.. _sphx_glr_auto_examples_foundations:

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
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  :doc:`/auto_examples/foundations/plot_01_constellations`

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      <div class="sphx-glr-thumbnail-title">Constellations and Gray labels</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
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  :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>
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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
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  :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>
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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
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  :doc:`/auto_examples/foundations/plot_04_rayleigh`

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


.. toctree::
   :hidden:

   /auto_examples/foundations/plot_01_constellations
   /auto_examples/foundations/plot_02_ber_qam_awgn
   /auto_examples/foundations/plot_03_ber_psk
   /auto_examples/foundations/plot_04_rayleigh
   /auto_examples/foundations/plot_05_information_rates