.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "gallery_1d/plot_synthetic.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        Click :ref:`here <sphx_glr_download_gallery_1d_plot_synthetic.py>`
        to download the full example code

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

.. _sphx_glr_gallery_1d_plot_synthetic.py:


Compute the scattering transform of a synthetic signal
======================================================
In this example we generate a harmonic signal of a few different frequencies
and analyze it with the 1D scattering transform.

.. GENERATED FROM PYTHON SOURCE LINES 10-12

Import the necessary packages
-----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 12-17

.. code-block:: default

    from kymatio.numpy import Scattering1D
    import matplotlib.pyplot as plt
    import numpy as np









.. GENERATED FROM PYTHON SOURCE LINES 18-28

Write a function that can generate a harmonic signal
----------------------------------------------------
Let's write a function that can generate some simple blip-type sounds with
decaying harmonics. It will take four arguments: T, the length of the output
vector; num_intervals, the number of different blips; gamma, the exponential
decay factor of the harmonic; random_state, a random seed to generate
random pitches and phase shifts.
The function proceeds by splitting the time length T into intervals, chooses
base frequencies and phases, generates sinusoidal sounds and harmonics,
and then adds a windowed version to the output signal.

.. GENERATED FROM PYTHON SOURCE LINES 28-54

.. code-block:: default

    def generate_harmonic_signal(T, num_intervals=4, gamma=0.9, random_state=42):
        """
        Generates a harmonic signal, which is made of piecewise constant notes
        (of random fundamental frequency), with half overlap
        """
        rng = np.random.RandomState(random_state)
        num_notes = 2 * (num_intervals - 1) + 1
        support = T // num_intervals
        half_support = support // 2

        base_freq = 0.1 * rng.rand(num_notes) + 0.05
        phase = 2 * np.pi * rng.rand(num_notes)
        window = np.hanning(support)
        x = np.zeros(T, dtype='float32')
        t = np.arange(0, support)
        u = 2 * np.pi * t
        for i in range(num_notes):
            ind_start = i * half_support
            note = np.zeros(support)
            for k in range(1):
                note += (np.power(gamma, k) *
                         np.cos(u * (k + 1) * base_freq[i] + phase[i]))
            x[ind_start:ind_start + support] += note * window

        return x








.. GENERATED FROM PYTHON SOURCE LINES 55-57

Let's take a look at what such a signal could look like
-------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 57-63

.. code-block:: default

    T = 2 ** 13
    x = generate_harmonic_signal(T)
    plt.figure(figsize=(8, 2))
    plt.plot(x)
    plt.title("Original signal")




.. image-sg:: /gallery_1d/images/sphx_glr_plot_synthetic_001.png
   :alt: Original signal
   :srcset: /gallery_1d/images/sphx_glr_plot_synthetic_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Original signal')



.. GENERATED FROM PYTHON SOURCE LINES 64-67

Spectrogram
-----------
Let's take a look at the signal spectrogram

.. GENERATED FROM PYTHON SOURCE LINES 67-71

.. code-block:: default

    plt.figure(figsize=(8, 4))
    plt.specgram(x, Fs=1024)
    plt.title("Time-Frequency spectrogram of signal")




.. image-sg:: /gallery_1d/images/sphx_glr_plot_synthetic_002.png
   :alt: Time-Frequency spectrogram of signal
   :srcset: /gallery_1d/images/sphx_glr_plot_synthetic_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Time-Frequency spectrogram of signal')



.. GENERATED FROM PYTHON SOURCE LINES 72-74

Doing the scattering transform
------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 74-98

.. code-block:: default

    J = 6
    Q = 16

    scattering = Scattering1D(J, T, Q)

    meta = scattering.meta()
    order0 = np.where(meta['order'] == 0)
    order1 = np.where(meta['order'] == 1)
    order2 = np.where(meta['order'] == 2)

    Sx = scattering(x)

    plt.figure(figsize=(8, 8))
    plt.subplot(3, 1, 1)
    plt.plot(Sx[order0][0])
    plt.title('Zeroth-order scattering')
    plt.subplot(3, 1, 2)
    plt.imshow(Sx[order1], aspect='auto')
    plt.title('First-order scattering')
    plt.subplot(3, 1, 3)
    plt.imshow(Sx[order2], aspect='auto')
    plt.title('Second-order scattering')
    plt.tight_layout()
    plt.show()



.. image-sg:: /gallery_1d/images/sphx_glr_plot_synthetic_003.png
   :alt: Zeroth-order scattering, First-order scattering, Second-order scattering
   :srcset: /gallery_1d/images/sphx_glr_plot_synthetic_003.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** ( 0 minutes  0.548 seconds)


.. _sphx_glr_download_gallery_1d_plot_synthetic.py:

.. only:: html

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


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

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

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

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


.. only:: html

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

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