.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/communications/detection/plot_02_cahn_quadrature_qpsk.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_detection_plot_02_cahn_quadrature_qpsk.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_detection_plot_02_cahn_quadrature_qpsk.py:


Quadrature modulation: QPSK doubles the rate at the same error rate
===================================================================

Charles Cahn's 1960 analysis of combined phase and amplitude modulation
formalised what quadrature carriers make possible. Cosine and sine carriers
at the same frequency are orthogonal, so each can carry an independent BPSK
stream. Gray-labelled QPSK (:func:`~engineeringkit.communications.qpsk`) sends
two bits per symbol in the bandwidth of one. Its bit-error rate per unit
:math:`E_b/N_0` equals that of BPSK
(:func:`~engineeringkit.communications.ber_qpsk_awgn`).

.. GENERATED FROM PYTHON SOURCE LINES 15-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.communications import ber_qpsk_awgn, qpsk
    from engineeringkit.communications.visualizers import plot_constellation








.. GENERATED FROM PYTHON SOURCE LINES 22-24

A noisy QPSK constellation
--------------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-42

.. code-block:: Python


    rng = np.random.default_rng(1960)
    ebn0 = 10 ** (6 / 10)
    bits = rng.integers(0, 2, 400_000)
    symbols = qpsk(bits)
    es_n0 = 2 * ebn0
    noise = (rng.normal(size=symbols.size) + 1j * rng.normal(size=symbols.size)) * np.sqrt(1 / (2 * es_n0))
    received = symbols + noise
    decided = np.column_stack((received.real > 0, received.imag > 0)).astype(int).reshape(-1)
    ber = np.mean(decided != bits)
    print(f"Eb/N0 = 6 dB: simulated QPSK BER {ber:.2e}, theory {float(ber_qpsk_awgn(ebn0)):.2e}")
    assert np.isclose(ber, ber_qpsk_awgn(ebn0), rtol=0.1)

    fig, axes = plt.subplots(1, 2, figsize=(9, 4))
    plot_constellation(received[:3000], ax=axes[0], label="received")
    plot_constellation(qpsk([0, 0, 0, 1, 1, 0, 1, 1]), ax=axes[0], label="ideal")
    axes[0].set_title("QPSK at Eb/N0 = 6 dB")




.. image-sg:: /api/gallery/communications/detection/images/sphx_glr_plot_02_cahn_quadrature_qpsk_001.png
   :alt: QPSK at Eb/N0 = 6 dB
   :srcset: /api/gallery/communications/detection/images/sphx_glr_plot_02_cahn_quadrature_qpsk_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Eb/N0 = 6 dB: simulated QPSK BER 2.41e-03, theory 2.39e-03

    Text(0.5, 1.0, 'QPSK at Eb/N0 = 6 dB')



.. GENERATED FROM PYTHON SOURCE LINES 43-45

Two bits per symbol in one bandwidth
------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 45-56

.. code-block:: Python


    t = np.linspace(0, 4, 800, endpoint=False)
    i_rail, q_rail = np.repeat(symbols[:4].real, 200), np.repeat(symbols[:4].imag, 200)
    passband = i_rail * np.cos(2 * np.pi * 3 * t) - q_rail * np.sin(2 * np.pi * 3 * t)
    recovered_i = 2 * np.mean((passband * np.cos(2 * np.pi * 3 * t)).reshape(4, -1), axis=1)
    assert np.allclose(recovered_i, symbols[:4].real)
    axes[1].plot(t, passband)
    axes[1].set(xlabel="time [symbols]", title="Quadrature carriers share one band")
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.192 seconds)


.. _sphx_glr_download_api_gallery_communications_detection_plot_02_cahn_quadrature_qpsk.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/detection/plot_02_cahn_quadrature_qpsk.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_cahn_quadrature_qpsk.ipynb <plot_02_cahn_quadrature_qpsk.ipynb>`

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

      :download:`Download Python source code: plot_02_cahn_quadrature_qpsk.py <plot_02_cahn_quadrature_qpsk.py>`

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

      :download:`Download zipped: plot_02_cahn_quadrature_qpsk.zip <plot_02_cahn_quadrature_qpsk.zip>`


.. only:: html

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

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