.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/communications/modulation/plot_02_carson_suppressed_carrier.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_02_carson_suppressed_carrier.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_02_carson_suppressed_carrier.py:


Carson's suppressed-carrier modulation
======================================

In 1915 John Renshaw Carson of AT&T showed that the carrier in AM carries no
information. Multiplying the message by the carrier alone,
:math:`s(t) = m(t)\cos 2\pi f_c t`
(:func:`~engineeringkit.communications.dsb_sc`), puts all the transmitted
power into the sidebands. The receiver must supply a coherent carrier: mix,
low-pass filter, and the message returns, scaled by :math:`\tfrac12\cos\phi`
for a phase error :math:`\phi`. Carson also showed that one sideband is
enough, the basis of single-sideband telephony.

.. GENERATED FROM PYTHON SOURCE LINES 16-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.signal import butter, sosfiltfilt

    from engineeringkit.communications import dsb_sc








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Coherent demodulation and phase error
-------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-43

.. code-block:: Python


    fs, fc = 200e3, 10e3
    t = np.arange(0, 0.02, 1 / fs)
    message = np.cos(2 * np.pi * 300 * t) + 0.5 * np.sin(2 * np.pi * 700 * t)
    s = dsb_sc(message, t, fc)
    lowpass = butter(6, 2e3, fs=fs, output="sos")
    phases = np.radians([0, 45, 90])
    fig, axes = plt.subplots(1, 2, figsize=(10, 3.6))
    for phi in phases:
        recovered = 2 * sosfiltfilt(lowpass, s * np.cos(2 * np.pi * fc * t + phi))
        gain = np.dot(recovered, message) / np.dot(message, message)
        print(f"phase error {np.degrees(phi):3.0f} deg: recovered gain {gain:+.3f} (cos phi = {np.cos(phi):+.3f})")
        assert np.isclose(gain, np.cos(phi), atol=0.01)
        axes[0].plot(t * 1e3, recovered, label=f"φ = {np.degrees(phi):.0f}°")
    axes[0].plot(t * 1e3, message, "k:", label="message")
    axes[0].set(xlim=(0, 10), xlabel="time [ms]", title="Coherent detection")
    axes[0].legend(fontsize=8)




.. image-sg:: /api/gallery/communications/modulation/images/sphx_glr_plot_02_carson_suppressed_carrier_001.png
   :alt: Coherent detection
   :srcset: /api/gallery/communications/modulation/images/sphx_glr_plot_02_carson_suppressed_carrier_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    phase error   0 deg: recovered gain +1.001 (cos phi = +1.000)
    phase error  45 deg: recovered gain +0.708 (cos phi = +0.707)
    phase error  90 deg: recovered gain -0.000 (cos phi = +0.000)

    <matplotlib.legend.Legend object at 0x7f0af4f6b5c0>



.. GENERATED FROM PYTHON SOURCE LINES 44-46

No carrier line in the spectrum
-------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 46-55

.. code-block:: Python


    freq = np.fft.rfftfreq(t.size, 1 / fs)
    axes[1].plot(freq / 1e3, np.abs(np.fft.rfft(s)) / t.size)
    axes[1].set(xlim=(8.5, 11.5), xlabel="frequency [kHz]", title="DSB-SC spectrum: sidebands only")
    carrier_bin = np.argmin(np.abs(freq - fc))
    assert np.abs(np.fft.rfft(s))[carrier_bin] / t.size < 1e-10
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.166 seconds)


.. _sphx_glr_download_api_gallery_communications_modulation_plot_02_carson_suppressed_carrier.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_02_carson_suppressed_carrier.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_carson_suppressed_carrier.ipynb <plot_02_carson_suppressed_carrier.ipynb>`

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

      :download:`Download Python source code: plot_02_carson_suppressed_carrier.py <plot_02_carson_suppressed_carrier.py>`

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

      :download:`Download zipped: plot_02_carson_suppressed_carrier.zip <plot_02_carson_suppressed_carrier.zip>`


.. only:: html

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

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