.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/communications/multiplexing/plot_01_ofdm_cyclic_prefix.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_multiplexing_plot_01_ofdm_cyclic_prefix.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_multiplexing_plot_01_ofdm_cyclic_prefix.py:


OFDM: many subcarriers from one inverse DFT
===========================================

Robert Chang (1966) showed that subcarriers whose spectra overlap remain
orthogonal if they are spaced by the symbol rate. Stephen Weinstein and Paul
Ebert (1971) generated the whole set with a single inverse DFT
(:func:`~engineeringkit.communications.ofdm_modulate`) and prefixed each block
with a copy of its tail. This *cyclic prefix* turns a dispersive channel's
linear convolution into a circular one, so after the receiver's DFT
(:func:`~engineeringkit.communications.ofdm_demodulate`) every subcarrier sees
one complex gain, removed by a single division. The idea now carries Wi-Fi,
LTE and digital television.

.. GENERATED FROM PYTHON SOURCE LINES 17-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.communications import ofdm_demodulate, ofdm_modulate, qpsk
    from engineeringkit.communications.visualizers import plot_constellation








.. GENERATED FROM PYTHON SOURCE LINES 24-26

A multipath channel with and without the cyclic prefix
------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 26-43

.. code-block:: Python


    rng = np.random.default_rng(1971)
    n_sc, n_blocks = 64, 40
    symbols = qpsk(rng.integers(0, 2, 2 * n_sc * n_blocks))
    channel = np.array([1.0, 0.0, 0.6 - 0.3j, 0.0, 0.0, 0.25j])
    response = np.fft.fft(channel, n_sc)
    fig, axes = plt.subplots(1, 3, figsize=(12, 3.8))
    for ax, cp in zip(axes[:2], [0, 8], strict=True):
        tx = ofdm_modulate(symbols, n_sc, cp)
        rx = np.convolve(tx, channel)[: tx.size]
        equalised = (ofdm_demodulate(rx, n_sc, cp).reshape(-1, n_sc) / response).reshape(-1)
        error = np.sqrt(np.mean(np.abs(equalised - symbols) ** 2))
        print(f"cyclic prefix {cp}: RMS error after one-tap equalisation = {error:.2e}")
        plot_constellation(equalised, ax=ax)
        ax.set_title(f"cyclic prefix = {cp} samples")
    assert error < 1e-12




.. image-sg:: /api/gallery/communications/multiplexing/images/sphx_glr_plot_01_ofdm_cyclic_prefix_001.png
   :alt: cyclic prefix = 0 samples, cyclic prefix = 8 samples
   :srcset: /api/gallery/communications/multiplexing/images/sphx_glr_plot_01_ofdm_cyclic_prefix_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    cyclic prefix 0: RMS error after one-tap equalisation = 3.26e-01
    cyclic prefix 8: RMS error after one-tap equalisation = 3.78e-16




.. GENERATED FROM PYTHON SOURCE LINES 44-49

The channel seen by each subcarrier
-----------------------------------

With the prefix, the demodulated pilot divided by the transmitted symbol is
exactly the channel's frequency response at that subcarrier.

.. GENERATED FROM PYTHON SOURCE LINES 49-59

.. code-block:: Python


    estimate = (ofdm_demodulate(np.convolve(ofdm_modulate(symbols[:n_sc], n_sc, 8), channel)[: n_sc + 8], n_sc, 8)) / symbols[:n_sc]
    assert np.allclose(estimate, response)
    axes[2].plot(np.abs(response), label="|H(k)|")
    axes[2].plot(np.abs(estimate), "k.", ms=3, label="pilot estimate")
    axes[2].set(xlabel="subcarrier", ylabel="gain", title="Frequency-selective fading")
    axes[2].legend()
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.198 seconds)


.. _sphx_glr_download_api_gallery_communications_multiplexing_plot_01_ofdm_cyclic_prefix.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/multiplexing/plot_01_ofdm_cyclic_prefix.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_ofdm_cyclic_prefix.ipynb <plot_01_ofdm_cyclic_prefix.ipynb>`

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

      :download:`Download Python source code: plot_01_ofdm_cyclic_prefix.py <plot_01_ofdm_cyclic_prefix.py>`

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

      :download:`Download zipped: plot_01_ofdm_cyclic_prefix.zip <plot_01_ofdm_cyclic_prefix.zip>`


.. only:: html

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

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