:orphan:

Gallery of classical results
============================

The textbook curves of digital and optical communications, simulated with ``comnum``
and compared with their closed forms. Every script runs as is (the figures of this
page are produced when the documentation is built), and the same results are checked
at a few points by ``tests/test_performance.py``.

Against the literature
----------------------

The scripts are grouped by theme, in the order of a link. Some reproduce one named
figure or table of a paper or a standard, with the same system parameters, and say
what the library's curve is expected to match: where the published points are
available as numbers, they are drawn on the figure; otherwise the caption names the
figure to open next to the plot. The others compare with closed forms and cite the
equations they use. The table below gathers every comparison of this gallery.

.. list-table::
   :header-rows: 1
   :widths: 30 42 28

   * - Reference
     - What is reproduced
     - Where
   * - S. M. Alamouti, *IEEE J. Sel. Areas Commun.* 16(8), 1998, Fig. 4
     - BER of coherent BPSK over Rayleigh fading: no diversity, MRRC with 2 and 4
       receive antennas, the two-branch transmit diversity scheme with 1 and 2
       receive antennas
     - :ref:`sphx_glr_auto_examples_mimo_plot_03_alamouti1998_fig4.py`
   * - X.-Y. Hu, E. Eleftheriou, D. M. Arnold, *IEEE Trans. Inf. Theory* 51(1), 2005,
       Fig. 5
     - Bit and block error rates of a regular (3, 6) PEG code of length 504 over
       AWGN with BPSK and sum-product decoding
     - :ref:`sphx_glr_auto_examples_coding_plot_02_hu2005_fig5.py`
   * - S. C. Thompson et al., *IEEE Trans. Commun.* 56(8), 2008, Fig. 2 and eq. (15)
     - BER of the CE-OFDM phase demodulator receiver for 8-PAM data and several
       modulation indices, against the high-CNR approximation
     - :ref:`sphx_glr_auto_examples_waveforms_plot_05_thompson2008_fig2.py`
   * - C. Häger, H. D. Pfister, *IEEE J. Sel. Areas Commun.* 39(1), 2021, Fig. 5
     - Effective SNR versus launch power over 25 × 80 km, linear equalisation and
       digital back-propagation with one step per span; the published points are
       drawn
     - :ref:`sphx_glr_auto_examples_optical_plot_03_hager2021_fig5.py`
   * - J. G. Proakis, M. Salehi, *Digital Communications*, 5th ed., 2008, Sec. 3.3-2
     - MSK detected by maximum-likelihood sequence detection has the BER of BPSK
       once precoded; without precoding, that of differentially decoded BPSK
     - :ref:`sphx_glr_auto_examples_waveforms_plot_04_proakis2008_msk.py`
   * - J. Guerreiro et al., *IEEE Commun. Lett.* 20(11), 2016, Fig. 1 and eq. (8);
       M. J. López Morales et al., *Globecom Workshops* 2022, Fig. 3; A. U. Ahmed,
       J. R. Zeidler, *IEEE Trans. Signal Process.* 63(17), 2015, Fig. 6; R. A. Pacheco,
       D. Hatzinakos, *ICASSP* 2006, Fig. 1; J. Guerreiro et al., *VTC Fall* 2015, Fig. 3
     - CE-OFDM: the phase detector, GAMP and the optimum BER at :math:`2\pi h = 1.4`;
       the arctangent, basic linear and enhanced receivers at 0.7; the 99 % bandwidth;
       the spectrum at :math:`2\pi h = 2`
     - :ref:`sphx_glr_auto_examples_waveforms_plot_06_ce_ofdm_receivers.py`
   * - O. Afisiadis et al., *IEEE Trans. Wireless Commun.* 19(2), 2020, Figs. 2 and 7;
       L. Vangelista, *IEEE Signal Process. Lett.* 24(12), 2017, Fig. 1; M. Chiani,
       A. Elzanaty, *IEEE Internet Things J.* 6(5), 2019, Fig. 3a
     - The LoRa modulation: the SER over AWGN for SF 7 to 12, FSCM against FSK over a
       frequency-selective channel, the continuous spectrum, a same-SF interferer
     - :ref:`sphx_glr_auto_examples_waveforms_plot_07_lora.py`
   * - ETSI EN 302 307-1 V1.4.1, Table 13
     - The :math:`E_s/N_0` thresholds of the 28 DVB-S2 MODCODs against the simulated
       frame error rates
     - :ref:`sphx_glr_auto_examples_systems_plot_01_dvbs2.py`
   * - Proakis & Salehi, 5th ed., Secs. 4.3 and 13.3; eq. (13.4-15)
     - Error rates of QAM, PSK and PAM over AWGN, BPSK over Rayleigh fading,
       maximal-ratio combining
     - :ref:`sphx_glr_auto_examples_foundations_plot_02_ber_qam_awgn.py`,
       :ref:`sphx_glr_auto_examples_foundations_plot_03_ber_psk.py`,
       :ref:`sphx_glr_auto_examples_foundations_plot_04_rayleigh.py`,
       :ref:`sphx_glr_auto_examples_mimo_plot_02_space_time_diversity.py`
   * - H. Ochiai, H. Imai, *IEEE Trans. Commun.* 49(2), 2001, eqs. (11) and (27)
     - The distribution of the PAPR of OFDM at the Nyquist rate and oversampled
     - :ref:`sphx_glr_auto_examples_waveforms_plot_01_papr_ccdf.py`
   * - Y. Polyanskiy, H. V. Poor, S. Verdú, *IEEE Trans. Inf. Theory* 56(5), 2010
     - The normal approximation of the finite-length limit, against polar and LDPC
       codes of length 256
     - :ref:`sphx_glr_auto_examples_coding_plot_01_channel_coding.py`
   * - P. Poggiolini, *J. Lightw. Technol.* 30(24), 2012; A. Carena et al., *Opt.
       Express* 22(13), 2014
     - The GN and EGN models of nonlinear interference against split-step
       simulations of a WDM link
     - :ref:`sphx_glr_auto_examples_optical_plot_04_wdm_nonlinearity.py`
   * - G. Böcherer, F. Steiner, P. Schulte, *IEEE Trans. Commun.* 63(12), 2015
     - Probabilistic amplitude shaping of 64-QAM at 3 bits per symbol
     - :ref:`sphx_glr_auto_examples_coding_plot_04_probabilistic_shaping.py`

The README of ``references/`` in the repository lists every document the library is
validated against, with its status, and ``specs/validation/`` holds the campaigns that
CI checks against these sources.


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Foundations
-----------

Constellations, error rates over AWGN and Rayleigh fading against their closed forms,
information rates.


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    <div class="sphx-glr-thumbcontainer" tooltip="The constellations of comnum carry the Gray labels of OptiCommPy: point i of c.points carries the bits of i, most significant bit first, and two nearest neighbours differ by a single bit. A symbol error to a neighbour therefore costs one bit error, which is what makes the approximation P_b \approx P_s / \log_2 M of the bit error rate accurate at high SNR.">

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  .. image:: /auto_examples/foundations/images/thumb/sphx_glr_plot_01_constellations_thumb.png
    :alt:

  :doc:`/auto_examples/foundations/plot_01_constellations`

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      <div class="sphx-glr-thumbnail-title">Constellations and Gray labels</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The waterfall curves of Gray-mapped 4-, 16- and 64-QAM over the additive white Gaussian noise channel, against the closed form">

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  .. image:: /auto_examples/foundations/images/thumb/sphx_glr_plot_02_ber_qam_awgn_thumb.png
    :alt:

  :doc:`/auto_examples/foundations/plot_02_ber_qam_awgn`

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      <div class="sphx-glr-thumbnail-title">Bit error rate of M-QAM over AWGN</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The bit error rate of Gray-mapped BPSK, QPSK, 8-PSK and 16-PSK, simulated with the Monte Carlo runner of comnum.sim: the trial is a Python function, every point runs until 200 bit errors (or 400 trials), and the result of every point is an error rate with its Wilson interval. BPSK and QPSK share the same curve Q(\sqrt{2 E_b/N_0}); beyond four points, every doubling of the order costs 3.5 to 4.5 dB at a BER of 10^{-5}, since the points get closer on the circle:">

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  .. image:: /auto_examples/foundations/images/thumb/sphx_glr_plot_03_ber_psk_thumb.png
    :alt:

  :doc:`/auto_examples/foundations/plot_03_ber_psk`

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      <div class="sphx-glr-thumbnail-title">M-PSK over AWGN with the Python runner</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A narrowband channel whose gain h \sim \mathcal{CN}(0, 1) changes from symbol to symbol: with coherent detection and an average SNR per bit \bar\gamma,">

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  .. image:: /auto_examples/foundations/images/thumb/sphx_glr_plot_04_rayleigh_thumb.png
    :alt:

  :doc:`/auto_examples/foundations/plot_04_rayleigh`

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      <div class="sphx-glr-thumbnail-title">BPSK over flat Rayleigh fading</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The mutual information I(X; Y) of a constellation over AWGN is the rate a receiver that decodes symbols can achieve; the generalised mutual information (GMI) is the rate of a bit-interleaved receiver that decodes bits independently, the usual coded-modulation receiver. Both stay below the Shannon capacity \log_2(1 + \mathrm{SNR}) and saturate at \log_2 M; with Gray labels the gap between them is small.">

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  .. image:: /auto_examples/foundations/images/thumb/sphx_glr_plot_05_information_rates_thumb.png
    :alt:

  :doc:`/auto_examples/foundations/plot_05_information_rates`

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      <div class="sphx-glr-thumbnail-title">Information rates of QAM</div>
    </div>


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Coding and shaping
------------------

Channel codes against the finite-length limits and the literature, the codes of 5G NR,
probabilistic amplitude shaping.


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    <div class="sphx-glr-thumbcontainer" tooltip="BPSK over AWGN, decoded from channel LLRs 2y/\sigma^2.">

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  .. image:: /auto_examples/coding/images/thumb/sphx_glr_plot_01_channel_coding_thumb.png
    :alt:

  :doc:`/auto_examples/coding/plot_01_channel_coding`

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      <div class="sphx-glr-thumbnail-title">Channel coding: from convolutional codes to the finite-length limit</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Fig. 5 of X.-Y. Hu, E. Eleftheriou and D. M. Arnold, &quot;Regular and irregular progressive edge-growth Tanner graphs&quot;, IEEE Trans. Inf. Theory 51(1), 2005, gives the bit and block error rates of a regular LDPC code built by progressive edge growth, with n = 504, m = 252 parity checks, symbol-node degree d_s = 3 and check-node degree d_c = 6, over the AWGN channel with BPSK and sum-product decoding, next to MacKay&#x27;s code and a random graph of the same parameters.">

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  .. image:: /auto_examples/coding/images/thumb/sphx_glr_plot_02_hu2005_fig5_thumb.png
    :alt:

  :doc:`/auto_examples/coding/plot_02_hu2005_fig5`

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      <div class="sphx-glr-thumbnail-title">Hu, Eleftheriou & Arnold 2005, Fig. 5: a regular PEG code of length 504</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The data channels of 5G NR use a quasi-cyclic LDPC code, the control channels a polar code (3GPP TS 38.212; T. Richardson and S. Kudekar, &quot;Design of low-density parity check codes for 5G new radio,&quot; IEEE Commun. Mag. 56(3), 2018; V. Bioglio, C. Condo and I. Land, &quot;Design of polar codes in 5G New Radio,&quot; IEEE Commun. Surveys Tuts. 23(1), 2021). comnum.NrLdpc and comnum.NrPolar build them from the tables of the standard — the two LDPC base graphs, the polar sequence of 1024 positions, the interleavers — with their rate matching to any number of transmitted bits.">

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  .. image:: /auto_examples/coding/images/thumb/sphx_glr_plot_03_nr_codes_thumb.png
    :alt:

  :doc:`/auto_examples/coding/plot_03_nr_codes`

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      <div class="sphx-glr-thumbnail-title">5G NR channel codes: LDPC and polar</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Rate. A uniform QAM wastes energy on its outer points: at a given SNR its rate stays below the capacity \log_2(1 + \mathrm{SNR}) of the AWGN channel, by up to the 1.53 dB of the ultimate shaping gain. Maxwell-Boltzmann probabilities p(x) \propto e^{-\lambda |x|^2} (Kschischang &amp; Pasupathy 1993) close most of that gap: the curve is the GMI of bit-metric decoding of 64-QAM with the priors, H(X) - \sum_k H(B_k \mid Y) (Böcherer 2014), with the best \lambda at every SNR, and the points are the estimates of simulated links (``specs/shaping_awgn.yaml`` checks them against the closed forms).">

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  .. image:: /auto_examples/coding/images/thumb/sphx_glr_plot_04_probabilistic_shaping_thumb.png
    :alt:

  :doc:`/auto_examples/coding/plot_04_probabilistic_shaping`

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      <div class="sphx-glr-thumbnail-title">Probabilistic shaping: the rate gain and probabilistic amplitude shaping</div>
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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>
    </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>
    </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>
    </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
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  :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>
    </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
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  :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>
    </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
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  :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>
    </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
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  :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>
    </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
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  :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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MIMO
----

Detection, space-time codes, capacity, message passing for large MIMO.


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    <div class="sphx-glr-thumbcontainer" tooltip="Four QPSK streams sent from four antennas and received on four, over i.i.d. Rayleigh fading with a new channel for every vector: y = H x + w. The detectors trade complexity for diversity:">

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  .. image:: /auto_examples/mimo/images/thumb/sphx_glr_plot_01_mimo_detection_thumb.png
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  :doc:`/auto_examples/mimo/plot_01_mimo_detection`

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      <div class="sphx-glr-thumbnail-title">MIMO detection: linear, successive cancellation and maximum likelihood</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Diversity is the slope of the error rate at high SNR: with L independent fading branches, P_b \propto \bar\gamma^{-L}. Receive diversity is obtained by maximal-ratio combining (MRC); Alamouti&#x27;s code obtains the same order at the transmitter, without channel knowledge there, at the price of splitting the power over the two antennas (3 dB). BPSK, i.i.d. Rayleigh fading constant over a code block; the lines are the closed form of MRC with L branches of mean SNR \gamma (Proakis &amp; Salehi, eq. 13.4-15):">

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  .. image:: /auto_examples/mimo/images/thumb/sphx_glr_plot_02_space_time_diversity_thumb.png
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  :doc:`/auto_examples/mimo/plot_02_space_time_diversity`

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      <div class="sphx-glr-thumbnail-title">Transmit diversity: Alamouti and space-time block codes</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Fig. 4 of S. M. Alamouti, &quot;A simple transmit diversity technique for wireless communications&quot;, IEEE J. Sel. Areas Commun. 16(8), 1998, compares the BER of uncoded coherent BPSK over Rayleigh fading for five systems: no diversity, maximal ratio receive combining (MRRC) with two and four receive antennas, and the two-branch transmit diversity scheme of the paper with one and two receive antennas. The assumptions of the paper&#x27;s Sec. IV are reproduced: the total transmit power of the two antennas equals that of the single antenna of MRRC, the fading amplitudes of every transmit-receive pair are uncorrelated Rayleigh with the same average power, and the receiver knows the channel.">

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  .. image:: /auto_examples/mimo/images/thumb/sphx_glr_plot_03_alamouti1998_fig4_thumb.png
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  :doc:`/auto_examples/mimo/plot_03_alamouti1998_fig4`

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      <div class="sphx-glr-thumbnail-title">Alamouti 1998, Fig. 4: transmit diversity against MRRC</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The capacity of a channel realisation with equal power on the n_t antennas is \log_2\det(I + \frac{\rho}{n_t} H H^H); its mean over Rayleigh fading grows like \min(n_t, n_r) \log_2 \rho at high SNR — one bit per 3 dB and per spatial stream (Foschini &amp; Gans 1998, Telatar 1999). Water-filling on the eigenmodes, which needs the channel at the transmitter, helps at low SNR, where only the strongest modes deserve power. Antenna correlation (exponential, \rho = 0.9) removes part of the multiplexing gain.">

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  .. image:: /auto_examples/mimo/images/thumb/sphx_glr_plot_04_mimo_capacity_thumb.png
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  :doc:`/auto_examples/mimo/plot_04_mimo_capacity`

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      <div class="sphx-glr-thumbnail-title">Ergodic capacity of MIMO channels</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="With tens of streams, the maximum-likelihood search is out of reach and the linear detectors lose several dB. Approximate message passing (AMP) alternates a linear step on the whole vector and a scalar MMSE denoiser on every stream: in the large-system limit, it decouples the MIMO channel into parallel AWGN channels whose noise variance a scalar recursion, the state evolution, predicts exactly.">

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  .. image:: /auto_examples/mimo/images/thumb/sphx_glr_plot_05_large_mimo_message_passing_thumb.png
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  :doc:`/auto_examples/mimo/plot_05_large_mimo_message_passing`

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      <div class="sphx-glr-thumbnail-title">Large MIMO: message passing (LAMA and OAMP)</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Fig. 4 of M. Biguesh and A. B. Gershman, &quot;Training-based MIMO channel estimation: a study of estimator tradeoffs and optimal training signals&quot;, IEEE Trans. Signal Process. 54(3), 2006, compares four estimators of a flat r \times t channel from S = H P + V, each needing more prior knowledge than the last: least squares (none), scaled LS (the trace of R_H = E\{H^H H\}), the relaxed MMSE (the same trace, R_H replaced by a scaled identity) and the LMMSE (all of R_H). The training is the DFT matrix of orthogonal rows (their eq. (10)), the transmit correlation exponential with \varepsilon = 0.8 (eq. (60)), the abscissa the training power over the noise in dB.">

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  .. image:: /auto_examples/mimo/images/thumb/sphx_glr_plot_06_biguesh2006_fig4_thumb.png
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  :doc:`/auto_examples/mimo/plot_06_biguesh2006_fig4`

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      <div class="sphx-glr-thumbnail-title">Biguesh & Gershman 2006, Fig. 4: LS, scaled LS, relaxed MMSE and LMMSE</div>
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Optical
-------

Fibre propagation and its compensation, nonlinear interference, Raman amplification,
optical modulators; optical wireless links (visible light indoors, free-space optics).


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    <div class="sphx-glr-thumbnails">

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.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Standard single-mode fibre spreads a 32 GBd signal over many symbols: D = 17 ps/(nm·km) accumulate 1360 ps/nm over 80 km, and the received 16-QAM constellation collapses into a cloud. Dispersion is an all-pass filter, e^{j \beta_2 \omega^2 L / 2}, so the receiver undoes it exactly in the frequency domain (comnum.cdc), or with an FIR whose length grows with the accumulated dispersion (comnum.cd_fir_taps).">

.. only:: html

  .. image:: /auto_examples/optical/images/thumb/sphx_glr_plot_01_chromatic_dispersion_thumb.png
    :alt:

  :doc:`/auto_examples/optical/plot_01_chromatic_dispersion`

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      <div class="sphx-glr-thumbnail-title">Chromatic dispersion and its compensation</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The classical launch-power sweep of a long-haul optical link: 16-QAM at 32 GBd over 10 spans of 80 km of standard single-mode fibre with an EDFA (noise figure 4.5 dB) after every span, simulated by the split-step Fourier method (1 km steps). At low power the amplified spontaneous emission limits the SNR, which grows by 1 dB per dB of launch power; at high power the Kerr nonlinearity (\gamma = 1.3 /(W·km)) takes over and the SNR falls. The optimum is the maximum of this &quot;bell&quot;.">

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  .. image:: /auto_examples/optical/images/thumb/sphx_glr_plot_02_launch_power_dbp_thumb.png
    :alt:

  :doc:`/auto_examples/optical/plot_02_launch_power_dbp`

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      <div class="sphx-glr-thumbnail-title">Nonlinear fibre propagation and digital back-propagation</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Fig. 5 of C. Häger and H. D. Pfister, &quot;Physics-based deep learning for fiber-optic communication systems&quot;, IEEE J. Sel. Areas Commun. 39(1), 2021 (arXiv:2010.14258), gives the effective SNR against the launch power of a single channel over 25 spans of 80 km of standard fibre at 10.7 GBd with Gaussian symbols, for linear equalisation (dispersion compensation alone) and digital back-propagation (DBP) with one step per span. The link is that of the paper: \beta_2 = -21.683 ps²/km at 194.6 THz, \gamma = 1.3 /W/km, 0.2 dB/km, a noise figure of 5 dB, root-raised-cosine pulses of roll-off 0.1, propagation at 6 samples per symbol and receivers at 2.">

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  .. image:: /auto_examples/optical/images/thumb/sphx_glr_plot_03_hager2021_fig5_thumb.png
    :alt:

  :doc:`/auto_examples/optical/plot_03_hager2021_fig5`

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      <div class="sphx-glr-thumbnail-title">Häger & Pfister 2021, Fig. 5: linear equalisation and back-propagation</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Dual-polarisation channels at 32 GBd propagated together by the Manakov equation (comnum.ManakovSsfm, the split step with the phase at the middle of the step), with EDFAs of noise figure 4.5 dB after every span of 80 km of standard fibre.">

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  .. image:: /auto_examples/optical/images/thumb/sphx_glr_plot_04_wdm_nonlinearity_thumb.png
    :alt:

  :doc:`/auto_examples/optical/plot_04_wdm_nonlinearity`

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      <div class="sphx-glr-thumbnail-title">WDM nonlinearity: the GN model and digital back-propagation</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The pumps of a distributed Raman amplifier transfer their power to the signals about 13 THz below them, all along the span. comnum.raman_profile solves the coupled power equations of stimulated Raman scattering (Bromage 2004) for the signals and any number of co- and counter-propagating pumps, with the measured gain spectrum of standard single-mode fibre.">

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  .. image:: /auto_examples/optical/images/thumb/sphx_glr_plot_05_raman_amplification_thumb.png
    :alt:

  :doc:`/auto_examples/optical/plot_05_raman_amplification`

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      <div class="sphx-glr-thumbnail-title">Raman amplification: pumps, gain flattening and the hybrid link</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A Mach–Zehnder modulator (MZM) splits the optical field into two arms, shifts their phases with the driving voltage and recombines them. Driven in push-pull, the two arms move in opposite directions and the output field is real (no chirp):">

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  .. image:: /auto_examples/optical/images/thumb/sphx_glr_plot_06_mach_zehnder_thumb.png
    :alt:

  :doc:`/auto_examples/optical/plot_06_mach_zehnder`

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      <div class="sphx-glr-thumbnail-title">Mach–Zehnder and IQ modulators</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Light as the medium outside the fibre, in three panels to open next to the figures of the papers they reproduce.">

.. only:: html

  .. image:: /auto_examples/optical/images/thumb/sphx_glr_plot_07_optical_wireless_thumb.png
    :alt:

  :doc:`/auto_examples/optical/plot_07_optical_wireless`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Optical wireless: an LED-lit room and a turbulent free-space link</div>
    </div>


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    </div>

Hardware impairments
--------------------

Power amplifiers and their predistortion, converters.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="A power amplifier compresses the large amplitudes: its AM/AM (and AM/PM) conversion is linear at low power and saturates. Driven by OFDM, whose peaks are rare but high, it creates in-band distortion (EVM) and out-of-band emission (spectral regrowth, measured by the adjacent channel leakage ratio, ACLR).">

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  .. image:: /auto_examples/hardware/images/thumb/sphx_glr_plot_01_amplifier_dpd_thumb.png
    :alt:

  :doc:`/auto_examples/hardware/plot_01_amplifier_dpd`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Power amplifiers and digital predistortion</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A converter of N bits and full scale A rounds every in-phase and quadrature sample to one of 2^N levels spaced by \Delta = 2A/2^N, and clips beyond \pm A. comnum.adc and comnum.dac model it, with an analogue bandwidth, the noise of an effective number of bits (ENOB), the sampling-clock jitter (ADC) and the zero-order hold (DAC); the adc and dac blocks put them in a link.">

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  .. image:: /auto_examples/hardware/images/thumb/sphx_glr_plot_02_converters_thumb.png
    :alt:

  :doc:`/auto_examples/hardware/plot_02_converters`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">DAC and ADC: quantisation, clipping and jitter</div>
    </div>


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    </div>

Receiver DSP
------------

Pulse shaping, channel estimation, synchronisation, blind modulation recognition.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A root-raised-cosine (RRC) transmit filter followed by the same matched filter at the receiver gives a raised-cosine overall response, whose zero crossings at every other symbol instant cancel the intersymbol interference (Nyquist&#x27;s criterion). The roll-off \beta trades bandwidth, (1 + \beta) R_s, against the length of the pulse tails and the sensitivity to timing errors — which the eye diagram shows.">

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  .. image:: /auto_examples/receiver/images/thumb/sphx_glr_plot_01_pulse_shaping_thumb.png
    :alt:

  :doc:`/auto_examples/receiver/plot_01_pulse_shaping`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Nyquist pulse shaping and the eye diagram</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A multipath channel is frequency selective, but OFDM with a cyclic prefix longer than its delay spread sees it as one flat fading coefficient per subcarrier, which a one-tap equaliser removes — provided the receiver knows it. The channel here is the Extended Vehicular A profile of 3GPP TS 36.101 (nine taps, 357 ns RMS delay spread) at a sampling rate of 3.84 MHz, with Clarke/Jakes Doppler (Zheng &amp; Xiao 2003).">

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  .. image:: /auto_examples/receiver/images/thumb/sphx_glr_plot_02_ofdm_channel_estimation_thumb.png
    :alt:

  :doc:`/auto_examples/receiver/plot_02_ofdm_channel_estimation`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">OFDM over multipath: channel estimation</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Timing and frequency. The preamble of Schmidl &amp; Cox (IEEE Trans. Commun. 45(12), 1997) has two identical halves: their correlation peaks — on a plateau as long as the cyclic prefix — at the start of the symbol, and its phase gives the carrier frequency offset, here in units of the subcarrier spacing. The error variance of the offset falls as 1/\mathrm{SNR}.">

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  .. image:: /auto_examples/receiver/images/thumb/sphx_glr_plot_03_sync_mimo_ofdm_thumb.png
    :alt:

  :doc:`/auto_examples/receiver/plot_03_sync_mimo_ofdm`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">OFDM synchronisation and MIMO-OFDM</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A receiver that does not know what it receives (spectrum monitoring, cognitive radio, an adaptive link before its first header) must find the modulation, the SNR and the symbol rate from the samples alone. comnum implements the classical methods:">

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  .. image:: /auto_examples/receiver/images/thumb/sphx_glr_plot_04_modulation_recognition_thumb.png
    :alt:

  :doc:`/auto_examples/receiver/plot_04_modulation_recognition`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Blind receiver: modulation recognition, SNR and symbol rate</div>
    </div>


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.. raw:: html

    </div>

Metrics and analysis
--------------------

The views of a signal and of the results of a campaign.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The tools of the analysis after a campaign: comnum.compare runs variants of a chain on the same grid, comnum.Results plots them with their confidence intervals and a closed form, probes measure inside the link during the campaign, and comnum.Chain.trace with comnum.plot shows the signal after every block.">

.. only:: html

  .. image:: /auto_examples/analysis/images/thumb/sphx_glr_plot_01_analysis_thumb.png
    :alt:

  :doc:`/auto_examples/analysis/plot_01_analysis`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Analysing a link: variants, probes and signals</div>
    </div>


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    </div>

Systems and standards
---------------------

DVB-S, DVB-S2, DVB-T, CCSDS and 5G NR.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="comnum.Dvbs2 implements the physical layer of ETSI EN 302 307-1: the BBFRAME, the BCH and LDPC codes of the standard (the 21 parity-check tables of its annexes), the bit interleaver, the QPSK, 8PSK, 16APSK and 32APSK constellations, and the PLFRAME with its π/2-BPSK header, its pilots and its Gold scrambling.">

.. only:: html

  .. image:: /auto_examples/systems/images/thumb/sphx_glr_plot_01_dvbs2_thumb.png
    :alt:

  :doc:`/auto_examples/systems/plot_01_dvbs2`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">DVB-S2: MODCODs, constellations and frame error rates</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="NR keeps the OFDM of LTE and scales it: the subcarrier spacing is \Delta f = 2^{\mu} \cdot 15 kHz, a 1 ms subframe holds 2^{\mu} slots of 14 symbols, and the cyclic prefix shrinks as 2^{-\mu} (X. Lin et al., &quot;5G New Radio: Unveiling the Essentials of the Next Generation Wireless Access Technology&quot;, IEEE Commun. Standards Mag. 3(3), 2019, Sec. II; \mu = 0 is the numerology of LTE, D. Astély et al., IEEE Commun. Mag. 47(4), 2009). This page builds the pieces a receiver meets first: the numerology (comnum.nr_numerology, comnum.nr_ofdm_modulate), the synchronisation signal block that carries the cell identity (comnum.nr_ssb_grid, comnum.nr_detect_cell_id) and the demodulation reference signals of the data channel (comnum.nr_pdsch_dmrs).">

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  .. image:: /auto_examples/systems/images/thumb/sphx_glr_plot_02_5g_nr_thumb.png
    :alt:

  :doc:`/auto_examples/systems/plot_02_5g_nr`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">5G NR: numerology, synchronisation block and reference signals</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="comnum.Dvbs implements the DVB-S transmitter of ETSI EN 300 421 V1.1.2 and a receiver aligned on the burst: the energy dispersal of the MPEG-2 transport stream (the PRBS 1 + X^{14} + X^{15} reset every 8 packets), the outer code RS(204, 188), the convolutional interleaver of 12 branches (depths 17 j), the inner code of constraint length 7 (X from 171_8, Y from 133_8) punctured to 2/3, 3/4, 5/6 or 7/8 by Table 2, and the Gray QPSK of Figure 5.">

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  .. image:: /auto_examples/systems/images/thumb/sphx_glr_plot_03_dvbs_thumb.png
    :alt:

  :doc:`/auto_examples/systems/plot_03_dvbs`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">DVB-S: the error rate after the Viterbi decoder against Table 3</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="comnum.Dvbt implements the non-hierarchical DVB-T transmitter of ETSI EN 300 744 V1.6.2 and a receiver that knows the channel: the outer coding of DVB-S (energy dispersal, RS(204, 188), the convolutional interleaver), the inner code (171, 133) punctured by Table 2, the inner interleaver (six bit interleavers of 126 bits and the symbol interleaver of the 2K or 8K mode), the constellations of Figure 9, and the OFDM frame of 68 symbols with its pilots and its transmission parameter signalling (TPS).">

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  .. image:: /auto_examples/systems/images/thumb/sphx_glr_plot_04_dvbt_thumb.png
    :alt:

  :doc:`/auto_examples/systems/plot_04_dvbt`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">DVB-T: the frame and the error rate after the Viterbi decoder against Annex A</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="CCSDS 131.0-B-5 (TM Synchronization and Channel Coding, 2023) specifies the channel codes of space telemetry: a convolutional code (Sec. 3), interleaved Reed–Solomon codeblocks (Sec. 4), turbo codes (Sec. 6) and LDPC codes (Sec. 7), then the markers and the randomizer that frame them on the channel (Secs. 9 and 10). comnum reproduces every table the standard prints exactly (card ccsds131_exact): the dual basis of Table F-1, the generators of Annex G, the lengths of Tables 6-2 and 7-5, the parity checks of the LDPC codes, the first bits of the randomizers.">

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  .. image:: /auto_examples/systems/images/thumb/sphx_glr_plot_05_ccsds_thumb.png
    :alt:

  :doc:`/auto_examples/systems/plot_05_ccsds`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">CCSDS: the codes of telemetry against the normal approximation</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The PDSCH of comnum (comnum.NrPdsch, comnum.NrTransportBlock, comnum.NrHarqProcess) run through test 1-2 of the demodulation requirements of 3GPP TS 38.101-4 (clause 5.2.2.1.1): the fixed reference channel R.PDSCH.1-1.2 (6 PRBs of a 10 MHz carrier at 15 kHz, QPSK, MCS 4, a transport block of 480 bits), the channel TDLC300-100 (Table B.2.1.1-4, 100 Hz of Doppler), 2 transmit antennas with a random Type I precoder per bundle of 2 PRBs at every slot, 2 or 4 receive antennas, HARQ with 4 processes and the redundancy versions 0, 2, 3, 1. The metric is the fraction of the maximum throughput; the requirement is 70 % at 0.2 dB with 2 antennas and −2.9 dB with 4.">

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  .. image:: /auto_examples/systems/images/thumb/sphx_glr_plot_06_nr_pdsch_conformance_thumb.png
    :alt:

  :doc:`/auto_examples/systems/plot_06_nr_pdsch_conformance`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">5G NR PDSCH: TS 38.101-4 test 1-2 against the RAN4 alignment</div>
    </div>


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.. raw:: html

    </div>

Simulation and specifications
-----------------------------

Chains of blocks and their campaigns, recordings of measured signals, the confidence
intervals of an error rate.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The declarative link specs/coherent_link.yaml — 16-QAM at 32 GBd over standard fibre, with laser phase noise, a detuned local oscillator, and the standard receiver DSP (dispersion compensation, matched filter, Gardner clock recovery, fourth-power frequency estimation, blind phase search, frame synchronisation) — read as a comnum.Chain.">

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  .. image:: /auto_examples/simulation/images/thumb/sphx_glr_plot_01_chains_thumb.png
    :alt:

  :doc:`/auto_examples/simulation/plot_01_chains`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">A coherent receiver as a chain of blocks</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Three wireless links written as comnum.Chain objects of native blocks and run on the parallel Rust runner.">

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  .. image:: /auto_examples/simulation/images/thumb/sphx_glr_plot_02_wireless_chains_thumb.png
    :alt:

  :doc:`/auto_examples/simulation/plot_02_wireless_chains`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Wireless links as chains: diversity, OFDM channel estimation, coded modulation</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="SigMF (the Signal Metadata Format, &lt;https://github.com/sigmf/SigMF&gt;) is the open format of the software-defined radio community: a JSON file name.sigmf-meta (the sample format, the sampling rate, the centre frequency of every capture, annotations) next to the raw samples name.sigmf-data, or both in a .sigmf archive. GNU Radio, inspectrum, SDRangel and most SDR tools read and write it.">

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  .. image:: /auto_examples/simulation/images/thumb/sphx_glr_plot_03_sigmf_recordings_thumb.png
    :alt:

  :doc:`/auto_examples/simulation/plot_03_sigmf_recordings`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Recorded signals: SigMF in, blind receiver out</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A Monte Carlo campaign estimates an error rate p from k errors among n trials and reports it with a confidence interval. The level of the interval, 95 % here, is only nominal: the probability that the interval actually contains p, its coverage, depends on n and p. For X \sim \mathrm{Bin}(n, p) it is the exact sum">

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  .. image:: /auto_examples/simulation/images/thumb/sphx_glr_plot_04_brown2001_fig5_thumb.png
    :alt:

  :doc:`/auto_examples/simulation/plot_04_brown2001_fig5`

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      <div class="sphx-glr-thumbnail-title">Coverage of the confidence intervals of an error rate</div>
    </div>


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    </div>


.. toctree::
   :hidden:
   :includehidden:


   /auto_examples/foundations/index.rst
   /auto_examples/coding/index.rst
   /auto_examples/waveforms/index.rst
   /auto_examples/mimo/index.rst
   /auto_examples/optical/index.rst
   /auto_examples/hardware/index.rst
   /auto_examples/receiver/index.rst
   /auto_examples/analysis/index.rst
   /auto_examples/systems/index.rst
   /auto_examples/simulation/index.rst



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

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