.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/special_functions/wavelets/plot_02_morlet_scalogram.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_special_functions_wavelets_plot_02_morlet_scalogram.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_special_functions_wavelets_plot_02_morlet_scalogram.py:


Grossmann and Morlet's continuous wavelet transform
===================================================

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
:math:`|W(s, t)|^2` shows which frequencies are present *when*, which the
Fourier transform alone cannot.

.. GENERATED FROM PYTHON SOURCE LINES 13-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.special_functions import morlet_cwt
    from mathematicskit.special_functions.visualizers.plots import plot_scalogram








.. GENERATED FROM PYTHON SOURCE LINES 20-22

A chirp plus a short burst
--------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-40

.. code-block:: Python


    dt = 0.002
    t = np.arange(0.0, 4.0, dt)
    chirp = np.sin(2 * np.pi * (5.0 * t + 5.0 * t**2))  # instantaneous frequency 5 + 10 t Hz
    burst = np.exp(-(((t - 3.0) / 0.05) ** 2)) * np.sin(2 * np.pi * 80.0 * t)
    x = chirp + burst

    scales = np.geomspace(0.005, 0.4, 120)
    result = morlet_cwt(x, scales, dt=dt)

    fig, (ax0, ax1) = plt.subplots(2, 1, figsize=(8, 7), sharex=True)
    ax0.plot(t, x, lw=0.6)
    ax0.set_title("Chirp (5 -> 45 Hz) and an 80 Hz burst at t = 3 s")
    plot_scalogram(result, ax=ax1)
    ax1.plot(t, 5.0 + 10.0 * t, "w--", lw=1, label="chirp frequency 5 + 10t")
    ax1.legend(loc="upper left")
    fig.tight_layout()




.. image-sg:: /api/gallery/special_functions/wavelets/images/sphx_glr_plot_02_morlet_scalogram_001.png
   :alt: Chirp (5 -> 45 Hz) and an 80 Hz burst at t = 3 s, Morlet scalogram
   :srcset: /api/gallery/special_functions/wavelets/images/sphx_glr_plot_02_morlet_scalogram_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 41-43

Ridge of the scalogram tracks the chirp
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 43-50

.. code-block:: Python


    power = np.abs(result.coefficients) ** 2
    chirp_band = result.frequencies < 60.0
    for time in (0.5, 1.5, 2.5):
        col = int(time / dt)
        ridge = result.frequencies[chirp_band][np.argmax(power[chirp_band, col])]
        print(f"t = {time}: ridge at {ridge:5.1f} Hz (true {5 + 10 * time:.1f} Hz)")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    t = 0.5: ridge at  10.2 Hz (true 10.0 Hz)
    t = 1.5: ridge at  19.7 Hz (true 20.0 Hz)
    t = 2.5: ridge at  29.6 Hz (true 30.0 Hz)





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.350 seconds)


.. _sphx_glr_download_api_gallery_special_functions_wavelets_plot_02_morlet_scalogram.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/special_functions/wavelets/plot_02_morlet_scalogram.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_morlet_scalogram.ipynb <plot_02_morlet_scalogram.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_morlet_scalogram.py <plot_02_morlet_scalogram.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_morlet_scalogram.zip <plot_02_morlet_scalogram.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_