.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/communications/detection/plot_01_kotelnikov_bpsk_detection.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_communications_detection_plot_01_kotelnikov_bpsk_detection.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_communications_detection_plot_01_kotelnikov_bpsk_detection.py:


Kotel'nikov's optimum receiver: BPSK in white Gaussian noise
============================================================

Vladimir Kotel'nikov's 1947 thesis, *The Theory of Optimum Noise Immunity*,
asked which receiver makes the fewest errors when known signals are buried in
white Gaussian noise. For two antipodal signals the answer is a correlator (or
matched filter) followed by a sign decision, with bit-error rate
:math:`Q(\sqrt{2E_b/N_0})` (:func:`~engineeringkit.communications.ber_bpsk_awgn`,
:func:`~engineeringkit.communications.q_function`). A Monte Carlo simulation of
:func:`~engineeringkit.communications.bpsk` through noise reproduces the curve.

.. GENERATED FROM PYTHON SOURCE LINES 15-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.communications import ber_bpsk_awgn, bpsk, q_function
    from engineeringkit.communications.visualizers import plot_ber_curve








.. GENERATED FROM PYTHON SOURCE LINES 22-24

Monte Carlo against theory
--------------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-46

.. code-block:: Python


    rng = np.random.default_rng(1947)
    ebn0_db = np.arange(0, 9)
    simulated = []
    for x_db in ebn0_db:
        ebn0 = 10 ** (x_db / 10)
        bits = rng.integers(0, 2, 400_000)
        received = bpsk(bits) + rng.normal(0, np.sqrt(1 / (2 * ebn0)), bits.size)
        simulated.append(np.mean((received > 0) != bits))
    theory = ber_bpsk_awgn(10 ** (ebn0_db / 10))
    for x, s, th in zip(ebn0_db, simulated, theory, strict=True):
        print(f"Eb/N0 = {x} dB: simulated {s:.2e}, theory {th:.2e}")
    assert np.allclose(simulated[:6], theory[:6], rtol=0.1)
    assert np.allclose(theory, q_function(np.sqrt(2 * 10 ** (ebn0_db / 10))))

    fine = np.linspace(0, 10, 200)
    ax = plot_ber_curve(fine, ber_bpsk_awgn(10 ** (fine / 10)), label="Q(sqrt(2 Eb/N0))")
    ax.semilogy(ebn0_db, simulated, "o", label="simulation")
    ax.legend()
    ax.set_title("Coherent BPSK in AWGN")

    plt.show()



.. image-sg:: /api/gallery/communications/detection/images/sphx_glr_plot_01_kotelnikov_bpsk_detection_001.png
   :alt: Coherent BPSK in AWGN
   :srcset: /api/gallery/communications/detection/images/sphx_glr_plot_01_kotelnikov_bpsk_detection_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Eb/N0 = 0 dB: simulated 7.85e-02, theory 7.86e-02
    Eb/N0 = 1 dB: simulated 5.61e-02, theory 5.63e-02
    Eb/N0 = 2 dB: simulated 3.74e-02, theory 3.75e-02
    Eb/N0 = 3 dB: simulated 2.30e-02, theory 2.29e-02
    Eb/N0 = 4 dB: simulated 1.24e-02, theory 1.25e-02
    Eb/N0 = 5 dB: simulated 6.01e-03, theory 5.95e-03
    Eb/N0 = 6 dB: simulated 2.41e-03, theory 2.39e-03
    Eb/N0 = 7 dB: simulated 7.45e-04, theory 7.73e-04
    Eb/N0 = 8 dB: simulated 1.72e-04, theory 1.91e-04





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

   **Total running time of the script:** (0 minutes 0.237 seconds)


.. _sphx_glr_download_api_gallery_communications_detection_plot_01_kotelnikov_bpsk_detection.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/communications/detection/plot_01_kotelnikov_bpsk_detection.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_kotelnikov_bpsk_detection.ipynb <plot_01_kotelnikov_bpsk_detection.ipynb>`

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

      :download:`Download Python source code: plot_01_kotelnikov_bpsk_detection.py <plot_01_kotelnikov_bpsk_detection.py>`

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

      :download:`Download zipped: plot_01_kotelnikov_bpsk_detection.zip <plot_01_kotelnikov_bpsk_detection.zip>`


.. only:: html

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

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