.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/communications/information/plot_02_hartley_law.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_information_plot_02_hartley_law.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_information_plot_02_hartley_law.py:


Hartley's law: information grows with the logarithm of the levels
=================================================================

Ralph Hartley (1928) argued that a message selected from :math:`M`
distinguishable symbols carries :math:`\log M` units of information, so a
channel of bandwidth :math:`B` carrying :math:`2B` symbols per second has a
line rate of :math:`2B\log_2 M` bits per second
(:func:`~engineeringkit.communications.hartley_capacity`). More levels raise
the rate only logarithmically, and noise limits how many levels can be told
apart. This example sends multilevel symbols through a noisy channel to
show that limit.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.communications import hartley_capacity, symbol_rate








.. GENERATED FROM PYTHON SOURCE LINES 22-24

Line rate of a 3 kHz telephone channel
--------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-30

.. code-block:: Python


    B = 3000.0
    for M in [2, 4, 8, 16, 64]:
        rate = float(hartley_capacity(B, M))
        print(f"M = {M:2d}: {rate:7.0f} bit/s at {float(symbol_rate(rate, M)):.0f} baud")





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

 .. code-block:: none

    M =  2:    6000 bit/s at 6000 baud
    M =  4:   12000 bit/s at 6000 baud
    M =  8:   18000 bit/s at 6000 baud
    M = 16:   24000 bit/s at 6000 baud
    M = 64:   36000 bit/s at 6000 baud




.. GENERATED FROM PYTHON SOURCE LINES 31-36

Noise sets how many levels survive
----------------------------------

Equally spaced levels in [-1, 1] with Gaussian noise of standard deviation
0.05: the symbol error rate jumps once the level spacing approaches the noise.

.. GENERATED FROM PYTHON SOURCE LINES 36-55

.. code-block:: Python


    rng = np.random.default_rng(1928)
    levels = [2, 4, 8, 16, 32, 64]
    errors = []
    for M in levels:
        alphabet = np.linspace(-1, 1, M)
        sent = rng.integers(0, M, 50_000)
        received = alphabet[sent] + rng.normal(0, 0.05, sent.size)
        decided = np.argmin(np.abs(received[:, None] - alphabet[None, :]), axis=1)
        errors.append(np.mean(decided != sent))
    fig, axes = plt.subplots(1, 2, figsize=(9, 3.6))
    axes[0].plot(levels, [float(hartley_capacity(B, m)) for m in levels], "o-")
    axes[0].set(xscale="log", xlabel="levels M", ylabel="line rate [bit/s]", title="2B log2 M")
    axes[1].semilogy(levels, np.maximum(errors, 1e-6), "o-")
    axes[1].set(xscale="log", xlabel="levels M", ylabel="symbol error rate", title="Noise σ = 0.05")
    fig.tight_layout()
    assert errors[0] < 1e-3 < errors[-1]

    plt.show()



.. image-sg:: /api/gallery/communications/information/images/sphx_glr_plot_02_hartley_law_001.png
   :alt: 2B log2 M, Noise σ = 0.05
   :srcset: /api/gallery/communications/information/images/sphx_glr_plot_02_hartley_law_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.482 seconds)


.. _sphx_glr_download_api_gallery_communications_information_plot_02_hartley_law.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/information/plot_02_hartley_law.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_hartley_law.ipynb <plot_02_hartley_law.ipynb>`

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

      :download:`Download Python source code: plot_02_hartley_law.py <plot_02_hartley_law.py>`

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

      :download:`Download zipped: plot_02_hartley_law.zip <plot_02_hartley_law.zip>`


.. only:: html

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

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