.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/mimo/plot_04_mimo_capacity.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_04_mimo_capacity.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_mimo_plot_04_mimo_capacity.py:


Ergodic capacity of MIMO channels
=================================

The capacity of a channel realisation with equal power on the :math:`n_t` antennas is
:math:`\log_2\det(I + \frac{\rho}{n_t} H H^H)`; its mean over Rayleigh fading grows
like :math:`\min(n_t, n_r) \log_2 \rho` at high SNR — one bit per 3 dB and per
spatial stream (Foschini & Gans 1998, Telatar 1999). Water-filling on the eigenmodes,
which needs the channel at the transmitter, helps at low SNR, where only the strongest
modes deserve power. Antenna correlation (exponential, :math:`\rho = 0.9`) removes
part of the multiplexing gain.

.. GENERATED FROM PYTHON SOURCE LINES 13-54

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    import comnum

    snr = np.arange(-10.0, 31.0, 2.5)
    N = 4000
    rng = comnum.Rng(17)
    fig, ax = plt.subplots()
    for n in [1, 2, 4]:
        h = comnum.mimo_channels(N, n, n, rng)
        (line,) = ax.plot(snr, [comnum.mimo_capacity(h, s).mean() for s in snr], label=f"{n} × {n}")
        if n == 4:
            ax.plot(
                snr,
                [comnum.mimo_capacity(h, s, water_filling=True).mean() for s in snr],
                color=line.get_color(),
                linestyle="--",
                linewidth=1.4,
                label="4 × 4, water-filling",
            )
            hc = comnum.mimo_channels(N, 4, 4, rng, rx_correlation=0.9, tx_correlation=0.9)
            ax.plot(
                snr,
                [comnum.mimo_capacity(hc, s).mean() for s in snr],
                color=line.get_color(),
                linestyle=":",
                linewidth=1.4,
                label=r"4 × 4, correlated ($\rho$ = 0.9)",
            )
    ax.set(
        xlabel=r"SNR $\rho$ (dB)",
        ylabel="ergodic capacity (bit/s/Hz)",
        xlim=(-10, 30),
        ylim=(0, None),
        title="i.i.d. Rayleigh fading",
    )
    ax.legend()
    fig.tight_layout()




.. image-sg:: /auto_examples/mimo/images/sphx_glr_plot_04_mimo_capacity_001.png
   :alt: i.i.d. Rayleigh fading
   :srcset: /auto_examples/mimo/images/sphx_glr_plot_04_mimo_capacity_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 55-56

At 20 dB, the capacity of 4 × 4 is close to four times that of a single antenna.

.. GENERATED FROM PYTHON SOURCE LINES 56-59

.. code-block:: Python

    h1, h4 = comnum.mimo_channels(N, 1, 1, rng), comnum.mimo_channels(N, 4, 4, rng)
    c1, c4 = comnum.mimo_capacity(h1, 20.0).mean(), comnum.mimo_capacity(h4, 20.0).mean()
    print(f"1 × 1: {c1:.2f} bit/s/Hz, 4 × 4: {c4:.2f} bit/s/Hz, ratio {c4 / c1:.2f}")




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

 .. code-block:: none

    1 × 1: 5.86 bit/s/Hz, 4 × 4: 22.19 bit/s/Hz, ratio 3.78





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

   **Total running time of the script:** (0 minutes 0.290 seconds)


.. _sphx_glr_download_auto_examples_mimo_plot_04_mimo_capacity.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_mimo_capacity.ipynb <plot_04_mimo_capacity.ipynb>`

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

      :download:`Download Python source code: plot_04_mimo_capacity.py <plot_04_mimo_capacity.py>`

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

      :download:`Download zipped: plot_04_mimo_capacity.zip <plot_04_mimo_capacity.zip>`


.. only:: html

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

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