n">b</span><span class="o">.</span><span class="n">crop</span><span class="p">(</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">span</span><span class="o">.</span><span class="n">contract</span><span class="p">(</span><span class="mi">1</span><span class="p">)),</span>
<span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span> <span class="n">sep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">sharex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="n">plot</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'LIGO-Hanford strain data around GW150914'</span><span class="p">)</span>
<span class="n">plot</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">text</span><span class="p">(</span>
<span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="s1">'Unfiltered data'</span><span class="p">,</span>
<span class="n">transform</span><span class="o">=</span><span class="n">plot</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">transAxes</span><span class="p">,</span> <span class="n">ha</span><span class="o">=</span><span class="s1">'right'</span><span class="p">)</span>
<span class="n">plot</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Amplitude [strain]'</span><span class="p">,</span> <span class="n">y</span><span class="o">=-</span><span class="mf">0.2</span><span class="p">)</span>
<span class="n">plot</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">text</span><span class="p">(</span>
<span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="s1">'50-250\,Hz bandpass, notches at 60, 120, 180 Hz'</span><span class="p">,</span>
<span class="n">transform</span><span class="o">=</span><span class="n">plot</span><span class="o">.</span><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">transAxes</span><span class="p">,</span> <span class="n">ha</span><span class="o">=</span><span class="s1">'right'</span><span class="p">)</span>
<span class="n">plot</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p>(<a class="reference external" href="../../examples/signal/gw150914-6.png">png</a>)</p>
<div class="figure">
<img alt="../../_images/gw150914-6.png" src="../../_images/gw150914-6.png" />
</div>
<p>We see now a spike around 16 seconds into the data, so let’s zoom into
that time by using <a class="reference internal" href="../../api/gwpy.timeseries.TimeSeries.html#gwpy.timeseries.TimeSeries.crop" title="gwpy.timeseries.TimeSeries.crop"><code class="xref py py-meth docutils literal"><span class="pre">crop()</span></code></a> and <a class="reference internal" href="../../api/gwpy.timeseries.TimeSeries.html#gwpy.timeseries.TimeSeries.plot" title="gwpy.timeseries.TimeSeries.plot"><code class="xref py py-meth docutils literal"><span class="pre">plot()</span></code></a>:</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">plot</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">crop</span><span class="p">(</span><span class="mi">1126259462</span><span class="p">,</span> <span class="mf">1126259462.6</span><span class="p">)</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">12</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
<span class="n">plot</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'LIGO-Hanford strain data around GW150914'</span><span class="p">)</span>
<span class="n">plot</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Amplitude [strain]'</span><span class="p">)</span>
<span class="n">plot</span><span class="o">.</span><span class="n">set_epoch</span><span class="p">(</span><span class="mf">1126259462.427</span><span class="p">)</span>
<span class="n">plot</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p>(<a class="reference external" href="../../examples/signal/gw150914-7.png">png</a>)</p>
<div class="figure">
<img alt="../../_images/gw150914-7.png" src="../../_images/gw150914-7.png" />
</div>
<p>Congratulations, you have succesfully filtered LIGO data to uncover the
first ever directly-detected gravitational wave signal, GW150914!
The above filtering is (almost) the same as what was applied to LIGO data
to produce Figure 1 in
<a class="reference external" href="https://doi.org/10.1103/PhysRevLett.116.061102">Abbott et al. (2016)</a>
(the joint LSC-Virgo publication announcing this detection).</p>
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