.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/communications/detection/plot_05_campopiano_glazer_qam.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_05_campopiano_glazer_qam.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_05_campopiano_glazer_qam.py:


Campopiano and Glazer's square QAM (1962)
=========================================

Phase-shift keying wastes the inside of the circle. C. N. Campopiano and
B. G. Glazer modulated the amplitudes of two carriers in quadrature
independently, so the points fill a square grid
(:func:`~engineeringkit.communications.qam_constellation`). Each rail is a
:math:`\sqrt M`-level amplitude decision, and the exact symbol-error rate
follows from one rail's error probability
(:func:`~engineeringkit.communications.ser_mqam_awgn`). At 16 points and
above, square QAM needs several decibels less power than M-PSK, which is
why every modem from the 1970s to Wi-Fi and 5G uses it.

.. GENERATED FROM PYTHON SOURCE LINES 17-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.communications import from_db, minimum_distance_detect, qam_constellation, ser_mpsk_awgn, ser_mqam_awgn








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Constellation and simulated error rate
--------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-56

.. code-block:: Python


    rng = np.random.default_rng(1962)
    fig, axes = plt.subplots(1, 2, figsize=(11, 4.2))
    points = qam_constellation(16)
    axes[0].plot(points.real, points.imag, "o")
    for label, point in enumerate(points):
        axes[0].annotate(f"{label:04b}", (point.real, point.imag), xytext=(0, 9), textcoords="offset points", ha="center", fontsize=8)
    axes[0].set(aspect="equal", title="16-QAM, Gray labelled", xlabel="in-phase", ylabel="quadrature")
    assert np.isclose(np.mean(np.abs(points) ** 2), 1)

    ebn0_db = np.arange(0, 19, 2.0)
    for M in (4, 16, 64):
        esn0 = from_db(ebn0_db) * np.log2(M)
        theory = ser_mqam_awgn(esn0, M)
        axes[1].semilogy(ebn0_db, theory, label=f"{M}-QAM")
        pts = qam_constellation(M)
        n = 40_000
        labels = rng.integers(0, M, n)
        measured = []
        for g in esn0:
            noise = (rng.standard_normal(n) + 1j * rng.standard_normal(n)) * np.sqrt(1 / (2 * g))
            measured.append(np.mean(minimum_distance_detect(pts[labels] + noise, pts) != labels))
        measured = np.array(measured)
        axes[1].semilogy(ebn0_db[measured > 0], measured[measured > 0], "k.", ms=4)
        assert np.allclose(measured[theory > 1e-2], theory[theory > 1e-2], rtol=0.1)
    axes[1].semilogy(ebn0_db, ser_mpsk_awgn(from_db(ebn0_db) * 4, 16), "C1--", label="16-PSK")
    axes[1].set(xlabel="$E_b/N_0$ [dB]", ylabel="symbol-error rate", ylim=(1e-5, 1), title="QAM in AWGN (dots: simulation)")
    axes[1].grid(True, which="both", alpha=0.3)
    axes[1].legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/communications/detection/images/sphx_glr_plot_05_campopiano_glazer_qam_001.png
   :alt: 16-QAM, Gray labelled, QAM in AWGN (dots: simulation)
   :srcset: /api/gallery/communications/detection/images/sphx_glr_plot_05_campopiano_glazer_qam_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 57-62

QAM against PSK at 16 points
----------------------------

The minimum distance of unit-energy 16-QAM is :math:`2\sqrt{1/10} = 0.632`;
that of 16-PSK is :math:`2\sin(\pi/16) = 0.390`, a ratio of 4.2 dB.

.. GENERATED FROM PYTHON SOURCE LINES 62-72

.. code-block:: Python


    gap = 20 * np.log10(2 * np.sqrt(0.1) / (2 * np.sin(np.pi / 16)))
    target = 1e-4
    grid = np.linspace(10, 30, 2001)
    need_qam = grid[np.argmax(ser_mqam_awgn(from_db(grid), 16) < target)]
    need_psk = grid[np.argmax(ser_mpsk_awgn(from_db(grid), 16) < target)]
    print(f"distance gap {gap:.2f} dB; Es/N0 for SER 1e-4: 16-QAM {need_qam:.2f} dB, 16-PSK {need_psk:.2f} dB")
    assert abs((need_psk - need_qam) - gap) < 0.3

    plt.show()




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

 .. code-block:: none

    distance gap 4.20 dB; Es/N0 for SER 1e-4: 16-QAM 19.01 dB, 16-PSK 22.99 dB





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

   **Total running time of the script:** (0 minutes 0.601 seconds)


.. _sphx_glr_download_api_gallery_communications_detection_plot_05_campopiano_glazer_qam.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_05_campopiano_glazer_qam.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_campopiano_glazer_qam.ipynb <plot_05_campopiano_glazer_qam.ipynb>`

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

      :download:`Download Python source code: plot_05_campopiano_glazer_qam.py <plot_05_campopiano_glazer_qam.py>`

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

      :download:`Download zipped: plot_05_campopiano_glazer_qam.zip <plot_05_campopiano_glazer_qam.zip>`


.. only:: html

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

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