.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "intro/scipy/auto_examples/plot_fftpack.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

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

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

.. _sphx_glr_intro_scipy_auto_examples_plot_fftpack.py:


=============================================
Plotting and manipulating FFTs for filtering
=============================================

Plot the power of the FFT of a signal and inverse FFT back to reconstruct
a signal.

This example demonstrate :func:`scipy.fftpack.fft`,
:func:`scipy.fftpack.fftfreq` and :func:`scipy.fftpack.ifft`. It
implements a basic filter that is very suboptimal, and should not be
used.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: default


    import numpy as np
    import scipy as sp
    import matplotlib.pyplot as plt








.. GENERATED FROM PYTHON SOURCE LINES 21-23

Generate the signal
###########################################################

.. GENERATED FROM PYTHON SOURCE LINES 23-37

.. code-block:: default


    # Seed the random number generator
    np.random.seed(1234)

    time_step = 0.02
    period = 5.

    time_vec = np.arange(0, 20, time_step)
    sig = (np.sin(2 * np.pi / period * time_vec)
           + 0.5 * np.random.randn(time_vec.size))

    plt.figure(figsize=(6, 5))
    plt.plot(time_vec, sig, label='Original signal')




.. image-sg:: /intro/scipy/auto_examples/images/sphx_glr_plot_fftpack_001.png
   :alt: plot fftpack
   :srcset: /intro/scipy/auto_examples/images/sphx_glr_plot_fftpack_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    [<matplotlib.lines.Line2D object at 0x7fe7b0790160>]



.. GENERATED FROM PYTHON SOURCE LINES 38-40

Compute and plot the power
###########################################################

.. GENERATED FROM PYTHON SOURCE LINES 40-74

.. code-block:: default


    # The FFT of the signal
    sig_fft = sp.fftpack.fft(sig)

    # And the power (sig_fft is of complex dtype)
    power = np.abs(sig_fft)**2

    # The corresponding frequencies
    sample_freq = sp.fftpack.fftfreq(sig.size, d=time_step)

    # Plot the FFT power
    plt.figure(figsize=(6, 5))
    plt.plot(sample_freq, power)
    plt.xlabel('Frequency [Hz]')
    plt.ylabel('plower')

    # Find the peak frequency: we can focus on only the positive frequencies
    pos_mask = np.where(sample_freq > 0)
    freqs = sample_freq[pos_mask]
    peak_freq = freqs[power[pos_mask].argmax()]

    # Check that it does indeed correspond to the frequency that we generate
    # the signal with
    np.allclose(peak_freq, 1./period)

    # An inner plot to show the peak frequency
    axes = plt.axes([0.55, 0.3, 0.3, 0.5])
    plt.title('Peak frequency')
    plt.plot(freqs[:8], power[pos_mask][:8])
    plt.setp(axes, yticks=[])

    # scipy.signal.find_peaks_cwt can also be used for more advanced
    # peak detection




.. image-sg:: /intro/scipy/auto_examples/images/sphx_glr_plot_fftpack_002.png
   :alt: Peak frequency
   :srcset: /intro/scipy/auto_examples/images/sphx_glr_plot_fftpack_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    []



.. GENERATED FROM PYTHON SOURCE LINES 75-80

Remove all the high frequencies
###########################################################

 We now remove all the high frequencies and transform back from
 frequencies to signal.

.. GENERATED FROM PYTHON SOURCE LINES 80-93

.. code-block:: default


    high_freq_fft = sig_fft.copy()
    high_freq_fft[np.abs(sample_freq) > peak_freq] = 0
    filtered_sig = sp.fftpack.ifft(high_freq_fft)

    plt.figure(figsize=(6, 5))
    plt.plot(time_vec, sig, label='Original signal')
    plt.plot(time_vec, filtered_sig, linewidth=3, label='Filtered signal')
    plt.xlabel('Time [s]')
    plt.ylabel('Amplitude')

    plt.legend(loc='best')




.. image-sg:: /intro/scipy/auto_examples/images/sphx_glr_plot_fftpack_003.png
   :alt: plot fftpack
   :srcset: /intro/scipy/auto_examples/images/sphx_glr_plot_fftpack_003.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    /opt/buildhome/python3.8/lib/python3.8/site-packages/matplotlib/cbook/__init__.py:1335: ComplexWarning: Casting complex values to real discards the imaginary part
      return np.asarray(x, float)

    <matplotlib.legend.Legend object at 0x7fe7b042c100>



.. GENERATED FROM PYTHON SOURCE LINES 94-98

**Note** This is actually a bad way of creating a filter: such brutal
cut-off in frequency space does not control distorsion on the signal.

Filters should be created using the SciPy filter design code

.. GENERATED FROM PYTHON SOURCE LINES 99-100

.. code-block:: default

    plt.show()








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

   **Total running time of the script:** ( 0 minutes  0.194 seconds)


.. _sphx_glr_download_intro_scipy_auto_examples_plot_fftpack.py:

.. only:: html

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




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

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

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

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


.. only:: html

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

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