.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/spectro/nmr/plot_04_fourier_transform_nmr.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_spectro_nmr_plot_04_fourier_transform_nmr.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_spectro_nmr_plot_04_fourier_transform_nmr.py:


Fourier-transform NMR: from a free-induction decay to a spectrum
=================================================================

Ernst and Anderson (1966) replaced the slow frequency sweep of
continuous-wave NMR with a single short radio-frequency pulse. The pulse
excites all resonances at once. The resulting free-induction decay
(FID) is a sum of decaying oscillations, one per resonance, and its
Fourier transform gives the whole spectrum. Because each FID takes about
as long as a single scan and many can be averaged, the method is far
more sensitive. This example simulates a noisy FID for three
resonances with
:func:`~chemistrykit.spectro.systems.nmr.free_induction_decay`,
transforms it with :func:`~chemistrykit.spectro.systems.nmr.fid_to_spectrum`,
and shows signal averaging: signal-to-noise grows as
:math:`\sqrt{N}` over N co-added scans.

.. GENERATED FROM PYTHON SOURCE LINES 20-49

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.spectro.systems.nmr import fid_to_spectrum, free_induction_decay

    rng = np.random.default_rng(1966)
    dwell, n_points, t2 = 5e-4, 8192, 0.15
    t = np.arange(n_points) * dwell
    offsets = np.array([-300.0, 120.0, 450.0])  # Hz from the carrier
    amplitudes = np.array([3.0, 2.0, 1.0])
    clean = free_induction_decay(t, offsets, amplitudes, t2)


    def noisy_scan():
        """One acquisition: the ideal FID plus complex white noise."""
        return clean + 1.5 * (rng.standard_normal(n_points) + 1j * rng.standard_normal(n_points))


    single = noisy_scan()
    averaged = np.mean([noisy_scan() for _ in range(64)], axis=0)

    freqs, spec_single = fid_to_spectrum(single, dwell)
    _, spec_avg = fid_to_spectrum(averaged, dwell)

    for offset in offsets:
        window = np.abs(freqs - offset) < 10.0
        print(f"line at {offset:+.0f} Hz recovered at {freqs[window][np.argmax(spec_avg[window])]:+.2f} Hz")
    print(f"Lorentzian FWHM expected 1/(pi T2) = {1.0 / (np.pi * t2):.2f} Hz")





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

 .. code-block:: none

    line at -300 Hz recovered at -300.05 Hz
    line at +120 Hz recovered at +120.12 Hz
    line at +450 Hz recovered at +449.95 Hz
    Lorentzian FWHM expected 1/(pi T2) = 2.12 Hz




.. GENERATED FROM PYTHON SOURCE LINES 50-52

Signal-to-noise ratio: peak height over the standard deviation of a
signal-free region. Averaging 64 scans should improve it by about 8.

.. GENERATED FROM PYTHON SOURCE LINES 52-62

.. code-block:: Python


    quiet = np.abs(freqs) > 700.0


    def snr(spec):
        return spec.max() / spec[quiet].std()


    print(f"SNR single scan: {snr(spec_single):.1f}; 64 scans: {snr(spec_avg):.1f}; ratio {snr(spec_avg) / snr(spec_single):.1f} (sqrt(64) = 8)")





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

 .. code-block:: none

    SNR single scan: 7.7; 64 scans: 54.5; ratio 7.1 (sqrt(64) = 8)




.. GENERATED FROM PYTHON SOURCE LINES 63-78

.. code-block:: Python

    fig, axes = plt.subplots(3, 1, figsize=(10, 8))
    axes[0].plot(t[:1600], single.real[:1600], color="0.5", lw=0.5, label="single scan")
    axes[0].plot(t[:1600], clean.real[:1600], color="black", lw=0.8, label="noise-free FID")
    axes[0].set_xlabel("time (s)")
    axes[0].set_title("Free-induction decay (real part)")
    axes[0].legend(loc="upper right")
    axes[1].plot(freqs, spec_single, color="tab:blue", lw=0.7)
    axes[1].set_title("Fourier transform of one scan")
    axes[2].plot(freqs, spec_avg, color="tab:green", lw=0.7)
    axes[2].set_title("Fourier transform of 64 co-added scans")
    for ax in axes[1:]:
        ax.set_xlim(-800.0, 800.0)
        ax.set_xlabel("frequency offset (Hz)")
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/spectro/nmr/images/sphx_glr_plot_04_fourier_transform_nmr_001.png
   :alt: Free-induction decay (real part), Fourier transform of one scan, Fourier transform of 64 co-added scans
   :srcset: /api/gallery/spectro/nmr/images/sphx_glr_plot_04_fourier_transform_nmr_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.117 seconds)


.. _sphx_glr_download_api_gallery_spectro_nmr_plot_04_fourier_transform_nmr.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_fourier_transform_nmr.ipynb <plot_04_fourier_transform_nmr.ipynb>`

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

      :download:`Download Python source code: plot_04_fourier_transform_nmr.py <plot_04_fourier_transform_nmr.py>`

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

      :download:`Download zipped: plot_04_fourier_transform_nmr.zip <plot_04_fourier_transform_nmr.zip>`


.. only:: html

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

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