.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/communications/modulation/plot_01_fessenden_amplitude_modulation.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_communications_modulation_plot_01_fessenden_amplitude_modulation.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_communications_modulation_plot_01_fessenden_amplitude_modulation.py:


Fessenden's amplitude modulation
================================

On Christmas Eve 1906 Reginald Fessenden is said to have broadcast voice and
music from Brant Rock, Massachusetts, by varying the amplitude of a
continuous high-frequency wave. Conventional AM,
:math:`s(t) = A_c[1+\mu m(t)]\cos 2\pi f_c t`
(:func:`~engineeringkit.communications.am_signal`), keeps the carrier so a
simple envelope detector can recover the message, provided the modulation
index stays at or below one. The spectrum shows the cost: a strong carrier
and two sidebands, so at most one third of the power carries the message.

.. GENERATED FROM PYTHON SOURCE LINES 16-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.signal import hilbert

    from engineeringkit.communications import am_signal








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Envelope detection works only without overmodulation
----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-43

.. code-block:: Python


    fs, fc, fm = 200e3, 10e3, 500.0
    t = np.arange(0, 0.01, 1 / fs)
    freq = np.fft.rfftfreq(t.size, 1 / fs)
    message = np.cos(2 * np.pi * fm * t)
    fig, axes = plt.subplots(2, 2, figsize=(10, 5.5))
    for col, mu in enumerate([0.7, 1.5]):
        s = am_signal(message, t, fc, mu=mu)
        envelope = np.abs(hilbert(s))
        error = np.max(np.abs(envelope[200:-200] - 1 - mu * message[200:-200]))
        print(f"mu = {mu}: max envelope-detector error = {error:.3f}")
        axes[0, col].plot(t * 1e3, s, lw=0.5)
        axes[0, col].plot(t * 1e3, envelope, "C3")
        axes[0, col].set(title=f"modulation index {mu}", xlabel="time [ms]")
        axes[1, col].stem(freq / 1e3, np.abs(np.fft.rfft(s)) / t.size, markerfmt=" ", basefmt=" ")
        axes[1, col].set(xlim=(8, 12), xlabel="frequency [kHz]")
    fig.tight_layout()




.. image-sg:: /api/gallery/communications/modulation/images/sphx_glr_plot_01_fessenden_amplitude_modulation_001.png
   :alt: modulation index 0.7, modulation index 1.5
   :srcset: /api/gallery/communications/modulation/images/sphx_glr_plot_01_fessenden_amplitude_modulation_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    mu = 0.7: max envelope-detector error = 0.000
    mu = 1.5: max envelope-detector error = 1.000




.. GENERATED FROM PYTHON SOURCE LINES 44-49

Carrier and sidebands
---------------------

For a tone, sideband power is :math:`\mu^2/(2+\mu^2)` of the total, at most
one third.

.. GENERATED FROM PYTHON SOURCE LINES 49-58

.. code-block:: Python


    mu = 1.0
    power = np.abs(np.fft.rfft(am_signal(message, t, fc, mu=mu))) ** 2
    carrier = power[np.argmin(np.abs(freq - fc))]
    efficiency = 1 - carrier / power.sum()
    print(f"mu = 1: sideband share of power = {efficiency:.3f} (theory 1/3)")
    assert np.isclose(efficiency, mu**2 / (2 + mu**2), atol=1e-3)

    plt.show()




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

 .. code-block:: none

    mu = 1: sideband share of power = 0.333 (theory 1/3)





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

   **Total running time of the script:** (0 minutes 0.331 seconds)


.. _sphx_glr_download_api_gallery_communications_modulation_plot_01_fessenden_amplitude_modulation.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/communications/modulation/plot_01_fessenden_amplitude_modulation.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_fessenden_amplitude_modulation.ipynb <plot_01_fessenden_amplitude_modulation.ipynb>`

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

      :download:`Download Python source code: plot_01_fessenden_amplitude_modulation.py <plot_01_fessenden_amplitude_modulation.py>`

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

      :download:`Download zipped: plot_01_fessenden_amplitude_modulation.zip <plot_01_fessenden_amplitude_modulation.zip>`


.. only:: html

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

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