.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/mimo/plot_02_space_time_diversity.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_mimo_plot_02_space_time_diversity.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_mimo_plot_02_space_time_diversity.py:


Transmit diversity: Alamouti and space-time block codes
=======================================================

Diversity is the slope of the error rate at high SNR: with :math:`L` independent
fading branches, :math:`P_b \propto \bar\gamma^{-L}`. Receive diversity is obtained
by maximal-ratio combining (MRC); Alamouti's code obtains the same order at the
transmitter, without channel knowledge there, at the price of splitting the power over
the two antennas (3 dB). BPSK, i.i.d. Rayleigh fading constant over a code block; the
lines are the closed form of MRC with :math:`L` branches of mean SNR :math:`\gamma`
(Proakis & Salehi, eq. 13.4-15):

.. math::

   P_b = \left(\frac{1 - \mu}{2}\right)^{L} \sum_{k=0}^{L-1}
   \binom{L - 1 + k}{k} \left(\frac{1 + \mu}{2}\right)^{k},
   \qquad \mu = \sqrt{\frac{\gamma}{1 + \gamma}}.

.. GENERATED FROM PYTHON SOURCE LINES 19-94

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    import comnum
    from comnum.plot import BACKGROUND, MUTED

    code = comnum.Ostbc.alamouti()
    points = np.arange(0.0, 31.0, 3.0)


    def simulate(n_tx, n_rx, eb_db, seed):
        """BER of BPSK: Alamouti for two transmit antennas, MRC otherwise."""
        g = comnum.db2lin(eb_db)
        errors = total = t = 0
        blocks = 20_000
        while errors < 200 and total < 4e6:
            rng = comnum.Rng.stream(seed, "diversity", int(eb_db), t)
            bits = comnum.bits(2 * blocks, rng)
            s = 1.0 - 2.0 * bits
            h = comnum.mimo_channels(blocks, n_rx, n_tx, rng)
            if n_tx == 2:
                x = code.encode(s)  # 2 antennas x 2 uses per block
                y = np.einsum("nij,njt->nit", h, x.reshape(2, blocks, 2).transpose(1, 0, 2))
                y = y + rng.complex_noise(y.size, 1 / g).reshape(y.shape)
                est, _ = code.decode(h, y.transpose(1, 0, 2).reshape(n_rx, 2 * blocks), 1 / g)
            else:  # one antenna: two symbols per block over the same channel, MRC
                x = s.reshape(blocks, 2)
                y = h[:, :, :1] * x[:, None, :] + rng.complex_noise(blocks * n_rx * 2, 1 / g).reshape(
                    blocks, n_rx, 2
                )
                est = (np.conj(h[:, :, :1]) * y).sum(axis=1).reshape(-1)
            errors += int(((est.real < 0) != (bits == 1)).sum())
            total += bits.size
            t += 1
        return errors / total


    configs = {
        "1 × 1": (1, 1, 1, 1.0),
        "1 × 2, MRC": (1, 2, 2, 1.0),
        "2 × 1, Alamouti": (2, 1, 2, 0.5),
        "2 × 2, Alamouti": (2, 2, 4, 0.5),
    }
    fig, ax = plt.subplots()
    eb = np.linspace(0, 30, 301)
    for (name, (nt, nr, branches, share)), marker in zip(
        configs.items(), ["o", "s", "D", "^"], strict=True
    ):
        (line,) = ax.semilogy(
            eb,
            comnum.theory_ber_rayleigh_mrc(eb + comnum.lin2db(share), branches=branches),
            linewidth=1.4,
        )
        sim = np.array([simulate(nt, nr, e, seed=nt * 10 + nr) for e in points])
        keep = sim > 0
        ax.semilogy(
            points[keep],
            sim[keep],
            marker,
            color=line.get_color(),
            markeredgecolor=BACKGROUND,
            label=name,
        )
    ax.plot([], [], color=MUTED, linewidth=1.4, label="theory")
    ax.set(
        ylim=(1e-6, 0.5),
        xlim=(0, 30),
        xlabel="$E_b/N_0$ (dB)",
        ylabel="bit error rate",
        title="BPSK over Rayleigh fading: diversity",
    )
    ax.legend()
    fig.tight_layout()




.. image-sg:: /auto_examples/mimo/images/sphx_glr_plot_02_space_time_diversity_001.png
   :alt: BPSK over Rayleigh fading: diversity
   :srcset: /auto_examples/mimo/images/sphx_glr_plot_02_space_time_diversity_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 95-96

The rate-3/4 codes extend transmit diversity to three and four antennas.

.. GENERATED FROM PYTHON SOURCE LINES 96-98

.. code-block:: Python

    for code34 in [comnum.Ostbc.rate_3_4(3), comnum.Ostbc.rate_3_4(4)]:
        print(code34, f"{code34.n_symbols} symbols over {code34.n_slots} channel uses")




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

 .. code-block:: none

    Ostbc("rate 3/4, 3 antennas", n_tx=3, rate=0.75) 3 symbols over 4 channel uses
    Ostbc("rate 3/4, 4 antennas", n_tx=4, rate=0.75) 3 symbols over 4 channel uses





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

   **Total running time of the script:** (0 minutes 19.433 seconds)


.. _sphx_glr_download_auto_examples_mimo_plot_02_space_time_diversity.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_space_time_diversity.ipynb <plot_02_space_time_diversity.ipynb>`

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

      :download:`Download Python source code: plot_02_space_time_diversity.py <plot_02_space_time_diversity.py>`

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

      :download:`Download zipped: plot_02_space_time_diversity.zip <plot_02_space_time_diversity.zip>`


.. only:: html

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

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