.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/waveforms/plot_01_papr_ccdf.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_waveforms_plot_01_papr_ccdf.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_waveforms_plot_01_papr_ccdf.py:


PAPR of OFDM: the CCDF and its closed forms
===========================================

An OFDM block is the sum of :math:`N` independent subcarriers: by the central limit
theorem its samples are nearly complex Gaussian, and rare blocks have large peaks. The
complementary cumulative distribution :math:`\Pr(\mathrm{PAPR} > \gamma)` is the
standard figure of merit.

- At the Nyquist rate the :math:`N` samples are nearly independent Rayleigh amplitudes:
  :math:`1 - (1 - e^{-\gamma})^{N}` (van Nee & Prasad, *OFDM for Wireless Multimedia
  Communications*, 2000).
- The peaks of the continuous signal fall between those samples: an oversampling by
  :math:`L = 4` captures them, and their distribution is
  :math:`1 - \exp\bigl(-N \sqrt{\pi\gamma/3}\, e^{-\gamma}\bigr)` (Ochiai & Imai,
  *IEEE Trans. Commun.* 49(2), 2001), an asymptotic result for large :math:`N`: tight
  from a few hundred subcarriers, slightly pessimistic for :math:`N = 64`.
- DFT-spread OFDM (localised SC-FDMA, the LTE uplink) transmits a single-carrier signal
  and removes most of the PAPR (Myung, Lim & Goodman, *IEEE Veh. Technol. Mag.* 1(3),
  2006).

.. GENERATED FROM PYTHON SOURCE LINES 22-86



.. image-sg:: /auto_examples/waveforms/images/sphx_glr_plot_01_papr_ccdf_001.png
   :alt: OFDM, oversampled (L = 4), N = 256: sampling and SC-FDMA
   :srcset: /auto_examples/waveforms/images/sphx_glr_plot_01_papr_ccdf_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, GREY, MUTED

    BLOCKS = 4000
    t = np.arange(4.0, 12.6, 0.1)
    c = comnum.Constellation("qam", 16).normalized()

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(9.5, 4.0), sharey=True)
    for n in [64, 256, 1024]:
        ofdm = comnum.Ofdm(n, n)
        _, d = comnum.symbols(c, n * BLOCKS, comnum.Rng(n))
        p = ofdm.papr_db(ofdm.modulate(d))
        (line,) = ax1.semilogy(t, comnum.theory_papr_ccdf(n, t), linewidth=1.4)
        ax1.semilogy(
            t[::4],
            comnum.ccdf(p, t[::4]),
            "o",
            color=line.get_color(),
            markeredgecolor=BACKGROUND,
            label=f"N = {n}",
        )
    ax1.plot([], [], color=MUTED, linewidth=1.4, label="Ochiai & Imai")
    ax1.set(
        xlabel=r"$\gamma$ (dB)",
        ylabel=r"$\Pr(\mathrm{PAPR} > \gamma)$",
        ylim=(1e-3, 1.1),
        xlim=(4, 12.5),
        title="OFDM, oversampled (L = 4)",
    )
    ax1.legend()

    n = 256
    _, d = comnum.symbols(c, n * BLOCKS, comnum.Rng(7))
    for L, marker in [(1, "s"), (4, "o")]:
        ofdm = comnum.Ofdm(n, n, oversampling=L)
        p = ofdm.papr_db(ofdm.modulate(d))
        ax2.semilogy(
            t,
            comnum.ccdf(p, t),
            marker=marker,
            markevery=6,
            markeredgecolor=BACKGROUND,
            label=f"OFDM, L = {L}",
        )
    ax2.semilogy(
        t,
        comnum.theory_papr_ccdf(n, t, oversampled=False),
        color=GREY,
        linestyle=":",
        linewidth=1.4,
        label="van Nee & Prasad",
    )
    sc = comnum.Ofdm(1024, 256, dft_spread=True)
    p = sc.papr_db(sc.modulate(d))
    ax2.semilogy(
        t, comnum.ccdf(p, t), marker="D", markevery=6, markeredgecolor=BACKGROUND, label="DFT-s-OFDM"
    )
    ax2.set(xlabel=r"$\gamma$ (dB)", xlim=(2, 12.5), title=f"N = {n}: sampling and SC-FDMA")
    ax2.legend()
    fig.tight_layout()


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

   **Total running time of the script:** (0 minutes 0.940 seconds)


.. _sphx_glr_download_auto_examples_waveforms_plot_01_papr_ccdf.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_papr_ccdf.ipynb <plot_01_papr_ccdf.ipynb>`

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

      :download:`Download Python source code: plot_01_papr_ccdf.py <plot_01_papr_ccdf.py>`

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

      :download:`Download zipped: plot_01_papr_ccdf.zip <plot_01_papr_ccdf.zip>`


.. only:: html

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

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