.. _sphx_glr_api_gallery_special_functions_wavelets:

Wavelets
--------

The Haar and Daubechies wavelets, and the Morlet continuous wavelet transform.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Haar&#x27;s 1910 orthonormal system of step functions splits a signal into a coarse average plus details at each scale: pairwise sums and differences over sqrt 2, repeated on the sums. A jump shows up as a single large coefficient per level, exactly where it happens.">

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  .. image:: /api/gallery/special_functions/wavelets/images/thumb/sphx_glr_plot_01_haar_wavelet_thumb.png
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  :doc:`/api/gallery/special_functions/wavelets/plot_01_haar_wavelet`

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      <div class="sphx-glr-thumbnail-title">Haar's wavelet: averages and differences at every scale</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Morlet, analysing seismic echoes, correlated the signal with shifted and dilated copies of a Gaussian-windowed complex exponential, and with Grossmann made the transform rigorous in 1984. The scalogram |W(s, t)|^2 shows which frequencies are present when, which the Fourier transform alone cannot.">

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  .. image:: /api/gallery/special_functions/wavelets/images/thumb/sphx_glr_plot_02_morlet_scalogram_thumb.png
    :alt:

  :doc:`/api/gallery/special_functions/wavelets/plot_02_morlet_scalogram`

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      <div class="sphx-glr-thumbnail-title">Grossmann and Morlet's continuous wavelet transform</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Daubechies&#x27; 1988 compactly supported orthonormal wavelets have p vanishing moments with a filter of only 2p taps, so polynomial pieces of degree below p produce near-zero detail coefficients. Keeping only the largest coefficients then compresses a smooth signal far better than Haar (p = 1) can.">

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  .. image:: /api/gallery/special_functions/wavelets/images/thumb/sphx_glr_plot_03_daubechies_compression_thumb.png
    :alt:

  :doc:`/api/gallery/special_functions/wavelets/plot_03_daubechies_compression`

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      <div class="sphx-glr-thumbnail-title">Daubechies wavelets: vanishing moments and compression</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/special_functions/wavelets/plot_01_haar_wavelet
   /api/gallery/special_functions/wavelets/plot_02_morlet_scalogram
   /api/gallery/special_functions/wavelets/plot_03_daubechies_compression