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<div class="section" id="continuous-wavelet-transform-cwt">
<span id="ref-cwt"></span><h1>Continuous Wavelet Transform (CWT)<a class="headerlink" href="#continuous-wavelet-transform-cwt" title="Permalink to this headline">¶</a></h1>
<p>This section describes functions used to perform single continuous wavelet
transforms.</p>
<div class="section" id="single-level-cwt">
<h2>Single level - <code class="docutils literal notranslate"><span class="pre">cwt</span></code><a class="headerlink" href="#single-level-cwt" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="pywt.cwt">
<code class="descclassname">pywt.</code><code class="descname">cwt</code><span class="sig-paren">(</span><em>data</em>, <em>scales</em>, <em>wavelet</em><span class="sig-paren">)</span><a class="headerlink" href="#pywt.cwt" title="Permalink to this definition">¶</a></dt>
<dd><p>One dimensional Continuous Wavelet Transform.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><dl class="first docutils">
<dt><strong>data</strong> <span class="classifier-delimiter">:</span> <span class="classifier">array_like</span></dt>
<dd><p class="first last">Input signal</p>
</dd>
<dt><strong>scales</strong> <span class="classifier-delimiter">:</span> <span class="classifier">array_like</span></dt>
<dd><p class="first last">The wavelet scales to use. One can use
<code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">=</span> <span class="pre">scale2frequency(scale,</span> <span class="pre">wavelet)/sampling_period</span></code> to determine
what physical frequency, <code class="docutils literal notranslate"><span class="pre">f</span></code>. Here, <code class="docutils literal notranslate"><span class="pre">f</span></code> is in hertz when the
<code class="docutils literal notranslate"><span class="pre">sampling_period</span></code> is given in seconds.</p>
</dd>
<dt><strong>wavelet</strong> <span class="classifier-delimiter">:</span> <span class="classifier">Wavelet object or name</span></dt>
<dd><p class="first last">Wavelet to use</p>
</dd>
<dt><strong>sampling_period</strong> <span class="classifier-delimiter">:</span> <span class="classifier">float</span></dt>
<dd><p class="first last">Sampling period for the frequencies output (optional).
The values computed for <code class="docutils literal notranslate"><span class="pre">coefs</span></code> are independent of the choice of
<code class="docutils literal notranslate"><span class="pre">sampling_period</span></code> (i.e. <code class="docutils literal notranslate"><span class="pre">scales</span></code> is not scaled by the sampling
period).</p>
</dd>
</dl>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><dl class="first last docutils">
<dt><strong>coefs</strong> <span class="classifier-delimiter">:</span> <span class="classifier">array_like</span></dt>
<dd><p class="first last">Continuous wavelet transform of the input signal for the given scales
and wavelet</p>
</dd>
<dt><strong>frequencies</strong> <span class="classifier-delimiter">:</span> <span class="classifier">array_like</span></dt>
<dd><p class="first last">If the unit of sampling period are seconds and given, than frequencies
are in hertz. Otherwise, a sampling period of 1 is assumed.</p>
</dd>
</dl>
</td>
</tr>
</tbody>
</table>
<p class="rubric">Notes</p>
<p>Size of coefficients arrays depends on the length of the input array and
the length of given scales.</p>
<p class="rubric">Examples</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">pywt</span>
<span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">512</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="mi">32</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">coef</span><span class="p">,</span> <span class="n">freqs</span><span class="o">=</span><span class="n">pywt</span><span class="o">.</span><span class="n">cwt</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">129</span><span class="p">),</span><span class="s1">'gaus1'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">matshow</span><span class="p">(</span><span class="n">coef</span><span class="p">)</span> <span class="c1"># doctest: +SKIP</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span> <span class="c1"># doctest: +SKIP</span>
<span class="go">----------</span>
<span class="gp">>>> </span><span class="kn">import</span> <span class="nn">pywt</span>
<span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="gp">>>> </span><span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">200</span><span class="p">,</span> <span class="n">endpoint</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">sig</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="mi">7</span> <span class="o">*</span> <span class="n">t</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mi">7</span><span class="o">*</span><span class="p">(</span><span class="n">t</span><span class="o">-</span><span class="mf">0.4</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="n">j</span><span class="o">*</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">t</span><span class="o">-</span><span class="mf">0.4</span><span class="p">)))</span>
<span class="gp">>>> </span><span class="n">widths</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">31</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">cwtmatr</span><span class="p">,</span> <span class="n">freqs</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">cwt</span><span class="p">(</span><span class="n">sig</span><span class="p">,</span> <span class="n">widths</span><span class="p">,</span> <span class="s1">'mexh'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">cwtmatr</span><span class="p">,</span> <span class="n">extent</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">31</span><span class="p">],</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'PRGn'</span><span class="p">,</span> <span class="n">aspect</span><span class="o">=</span><span class="s1">'auto'</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">vmax</span><span class="o">=</span><span class="nb">abs</span><span class="p">(</span><span class="n">cwtmatr</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">(),</span> <span class="n">vmin</span><span class="o">=-</span><span class="nb">abs</span><span class="p">(</span><span class="n">cwtmatr</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">())</span> <span class="c1"># doctest: +SKIP</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span> <span class="c1"># doctest: +SKIP</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="continuous-wavelet-families">
<h2>Continuous Wavelet Families<a class="headerlink" href="#continuous-wavelet-families" title="Permalink to this headline">¶</a></h2>
<p>A variety of continuous wavelets have been implemented. A list of the available
wavelet names compatible with <code class="docutils literal notranslate"><span class="pre">cwt</span></code> can be obtained by:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">wavlist</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">wavelist</span><span class="p">(</span><span class="n">kind</span><span class="o">=</span><span class="s1">'continuous'</span><span class="p">)</span>
</pre></div>
</div>
<div class="section" id="mexican-hat-wavelet">
<h3>Mexican Hat Wavelet<a class="headerlink" href="#mexican-hat-wavelet" title="Permalink to this headline">¶</a></h3>
<p>The mexican hat wavelet <code class="docutils literal notranslate"><span class="pre">"mexh"</span></code> is given by:</p>
<div class="math notranslate nohighlight">
\[\psi(t) = \frac{2}{\sqrt{3} \sqrt[4]{\pi}} \exp^{-\frac{t^2}{2}}
\left( 1 - t^2 \right)\]</div>
<p>where the constant out front is a normalization factor so that the wavelet has
unit energy.</p>
</div>
<div class="section" id="morlet-wavelet">
<h3>Morlet Wavelet<a class="headerlink" href="#morlet-wavelet" title="Permalink to this headline">¶</a></h3>
<p>The Morlet wavelet <code class="docutils literal notranslate"><span class="pre">"morl"</span></code> is given by:</p>
<div class="math notranslate nohighlight">
\[\psi(t) = \exp^{-\frac{t^2}{2}} \cos(5t)\]</div>
</div>
<div class="section" id="complex-morlet-wavelets">
<h3>Complex Morlet Wavelets<a class="headerlink" href="#complex-morlet-wavelets" title="Permalink to this headline">¶</a></h3>
<p>The complex Morlet wavelet (<code class="docutils literal notranslate"><span class="pre">"cmorB-C"</span></code> with floating point values B, C) is
given by:</p>
<div class="math notranslate nohighlight">
\[\psi(t) = \frac{1}{\sqrt{\pi B}} \exp^{-\frac{t^2}{B}}
\exp^{\mathrm{j} 2\pi C t}\]</div>
<p>where <span class="math notranslate nohighlight">\(B\)</span> is the bandwidth and <span class="math notranslate nohighlight">\(C\)</span> is the center frequency.</p>
</div>
<div class="section" id="gaussian-derivative-wavelets">
<h3>Gaussian Derivative Wavelets<a class="headerlink" href="#gaussian-derivative-wavelets" title="Permalink to this headline">¶</a></h3>
<p>The Gaussian wavelets (<code class="docutils literal notranslate"><span class="pre">"gausP"</span></code> where P is an integer between 1 and and 8)
correspond to the Pth order derivatives of the function:</p>
<div class="math notranslate nohighlight">
\[\psi(t) = C \exp^{-t^2}\]</div>
<p>where <span class="math notranslate nohighlight">\(C\)</span> is an order-dependent normalization constant.</p>
</div>
<div class="section" id="complex-gaussian-derivative-wavelets">
<h3>Complex Gaussian Derivative Wavelets<a class="headerlink" href="#complex-gaussian-derivative-wavelets" title="Permalink to this headline">¶</a></h3>
<p>The complex Gaussian wavelets (<code class="docutils literal notranslate"><span class="pre">"cgauP"</span></code> where P is an integer between 1 and
8) correspond to the Pth order derivatives of the function:</p>
<div class="math notranslate nohighlight">
\[\psi(t) = C \exp^{-\mathrm{j} t}\exp^{-t^2}\]</div>
<p>where <span class="math notranslate nohighlight">\(C\)</span> is an order-dependent normalization constant.</p>
</div>
<div class="section" id="shannon-wavelets">
<h3>Shannon Wavelets<a class="headerlink" href="#shannon-wavelets" title="Permalink to this headline">¶</a></h3>
<p>The Shannon wavelets (<code class="docutils literal notranslate"><span class="pre">"shanB-C"</span></code> with floating point values B and C)
correspond to the following wavelets:</p>
<div class="math notranslate nohighlight">
\[\psi(t) = \sqrt{B} \frac{\sin(\pi B t)}{\pi B t} \exp^{\mathrm{j}2 \pi C t}\]</div>
<p>where <span class="math notranslate nohighlight">\(B\)</span> is the bandwith and <span class="math notranslate nohighlight">\(C\)</span> is the center frequency.</p>
</div>
<div class="section" id="freuqency-b-spline-wavelets">
<h3>Freuqency B-Spline Wavelets<a class="headerlink" href="#freuqency-b-spline-wavelets" title="Permalink to this headline">¶</a></h3>
<p>The frequency B-spline wavelets (<code class="docutils literal notranslate"><span class="pre">"fpspM-B-C"</span></code> with integer M and floating
point B, C) correspond to the following wavelets:</p>
<div class="math notranslate nohighlight">
\[\psi(t) = \sqrt{B}
\left[\frac{\sin(\pi B \frac{t}{M})}{\pi B \frac{t}{M}}\right]^M
\exp^{2\mathrm{j} \pi C t}\]</div>
<p>where <span class="math notranslate nohighlight">\(M\)</span> is the spline order, <span class="math notranslate nohighlight">\(B\)</span> is the bandwidth and <span class="math notranslate nohighlight">\(C\)</span> is
the center frequency.</p>
</div>
</div>
<div class="section" id="choosing-the-scales-for-cwt">
<h2>Choosing the scales for <code class="docutils literal notranslate"><span class="pre">cwt</span></code><a class="headerlink" href="#choosing-the-scales-for-cwt" title="Permalink to this headline">¶</a></h2>
<p>For each of the wavelets described below, the implementation in PyWavelets
evaluates the wavelet function for <span class="math notranslate nohighlight">\(t\)</span> over the range
<code class="docutils literal notranslate"><span class="pre">[wavelet.lower_bound,</span> <span class="pre">wavelet.upper_bound]</span></code> (with default range
<span class="math notranslate nohighlight">\([-8, 8]\)</span>). <code class="docutils literal notranslate"><span class="pre">scale</span> <span class="pre">=</span> <span class="pre">1</span></code> corresponds to the case where the extent of the
wavelet is <code class="docutils literal notranslate"><span class="pre">(wavelet.upper_bound</span> <span class="pre">-</span> <span class="pre">wavelet.lower_bound</span> <span class="pre">+</span> <span class="pre">1)</span></code> samples of the
digital signal being analyzed. Larger scales correspond to stretching of the
wavelet. For example, at <code class="docutils literal notranslate"><span class="pre">scale=10</span></code> the wavelet is stretched by a factor of
10, making it sensitive to lower frequencies in the signal.</p>
<p>To relate a given scale to a specific signal frequency, the sampling period
of the signal must be known. <a class="reference internal" href="other-functions.html#pywt.scale2frequency" title="pywt.scale2frequency"><code class="xref py py-func docutils literal notranslate"><span class="pre">pywt.scale2frequency()</span></code></a> can be used to
convert a list of scales to their corresponding frequencies. The proper choice
of scales depends on the chosen wavelet, so <a class="reference internal" href="other-functions.html#pywt.scale2frequency" title="pywt.scale2frequency"><code class="xref py py-func docutils literal notranslate"><span class="pre">pywt.scale2frequency()</span></code></a> should
be used to get an idea of an appropriate range for the signal of interest.</p>
<p>For the <code class="docutils literal notranslate"><span class="pre">cmor</span></code>, <code class="docutils literal notranslate"><span class="pre">fbsp</span></code> and <code class="docutils literal notranslate"><span class="pre">shan</span></code> wavelets, the user can specify a
specific a normalized center frequency. A value of 1.0 corresponds to 1/dt
where dt is the sampling period. In other words, when analyzing a signal
sampled at 100 Hz, a center frequency of 1.0 corresponds to ~100 Hz at
<code class="docutils literal notranslate"><span class="pre">scale</span> <span class="pre">=</span> <span class="pre">1</span></code>. This is above the Nyquist rate of 50 Hz, so for this
particular wavelet, one would analyze a signal using <code class="docutils literal notranslate"><span class="pre">scales</span> <span class="pre">>=</span> <span class="pre">2</span></code>.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">import</span> <span class="nn">pywt</span>
<span class="gp">>>> </span><span class="n">dt</span> <span class="o">=</span> <span class="mf">0.01</span> <span class="c1"># 100 Hz sampling</span>
<span class="gp">>>> </span><span class="n">frequencies</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">scale2frequency</span><span class="p">(</span><span class="s1">'cmor1.5-1.0'</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span> <span class="o">/</span> <span class="n">dt</span>
<span class="gp">>>> </span><span class="n">frequencies</span>
<span class="go">array([ 100. , 50. , 33.33333333, 25. ])</span>
</pre></div>
</div>
<p>The CWT in PyWavelets is applied to discrete data by convolution with samples
of the integral of the wavelet. If <code class="docutils literal notranslate"><span class="pre">scale</span></code> is too low, this will result in
a discrete filter that is inadequately sampled leading to aliasing as shown
in the example below. Here the wavelet is <code class="docutils literal notranslate"><span class="pre">'cmor1.5-1.0'</span></code>. The left column of
the figure shows the discrete filters used in the convolution at various
scales. The right column are the corresponding Fourier power spectra of each
filter.. For scales 1 and 2 it can be seen that aliasing due to violation of
the Nyquist limit occurs.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pywt</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
<span class="n">wav</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">ContinuousWavelet</span><span class="p">(</span><span class="s1">'cmor1.5-1.0'</span><span class="p">)</span>
<span class="c1"># print the range over which the wavelet will be evaluated</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"Continuous wavelet will be evaluated over the range [{}, {}]"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
<span class="n">wav</span><span class="o">.</span><span class="n">lower_bound</span><span class="p">,</span> <span class="n">wav</span><span class="o">.</span><span class="n">upper_bound</span><span class="p">))</span>
<span class="n">width</span> <span class="o">=</span> <span class="n">wav</span><span class="o">.</span><span class="n">upper_bound</span> <span class="o">-</span> <span class="n">wav</span><span class="o">.</span><span class="n">lower_bound</span>
<span class="n">scales</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">15</span><span class="p">]</span>
<span class="n">max_len</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">scales</span><span class="p">)</span><span class="o">*</span><span class="n">width</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">max_len</span><span class="p">)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">scales</span><span class="p">),</span> <span class="mi">2</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">6</span><span class="p">))</span>
<span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">scale</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">scales</span><span class="p">):</span>
<span class="c1"># The following code is adapted from the internals of cwt</span>
<span class="n">int_psi</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">integrate_wavelet</span><span class="p">(</span><span class="n">wav</span><span class="p">,</span> <span class="n">precision</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
<span class="n">step</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">j</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">floor</span><span class="p">(</span>
<span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">scale</span> <span class="o">*</span> <span class="n">width</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">scale</span> <span class="o">*</span> <span class="n">step</span><span class="p">))</span>
<span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">j</span><span class="p">)</span> <span class="o">>=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">int_psi</span><span class="p">):</span>
<span class="n">j</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">delete</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">((</span><span class="n">j</span> <span class="o">>=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">int_psi</span><span class="p">)))[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">j</span> <span class="o">=</span> <span class="n">j</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">int</span><span class="p">)</span>
<span class="c1"># normalize int_psi for easier plotting</span>
<span class="n">int_psi</span> <span class="o">/=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">int_psi</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
<span class="c1"># discrete samples of the integrated wavelet</span>
<span class="n">filt</span> <span class="o">=</span> <span class="n">int_psi</span><span class="p">[</span><span class="n">j</span><span class="p">][::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="c1"># The CWT consists of convolution of filt with the signal at this scale</span>
<span class="c1"># Here we plot this discrete convolution kernel at each scale.</span>
<span class="n">nt</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">filt</span><span class="p">)</span>
<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="n">nt</span><span class="o">//</span><span class="mi">2</span><span class="p">,</span> <span class="n">nt</span><span class="o">//</span><span class="mi">2</span><span class="p">,</span> <span class="n">nt</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">filt</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">filt</span><span class="o">.</span><span class="n">imag</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="o">-</span><span class="n">max_len</span><span class="o">//</span><span class="mi">2</span><span class="p">,</span> <span class="n">max_len</span><span class="o">//</span><span class="mi">2</span><span class="p">])</span>
<span class="n">axes</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">axes</span><span class="p">[</span><span class="n">n</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="mi">50</span><span class="p">,</span> <span class="mf">0.35</span><span class="p">,</span> <span class="s1">'scale = {}'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">scale</span><span class="p">))</span>
<span class="n">f</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">max_len</span><span class="p">)</span>
<span class="n">filt_fft</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fftshift</span><span class="p">(</span><span class="n">np</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">filt</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">max_len</span><span class="p">))</span>
<span class="n">filt_fft</span> <span class="o">/=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">filt_fft</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
<span class="n">axes</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">filt_fft</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">])</span>
<span class="n">axes</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">axes</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">])</span>
<span class="n">axes</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="sa">r</span><span class="s1">'$-\pi$'</span><span class="p">,</span> <span class="s1">'0'</span><span class="p">,</span> <span class="sa">r</span><span class="s1">'$\pi$'</span><span class="p">])</span>
<span class="n">axes</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="bp">True</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="s1">'x'</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="n">n</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="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="s1">'scale = {}'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">scale</span><span class="p">))</span>
<span class="n">axes</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'time (samples)'</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'frequency (radians)'</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'real'</span><span class="p">,</span> <span class="s1">'imaginary'</span><span class="p">],</span> <span class="n">loc</span><span class="o">=</span><span class="s1">'upper left'</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'Power'</span><span class="p">],</span> <span class="n">loc</span><span class="o">=</span><span class="s1">'upper left'</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</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">'filter'</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="sa">r</span><span class="s1">'|FFT(filter)|$^2$'</span><span class="p">)</span>
</pre></div>
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<li><a class="reference internal" href="#">Continuous Wavelet Transform (CWT)</a><ul>
<li><a class="reference internal" href="#single-level-cwt">Single level - <code class="docutils literal notranslate"><span class="pre">cwt</span></code></a></li>
<li><a class="reference internal" href="#continuous-wavelet-families">Continuous Wavelet Families</a><ul>
<li><a class="reference internal" href="#mexican-hat-wavelet">Mexican Hat Wavelet</a></li>
<li><a class="reference internal" href="#morlet-wavelet">Morlet Wavelet</a></li>
<li><a class="reference internal" href="#complex-morlet-wavelets">Complex Morlet Wavelets</a></li>
<li><a class="reference internal" href="#gaussian-derivative-wavelets">Gaussian Derivative Wavelets</a></li>
<li><a class="reference internal" href="#complex-gaussian-derivative-wavelets">Complex Gaussian Derivative Wavelets</a></li>
<li><a class="reference internal" href="#shannon-wavelets">Shannon Wavelets</a></li>
<li><a class="reference internal" href="#freuqency-b-spline-wavelets">Freuqency B-Spline Wavelets</a></li>
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<li><a class="reference internal" href="#choosing-the-scales-for-cwt">Choosing the scales for <code class="docutils literal notranslate"><span class="pre">cwt</span></code></a></li>
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