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

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_examples_filtering.py>`
        to download the full example code.

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

.. _sphx_glr_examples_filtering.py:


Filtering a signal
==================

A graph signal is filtered by transforming it to the spectral domain (via the
Fourier transform), performing a point-wise multiplication (motivated by the
convolution theorem), and transforming it back to the vertex domain (via the
inverse graph Fourier transform).

.. note::

    In practice, filtering is implemented in the vertex domain to avoid the
    computationally expensive graph Fourier transform. To do so, filters are
    implemented as polynomials of the eigenvalues / Laplacian. Hence, filtering
    a signal reduces to its multiplications with sparse matrices (the graph
    Laplacian).

.. GENERATED FROM PYTHON SOURCE LINES 19-68



.. image-sg:: /examples/images/sphx_glr_filtering_001.png
   :alt: input signal $x$ in the vertex domain, signals in the spectral domain, filtered signal $y$ in the vertex domain
   :srcset: /examples/images/sphx_glr_filtering_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    import numpy as np
    from matplotlib import pyplot as plt

    import pygsp as pg

    G = pg.graphs.Sensor(seed=42)
    G.compute_fourier_basis()

    # g = pg.filters.Rectangular(G, band_max=0.2)
    g = pg.filters.Expwin(G, band_max=0.5)

    fig, axes = plt.subplots(1, 3, figsize=(12, 4))
    fig.subplots_adjust(hspace=0.5)

    x = np.random.default_rng(1).normal(size=G.N)
    # x = np.random.default_rng(42).uniform(-1, 1, size=G.N)
    x = 3 * x / np.linalg.norm(x)
    y = g.filter(x)
    x_hat = G.gft(x).squeeze()
    y_hat = G.gft(y).squeeze()

    limits = [x.min(), x.max()]

    G.plot(x, limits=limits, ax=axes[0], title="input signal $x$ in the vertex domain")
    axes[0].text(0, -0.1, f"$x^T L x = {G.dirichlet_energy(x):.2f}$")
    axes[0].set_axis_off()

    g.plot(ax=axes[1], alpha=1)
    line_filt = axes[1].lines[-2]
    (line_in,) = axes[1].plot(G.e, np.abs(x_hat), ".-")
    (line_out,) = axes[1].plot(G.e, np.abs(y_hat), ".-")
    # axes[1].set_xticks(range(0, 16, 4))
    axes[1].set_xlabel(r"graph frequency $\lambda$")
    axes[1].set_ylabel(r"frequency content $\hat{x}(\lambda)$")
    axes[1].set_title(r"signals in the spectral domain")
    axes[1].legend([r"input signal $\hat{x}$"])
    labels = [
        r"input signal $\hat{x}$",
        "kernel $g$",
        r"filtered signal $\hat{y}$",
    ]
    axes[1].legend([line_in, line_filt, line_out], labels, loc="upper right")

    G.plot(y, limits=limits, ax=axes[2], title="filtered signal $y$ in the vertex domain")
    axes[2].text(0, -0.1, f"$y^T L y = {G.dirichlet_energy(y):.2f}$")
    axes[2].set_axis_off()

    fig.tight_layout()


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

   **Total running time of the script:** (0 minutes 0.419 seconds)

**Estimated memory usage:**  160 MB


.. _sphx_glr_download_examples_filtering.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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