.. _sphx_glr_auto_examples_waveforms:

Waveforms
---------

OFDM and its PAPR, continuous-phase modulation, constant-envelope OFDM, chirp spread
spectrum, filtered multicarrier (FBMC/OQAM, UFMC) and OTFS.


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    <div class="sphx-glr-thumbcontainer" tooltip="An OFDM block is the sum of N independent subcarriers: by the central limit theorem its samples are nearly complex Gaussian, and rare blocks have large peaks. The complementary cumulative distribution \Pr(\mathrm{PAPR} &gt; \gamma) is the standard figure of merit.">

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  .. image:: /auto_examples/waveforms/images/thumb/sphx_glr_plot_01_papr_ccdf_thumb.png
    :alt:

  :doc:`/auto_examples/waveforms/plot_01_papr_ccdf`

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      <div class="sphx-glr-thumbnail-title">PAPR of OFDM: the CCDF and its closed forms</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="The classical techniques (Han &amp; Lee, IEEE Wireless Commun. 12(2), 2005), applied to 16-QAM OFDM with 128 subcarriers (oversampling by 4):">

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  .. image:: /auto_examples/waveforms/images/thumb/sphx_glr_plot_02_papr_reduction_thumb.png
    :alt:

  :doc:`/auto_examples/waveforms/plot_02_papr_reduction`

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      <div class="sphx-glr-thumbnail-title">PAPR reduction techniques</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Two classical analyses of OFDM links, reproduced with comnum and checked against their own figures:">

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  .. image:: /auto_examples/waveforms/images/thumb/sphx_glr_plot_03_ofdm_theory_with_impairments_thumb.png
    :alt:

  :doc:`/auto_examples/waveforms/plot_03_ofdm_theory_with_impairments`

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      <div class="sphx-glr-thumbnail-title">OFDM with impairments: imperfect channel estimates and nonlinear amplifiers</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Minimum-shift keying is continuous-phase FSK with a modulation index of 1/2. Section 3.3-2 of J. G. Proakis and M. Salehi, Digital Communications, 5th ed., 2008, shows that its signal is an offset QPSK whose quadrature pulses are half sinusoids, so that coherent detection over AWGN decides the bits with the BER of BPSK, Q(\sqrt{2 E_b/N_0}), once the bits are precoded so that each decides a sign of the equivalent offset QPSK signal.">

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  .. image:: /auto_examples/waveforms/images/thumb/sphx_glr_plot_04_proakis2008_msk_thumb.png
    :alt:

  :doc:`/auto_examples/waveforms/plot_04_proakis2008_msk`

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      <div class="sphx-glr-thumbnail-title">Proakis & Salehi 2008, Sec. 3.3-2: MSK has the error rate of BPSK</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Fig. 2 of S. C. Thompson, A. U. Ahmed, J. G. Proakis, J. R. Zeidler and M. J. Geile, &quot;Constant envelope OFDM&quot;, IEEE Trans. Commun. 56(8), 2008, shows the BER of the phase demodulator receiver of constant-envelope OFDM for 8-PAM data symbols and N = 126 subcarriers, at modulation indices 2\pi h from 0.1 to 1.5, against the approximation of eq. (15) of the paper, valid above the threshold of the phase demodulator:">

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  .. image:: /auto_examples/waveforms/images/thumb/sphx_glr_plot_05_thompson2008_fig2_thumb.png
    :alt:

  :doc:`/auto_examples/waveforms/plot_05_thompson2008_fig2`

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      <div class="sphx-glr-thumbnail-title">Thompson et al. 2008, Fig. 2: the CE-OFDM phase demodulator</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Four panels, each to open next to the figure of the paper it reproduces:">

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  .. image:: /auto_examples/waveforms/images/thumb/sphx_glr_plot_06_ce_ofdm_receivers_thumb.png
    :alt:

  :doc:`/auto_examples/waveforms/plot_06_ce_ofdm_receivers`

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      <div class="sphx-glr-thumbnail-title">CE-OFDM receivers, spectrum and bandwidth</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Chirp spread spectrum, the modulation of LoRa (``comnum.Css``), in four panels, each to open next to the figure of the paper it reproduces. The SNR is that of a sample at one sample per chip, the abscissa of the papers.">

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  .. image:: /auto_examples/waveforms/images/thumb/sphx_glr_plot_07_lora_thumb.png
    :alt:

  :doc:`/auto_examples/waveforms/plot_07_lora`

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      <div class="sphx-glr-thumbnail-title">The LoRa modulation: error rates, spectrum and interference</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Three answers to the weaknesses of CP-OFDM, side by side.">

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  .. image:: /auto_examples/waveforms/images/thumb/sphx_glr_plot_08_filtered_multicarrier_otfs_thumb.png
    :alt:

  :doc:`/auto_examples/waveforms/plot_08_filtered_multicarrier_otfs`

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      <div class="sphx-glr-thumbnail-title">Filtered multicarrier and OTFS</div>
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.. toctree::
   :hidden:

   /auto_examples/waveforms/plot_01_papr_ccdf
   /auto_examples/waveforms/plot_02_papr_reduction
   /auto_examples/waveforms/plot_03_ofdm_theory_with_impairments
   /auto_examples/waveforms/plot_04_proakis2008_msk
   /auto_examples/waveforms/plot_05_thompson2008_fig2
   /auto_examples/waveforms/plot_06_ce_ofdm_receivers
   /auto_examples/waveforms/plot_07_lora
   /auto_examples/waveforms/plot_08_filtered_multicarrier_otfs