.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/waveforms/plot_04_proakis2008_msk.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_04_proakis2008_msk.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_waveforms_plot_04_proakis2008_msk.py:


Proakis & Salehi 2008, Sec. 3.3-2: MSK has the error rate of BPSK
=================================================================

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,
:math:`Q(\sqrt{2 E_b/N_0})`, once the bits are precoded so that each decides a sign
of the equivalent offset QPSK signal.

Without precoding, the information bits are the phase increments, and a symbol-by-
symbol decision on the phase states of the maximum-likelihood path decides each bit
from two consecutive phases: an error in one phase affects two bits, and the BER is
that of differentially decoded BPSK, :math:`2p(1 - p)` with :math:`p` the BER of
BPSK. The blocks ``cpm_tx`` and ``cpm_rx`` run the maximum-likelihood sequence
detector on the phase trellis in both cases; ``specs/validation/msk_bpsk.yaml``
checks the precoded case in CI within 4 % at 2, 4 and 6 dB.

.. GENERATED FROM PYTHON SOURCE LINES 20-84

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    import comnum
    from comnum.plot import BACKGROUND, MUTED

    EBN0 = np.arange(0.0, 9.0, 1.0)
    results = comnum.run_spec(
        {
            "name": "msk_bpsk",
            "seed": 3,
            "cases": [{"precode": True}, {"precode": False}],
            "grid": {"ebn0_db": list(EBN0)},
            "link": [
                {"type": "source", "constellation": "BPSK", "n_symbols": 4000},
                {
                    "type": "cpm_tx",
                    "h": "1/2",
                    "pulse": "rec",
                    "length": 1,
                    "sps": 4,
                    "precode": "$precode",
                },
                {"type": "awgn_ebn0", "ebn0_db": "$ebn0_db"},
                {
                    "type": "cpm_rx",
                    "h": "1/2",
                    "pulse": "rec",
                    "length": 1,
                    "sps": 4,
                    "detector": "ml",
                    "precode": "$precode",
                },
            ],
            "metrics": {"report": ["ber"], "scale": "none"},
            "stop": {"min_errors": {"ber": 400}, "min_trials": 2, "max_trials": 2000},
        }
    )


    def bpsk(ebn0_db):
        return comnum.theory_ber(2, ebn0_db, "psk")


    fig, ax = plt.subplots()
    fine = np.linspace(0, 8, 81)
    ax.semilogy(fine, bpsk(fine), color=MUTED, linewidth=1.4, label="BPSK, $Q(\\sqrt{2E_b/N_0})$")
    ax.semilogy(fine, 2 * bpsk(fine) * (1 - bpsk(fine)), ":", color=MUTED, linewidth=1.4,
                label="differentially decoded BPSK, $2p(1-p)$")  # fmt: skip
    for precode, marker, label in [(True, "o", "MSK, precoded"), (False, "s", "MSK, not precoded")]:
        _, ber, lo, hi = results.series("ebn0_db", "ber", interval=True, precode=precode)
        ax.errorbar(EBN0, ber, yerr=[ber - lo, hi - ber], fmt=marker, markeredgecolor=BACKGROUND,
                    capsize=0, label=label)  # fmt: skip
    ax.set(
        xlim=(0, 8),
        ylim=(1e-4, 0.5),
        xlabel="$E_b/N_0$ (dB)",
        ylabel="bit error rate",
        title="MSK with maximum-likelihood sequence detection over AWGN",
    )
    ax.legend(loc="lower left")
    fig.tight_layout()




.. image-sg:: /auto_examples/waveforms/images/sphx_glr_plot_04_proakis2008_msk_001.png
   :alt: MSK with maximum-likelihood sequence detection over AWGN
   :srcset: /auto_examples/waveforms/images/sphx_glr_plot_04_proakis2008_msk_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 85-86

The precoded points lie on the BPSK curve within their intervals.

.. GENERATED FROM PYTHON SOURCE LINES 86-88

.. code-block:: Python

    for e, ber in zip(*results.series("ebn0_db", "ber", precode=True), strict=True):
        print(f"Eb/N0 = {e:.0f} dB: MSK {ber:.3e}, BPSK {bpsk(e):.3e}")




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

 .. code-block:: none

    Eb/N0 = 0 dB: MSK 7.935e-02, BPSK 7.865e-02
    Eb/N0 = 1 dB: MSK 5.653e-02, BPSK 5.628e-02
    Eb/N0 = 2 dB: MSK 3.664e-02, BPSK 3.751e-02
    Eb/N0 = 3 dB: MSK 2.270e-02, BPSK 2.288e-02
    Eb/N0 = 4 dB: MSK 1.250e-02, BPSK 1.250e-02
    Eb/N0 = 5 dB: MSK 5.898e-03, BPSK 5.954e-03
    Eb/N0 = 6 dB: MSK 2.266e-03, BPSK 2.388e-03
    Eb/N0 = 7 dB: MSK 8.418e-04, BPSK 7.727e-04
    Eb/N0 = 8 dB: MSK 1.992e-04, BPSK 1.909e-04





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

   **Total running time of the script:** (0 minutes 0.682 seconds)


.. _sphx_glr_download_auto_examples_waveforms_plot_04_proakis2008_msk.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_04_proakis2008_msk.ipynb <plot_04_proakis2008_msk.ipynb>`

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

      :download:`Download Python source code: plot_04_proakis2008_msk.py <plot_04_proakis2008_msk.py>`

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

      :download:`Download zipped: plot_04_proakis2008_msk.zip <plot_04_proakis2008_msk.zip>`


.. only:: html

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

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