.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/mimo/plot_06_biguesh2006_fig4.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_06_biguesh2006_fig4.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_mimo_plot_06_biguesh2006_fig4.py:


Biguesh & Gershman 2006, Fig. 4: LS, scaled LS, relaxed MMSE and LMMSE
=======================================================================

Fig. 4 of M. Biguesh and A. B. Gershman, "Training-based MIMO channel estimation: a
study of estimator tradeoffs and optimal training signals", *IEEE Trans. Signal
Process.* 54(3), 2006, compares four estimators of a flat :math:`r \times t` channel
from :math:`S = H P + V`, each needing more prior knowledge than the last: least
squares (none), scaled LS (the trace of :math:`R_H = E\{H^H H\}`), the relaxed MMSE
(the same trace, :math:`R_H` replaced by a scaled identity) and the LMMSE (all of
:math:`R_H`). The training is the DFT matrix of orthogonal rows (their eq. (10)), the
transmit correlation exponential with :math:`\varepsilon = 0.8` (eq. (60)), the
abscissa the training power over the noise in dB.

The lines are the closed-form MSE of :class:`comnum.MimoTraining` (their eqs. (12),
(20), (49) and (28)); the dots a Monte Carlo of 1000 channels; the crosses the values
read on the published figure. Scaled LS and relaxed MMSE coincide under orthogonal
training, as eq. (48) shows. The card ``validation/cards/biguesh2006.yaml`` checks 121
readings of Figs. 1-4 within 7.4 %; for :math:`t = r = 2` below 6 dB the published
SLS and LMMSE markers lie 11-18 % above the paper's own equations (left of the
figure), the only points where the figure and its equations disagree.

.. GENERATED FROM PYTHON SOURCE LINES 23-90



.. image-sg:: /auto_examples/mimo/images/sphx_glr_plot_06_biguesh2006_fig4_001.png
   :alt: $t = r = 2$, $\varepsilon = 0.8$, $t = r = 8$, $\varepsilon = 0.8$
   :srcset: /auto_examples/mimo/images/sphx_glr_plot_06_biguesh2006_fig4_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    import comnum
    from comnum.plot import BACKGROUND

    EPS, RUNS = 0.8, 1000
    x_db = np.linspace(3, 19, 81)
    mc_db = [3, 6, 10, 13, 16, 19]
    # readings of Fig. 4 (vector paths of the figure), SLS = RMMSE
    readings = {
        8: {
            "ls": [4.0, 2.0, 0.79, 0.100],
            "sls": [0.82, 0.68, 0.45, 0.091],
            "mmse": [0.60, 0.47, 0.29, 0.074],
        },
        2: {
            "ls": [0.98, 0.485, 0.20, 0.024],
            "sls": [0.59, 0.38, 0.18, 0.024],
            "mmse": [0.45, 0.29, 0.143, 0.024],
        },
    }
    names = {"ls": "LS", "sls": "scaled LS", "rmmse": "relaxed MMSE", "mmse": "LMMSE"}


    def correlation(t, r):
        return r * comnum.exponential_correlation(t, EPS)


    def monte_carlo(t, r, estimator, snr_db, rng):
        p = comnum.MimoTraining.orthogonal(t, total_power=10 ** (snr_db / 10))
        r_h, err = correlation(t, r), 0.0
        for _ in range(RUNS):
            h = comnum.mimo_channel(r, t, rng, tx_correlation=EPS)
            v = rng.complex_noise(r * t, 1.0).reshape(r, t)
            h_hat = p.estimate(h @ p.symbols + v, estimator, 1.0, r_h=r_h, trace=float(r * t))
            err += np.sum(np.abs(h - h_hat) ** 2)
        return err / RUNS / (r * t)


    fig, axes = plt.subplots(1, 2, figsize=(10, 4.6), sharey=True)
    rng = comnum.Rng(2006)
    for ax, t in zip(axes, [2, 8], strict=True):
        r = t
        for est, marker in zip(["ls", "sls", "rmmse", "mmse"], ["o", "s", "", "D"], strict=True):
            cf = [
                comnum.MimoTraining.orthogonal(t, total_power=10 ** (x / 10)).mse(
                    est, correlation(t, r), 1.0, r
                )
                / (r * t)
                for x in x_db
            ]
            style = "--" if est == "rmmse" else "-"
            (line,) = ax.semilogy(x_db, cf, style, linewidth=1.4, label=names[est])
            if est == "rmmse":
                continue
            mc = [monte_carlo(t, r, est, x, rng) for x in mc_db]
            ax.semilogy(mc_db, mc, marker, color=line.get_color(), markeredgecolor=BACKGROUND)
            ax.semilogy([3, 6, 10, 19], readings[t][est], "x", color=line.get_color(), markersize=7)
        ax.plot([], [], "ok", label="Monte Carlo (1000 channels)")
        ax.plot([], [], "xk", label="read on the published Fig. 4")
        ax.set(xlabel="training power / noise (dB)", title=f"$t = r = {t}$, $\\varepsilon = {EPS}$")
        ax.grid(True, which="both", alpha=0.3)
    axes[0].set_ylabel("normalized MSE")
    axes[1].legend(loc="lower left")
    fig.tight_layout()


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

   **Total running time of the script:** (0 minutes 0.458 seconds)


.. _sphx_glr_download_auto_examples_mimo_plot_06_biguesh2006_fig4.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_06_biguesh2006_fig4.ipynb <plot_06_biguesh2006_fig4.ipynb>`

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

      :download:`Download Python source code: plot_06_biguesh2006_fig4.py <plot_06_biguesh2006_fig4.py>`

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

      :download:`Download zipped: plot_06_biguesh2006_fig4.zip <plot_06_biguesh2006_fig4.zip>`


.. only:: html

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

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