.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/foundations/plot_02_ber_qam_awgn.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_foundations_plot_02_ber_qam_awgn.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_foundations_plot_02_ber_qam_awgn.py:


Bit error rate of M-QAM over AWGN
=================================

The waterfall curves of Gray-mapped 4-, 16- and 64-QAM over the additive white
Gaussian noise channel, against the closed form

.. math::

   P_b \approx \frac{4}{\log_2 M}\left(1 - \frac{1}{\sqrt{M}}\right)
   Q\left(\sqrt{\frac{3 \log_2 M}{M - 1}\frac{E_b}{N_0}}\right)

(Proakis & Salehi, *Digital Communications*, Sec. 4.3). The campaign is a declarative
specification run on every core by the Rust runner: each point stops after 200 bit
errors, and the sweep of a constellation stops once its BER falls below
:math:`10^{-6}`. The error bars are the 95 % Wilson intervals.

.. GENERATED FROM PYTHON SOURCE LINES 18-42

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    import comnum
    from comnum.plot import BACKGROUND, MUTED

    results = comnum.run_spec(
        """
    name: qam_awgn
    seed: 2024
    grid:
      m: [4, 16, 64]
      ebn0_db: {from: 0, to: 20, step: 1}
    link:
      - {type: source, constellation: {kind: qam, m: $m}, n_symbols: 8192}
      - {type: awgn_ebn0, ebn0_db: $ebn0_db}
    metrics: {report: [ber]}
    stop: {min_errors: {ber: 200}, min_trials: 8, max_trials: 4000}
    callbacks:
      skip_below: {metric: ber, threshold: 1e-6, along: ebn0_db}
    """
    )








.. GENERATED FROM PYTHON SOURCE LINES 43-44

Simulated points with their intervals, and the closed form.

.. GENERATED FROM PYTHON SOURCE LINES 44-72

.. code-block:: Python

    fig, ax = plt.subplots()
    eb = np.linspace(0, 20, 201)
    for m in [4, 16, 64]:
        # the conditions skipped below 1e-6 have no metric: series leaves them out
        x, ber, lo, hi = results.series("ebn0_db", "ber", interval=True, m=m)
        keep = ber > 0
        (line,) = ax.semilogy(eb, comnum.theory_ber(m, eb, "qam"), linewidth=1.4)
        ax.errorbar(
            x[keep],
            ber[keep],
            yerr=[ber[keep] - lo[keep], hi[keep] - ber[keep]],
            fmt="o",
            color=line.get_color(),
            markeredgecolor=BACKGROUND,
            capsize=0,
            label=f"{m}-QAM",
        )
    ax.plot([], [], color=MUTED, linewidth=1.4, label="theory")
    ax.set(
        ylim=(1e-6, 0.5),
        xlim=(0, 20),
        xlabel="$E_b/N_0$ (dB)",
        ylabel="bit error rate",
        title="Gray-mapped QAM over AWGN",
    )
    ax.legend()
    fig.tight_layout()




.. image-sg:: /auto_examples/foundations/images/sphx_glr_plot_02_ber_qam_awgn_001.png
   :alt: Gray-mapped QAM over AWGN
   :srcset: /auto_examples/foundations/images/sphx_glr_plot_02_ber_qam_awgn_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 73-75

The Eb/N0 needed for a BER of 1e-5 grows by 3.8 dB from QPSK to 16-QAM, and by
4.4 dB more to 64-QAM: the price of two more bits per symbol.

.. GENERATED FROM PYTHON SOURCE LINES 75-81

.. code-block:: Python

    for m in [4, 16, 64]:
        lo, hi = 0.0, 30.0
        for _ in range(60):
            mid = (lo + hi) / 2
            lo, hi = (mid, hi) if comnum.theory_ber(m, mid, "qam") > 1e-5 else (lo, mid)
        print(f"{m:>2}-QAM: Eb/N0 = {lo:.2f} dB at a BER of 1e-5")




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

 .. code-block:: none

     4-QAM: Eb/N0 = 9.59 dB at a BER of 1e-5
    16-QAM: Eb/N0 = 13.43 dB at a BER of 1e-5
    64-QAM: Eb/N0 = 17.79 dB at a BER of 1e-5





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

   **Total running time of the script:** (0 minutes 2.871 seconds)


.. _sphx_glr_download_auto_examples_foundations_plot_02_ber_qam_awgn.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_ber_qam_awgn.ipynb <plot_02_ber_qam_awgn.ipynb>`

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

      :download:`Download Python source code: plot_02_ber_qam_awgn.py <plot_02_ber_qam_awgn.py>`

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

      :download:`Download zipped: plot_02_ber_qam_awgn.zip <plot_02_ber_qam_awgn.zip>`


.. only:: html

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

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