.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/receiver/plot_03_sync_mimo_ofdm.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_receiver_plot_03_sync_mimo_ofdm.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_receiver_plot_03_sync_mimo_ofdm.py:


OFDM synchronisation and MIMO-OFDM
==================================

**Timing and frequency.** The preamble of Schmidl & 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 :math:`1/\mathrm{SNR}`.

**MIMO-OFDM.** With two transmit and two receive antennas over the EPA profile, every
subcarrier carries a flat 2 × 2 MIMO channel; the detectors of the MIMO module run
subcarrier by subcarrier (Stüber et al., *Proc. IEEE* 92(2), 2004). ML (here the sphere
decoder) collects the receive diversity that the linear detectors lose.

.. GENERATED FROM PYTHON SOURCE LINES 16-73

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np
    from _gallery_utils import mimo_ofdm_ser

    import comnum
    from comnum.plot import AMBER, BACKGROUND, GREY

    ofdm = comnum.Ofdm(128, 100, n_cp=16)
    c = comnum.Constellation("qam", 4).normalized()
    pre = comnum.ofdm_preamble(ofdm, seed=3)
    rng = comnum.Rng(1)
    data = ofdm.modulate(c.symbols(rng.integers(4, 2 * 100)))
    r = np.concatenate([np.zeros(300), pre, data])
    r = r + rng.complex_noise(r.size, 0.1)
    metric = comnum.schmidl_cox(r, ofdm.fft_len // 2)

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(9.5, 3.8))
    ax1.plot(metric, linewidth=1.4)
    ax1.axvspan(300, 300 + 4 * 16, color=AMBER, alpha=0.25, label="cyclic prefix")
    ax1.set(xlabel="sample", ylabel="M(d)", title="Schmidl & Cox timing metric, 10 dB", xlim=(0, 1100))
    ax1.legend()


    def cfo_rmse(snr_db, trials=200):
        s2 = comnum.db2lin(-snr_db)
        err = []
        for t in range(trials):
            g = comnum.Rng.stream(2, "cfo", int(snr_db + 100), t)
            eps = g.uniform(1)[0] - 0.5
            d = ofdm.modulate(c.symbols(g.integers(4, 100)))
            x = np.concatenate([np.zeros(150), pre, d])
            x = (x + g.complex_noise(x.size, s2)) * np.exp(
                2j * np.pi * eps * np.arange(x.size) / ofdm.fft_len
            )
            err.append(comnum.ofdm_synchronize(x, ofdm)["cfo"] - eps)
        return np.sqrt(np.mean(np.square(err)))


    snrs = np.arange(-5.0, 26.0, 5.0)
    ax2.semilogy(
        snrs, [cfo_rmse(s) for s in snrs], marker="o", markeredgecolor=BACKGROUND, label="simulated"
    )
    ax2.semilogy(
        snrs,
        cfo_rmse(10.0) * 10 ** (-(snrs - 10) / 20),
        color=GREY,
        linestyle=":",
        linewidth=1.2,
        label=r"$\propto 1/\sqrt{\mathrm{SNR}}$",
    )
    ax2.set(
        xlabel="SNR (dB)", ylabel="RMS error (subcarrier spacings)", title="frequency offset estimate"
    )
    ax2.legend()
    fig.tight_layout()




.. image-sg:: /auto_examples/receiver/images/sphx_glr_plot_03_sync_mimo_ofdm_001.png
   :alt: Schmidl & Cox timing metric, 10 dB, frequency offset estimate
   :srcset: /auto_examples/receiver/images/sphx_glr_plot_03_sync_mimo_ofdm_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 74-75

2 × 2 MIMO-OFDM over EPA (1.92 MHz, 96 of 128 subcarriers), perfect CSI.

.. GENERATED FROM PYTHON SOURCE LINES 75-102

.. code-block:: Python

    mofdm = comnum.Ofdm(128, 96, oversampling=1, n_cp=16)
    ch = comnum.TdlChannel("epa", 1.92e6, n_rx=2, n_tx=2)


    def ser(snr_db, method, frames=150):
        """Four OFDM symbols per frame, one channel per frame, per-subcarrier detection
        (shared with the test that checks these curves)."""
        return mimo_ofdm_ser(mofdm, ch, c, snr_db, method, frames, seed=3, condition=int(snr_db))


    fig, ax = plt.subplots()
    snr = np.arange(0.0, 31.0, 5.0)
    for (name, m), marker in zip(
        {"ZF": "zf", "MMSE": "mmse", "MMSE-SIC": "mmse_sic", "ML (sphere)": "sphere"}.items(),
        ["o", "s", "D", "^"],
        strict=True,
    ):
        v = np.array([ser(s, m) for s in snr])
        keep = v > 0
        ax.semilogy(snr[keep], v[keep], marker=marker, markeredgecolor=BACKGROUND, label=name)
    ax.set(
        xlabel=r"$E_s/N_0$ per receive antenna (dB)",
        ylabel="symbol error rate",
        title="2 × 2 MIMO-OFDM, QPSK, EPA",
    )
    ax.legend()
    fig.tight_layout()



.. image-sg:: /auto_examples/receiver/images/sphx_glr_plot_03_sync_mimo_ofdm_002.png
   :alt: 2 × 2 MIMO-OFDM, QPSK, EPA
   :srcset: /auto_examples/receiver/images/sphx_glr_plot_03_sync_mimo_ofdm_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 1.253 seconds)


.. _sphx_glr_download_auto_examples_receiver_plot_03_sync_mimo_ofdm.py:

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