.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/foundations/plot_04_rayleigh.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_04_rayleigh.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_foundations_plot_04_rayleigh.py:


BPSK over flat Rayleigh fading
==============================

A narrowband channel whose gain :math:`h \sim \mathcal{CN}(0, 1)` changes from symbol
to symbol: with coherent detection and an average SNR per bit :math:`\bar\gamma`,

.. math::

   P_b = \frac{1}{2}\left(1 - \sqrt{\frac{\bar\gamma}{1 + \bar\gamma}}\right)
   \approx \frac{1}{4\bar\gamma},

a decay of one decade per 10 dB instead of the exponential waterfall of AWGN: deep
fades dominate the errors (Proakis & Salehi, Sec. 13.3). The fading gains are drawn
with the same generator as the noise, so the whole link is reproducible from its
seed.

.. GENERATED FROM PYTHON SOURCE LINES 18-46

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    import comnum
    from comnum.plot import BACKGROUND, GREY

    N = 200_000
    points = np.arange(0.0, 41.0, 4.0)
    sim_awgn, sim_ray = [], []
    for i, eb_db in enumerate(points):
        g = comnum.db2lin(eb_db)
        errors = {"awgn": 0, "rayleigh": 0}
        total = 0
        t = 0
        while min(errors.values()) < 200 and total < 4e7:
            rng = comnum.Rng.stream(5, "rayleigh", i, t)
            bits = comnum.bits(N, rng)
            x = 1.0 - 2.0 * bits
            w = rng.complex_noise(N, 1.0 / g)
            h = rng.complex_noise(N, 1.0)
            errors["awgn"] += int((((x + w).real < 0) != (bits == 1)).sum())
            errors["rayleigh"] += int((((np.conj(h) * (h * x + w)).real < 0) != (bits == 1)).sum())
            total += N
            t += 1
        sim_awgn.append(errors["awgn"] / total)
        sim_ray.append(errors["rayleigh"] / total)








.. GENERATED FROM PYTHON SOURCE LINES 47-89

.. code-block:: Python

    eb = np.linspace(0, 40, 401)
    fig, ax = plt.subplots()
    (l1,) = ax.semilogy(eb, comnum.theory_ber(2, eb, "psk"), linewidth=1.4, label="AWGN, theory")
    # (1 - sqrt(g / (1 + g))) / 2, the maximal-ratio combining of one branch
    (l2,) = ax.semilogy(
        eb, comnum.theory_ber_rayleigh_mrc(eb, branches=1), linewidth=1.4, label="Rayleigh, theory"
    )
    ax.semilogy(
        eb,
        1 / (4 * comnum.db2lin(eb)),
        color=GREY,
        linestyle=":",
        linewidth=1.2,
        label=r"$1/(4\bar\gamma)$",
    )
    keep = np.array(sim_awgn) > 0
    ax.plot(
        points[keep],
        np.array(sim_awgn)[keep],
        "o",
        color=l1.get_color(),
        markeredgecolor=BACKGROUND,
        label="AWGN, simulated",
    )
    ax.plot(
        points,
        sim_ray,
        "s",
        color=l2.get_color(),
        markeredgecolor=BACKGROUND,
        label="Rayleigh, simulated",
    )
    ax.set(
        ylim=(1e-6, 0.5),
        xlim=(0, 40),
        xlabel=r"average $E_b/N_0$ (dB)",
        ylabel="bit error rate",
        title="BPSK with coherent detection",
    )
    ax.legend()
    fig.tight_layout()




.. image-sg:: /auto_examples/foundations/images/sphx_glr_plot_04_rayleigh_001.png
   :alt: BPSK with coherent detection
   :srcset: /auto_examples/foundations/images/sphx_glr_plot_04_rayleigh_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 90-91

At a BER of 1e-4, fading costs about 25 dB.

.. GENERATED FROM PYTHON SOURCE LINES 91-100

.. code-block:: Python

    for name, f in [
        ("AWGN", lambda e: comnum.theory_ber(2, e, "psk")),
        ("Rayleigh", lambda e: comnum.theory_ber_rayleigh_mrc(e, branches=1)),
    ]:
        lo, hi = 0.0, 60.0
        for _ in range(60):
            mid = (lo + hi) / 2
            lo, hi = (mid, hi) if f(mid) > 1e-4 else (lo, mid)
        print(f"{name:>8}: {lo:.1f} dB at a BER of 1e-4")




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

 .. code-block:: none

        AWGN: 8.4 dB at a BER of 1e-4
    Rayleigh: 34.0 dB at a BER of 1e-4





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

   **Total running time of the script:** (0 minutes 17.110 seconds)


.. _sphx_glr_download_auto_examples_foundations_plot_04_rayleigh.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_04_rayleigh.ipynb <plot_04_rayleigh.ipynb>`

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

      :download:`Download Python source code: plot_04_rayleigh.py <plot_04_rayleigh.py>`

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

      :download:`Download zipped: plot_04_rayleigh.zip <plot_04_rayleigh.zip>`


.. only:: html

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

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