

.. _sphx_glr_gallery_pylab_examples_psd_demo.py:


========
Psd Demo
========

Plotting Power Spectral Density (PSD) in Matplotlib.

The PSD is a common plot in the field of signal processing. NumPy has
many useful libraries for computing a PSD. Below we demo a few examples
of how this can be accomplished and visualized with Matplotlib.



.. code-block:: python

    import matplotlib.pyplot as plt
    import numpy as np
    import matplotlib.mlab as mlab

    # Fixing random state for reproducibility
    np.random.seed(19680801)


    dt = 0.01
    t = np.arange(0, 10, dt)
    nse = np.random.randn(len(t))
    r = np.exp(-t/0.05)

    cnse = np.convolve(nse, r)*dt
    cnse = cnse[:len(t)]
    s = 0.1*np.sin(2*np.pi*t) + cnse

    plt.subplot(211)
    plt.plot(t, s)
    plt.subplot(212)
    plt.psd(s, 512, 1/dt)

    plt.show()




.. image:: /gallery/pylab_examples/images/sphx_glr_psd_demo_001.png
    :align: center




Compare this with the equivalent Matlab code to accomplish the same thing::

    dt = 0.01;
    t = [0:dt:10];
    nse = randn(size(t));
    r = exp(-t/0.05);
    cnse = conv(nse, r)*dt;
    cnse = cnse(1:length(t));
    s = 0.1*sin(2*pi*t) + cnse;

    subplot(211)
    plot(t,s)
    subplot(212)
    psd(s, 512, 1/dt)

Below we'll show a slightly more complex example that demonstrates
how padding affects the resulting PSD.



.. code-block:: python


    dt = np.pi / 100.
    fs = 1. / dt
    t = np.arange(0, 8, dt)
    y = 10. * np.sin(2 * np.pi * 4 * t) + 5. * np.sin(2 * np.pi * 4.25 * t)
    y = y + np.random.randn(*t.shape)

    # Plot the raw time series
    fig = plt.figure()
    fig.subplots_adjust(hspace=0.45, wspace=0.3)
    ax = fig.add_subplot(2, 1, 1)
    ax.plot(t, y)

    # Plot the PSD with different amounts of zero padding. This uses the entire
    # time series at once
    ax2 = fig.add_subplot(2, 3, 4)
    ax2.psd(y, NFFT=len(t), pad_to=len(t), Fs=fs)
    ax2.psd(y, NFFT=len(t), pad_to=len(t)*2, Fs=fs)
    ax2.psd(y, NFFT=len(t), pad_to=len(t)*4, Fs=fs)
    plt.title('zero padding')

    # Plot the PSD with different block sizes, Zero pad to the length of the
    # original data sequence.
    ax3 = fig.add_subplot(2, 3, 5, sharex=ax2, sharey=ax2)
    ax3.psd(y, NFFT=len(t), pad_to=len(t), Fs=fs)
    ax3.psd(y, NFFT=len(t)//2, pad_to=len(t), Fs=fs)
    ax3.psd(y, NFFT=len(t)//4, pad_to=len(t), Fs=fs)
    ax3.set_ylabel('')
    plt.title('block size')

    # Plot the PSD with different amounts of overlap between blocks
    ax4 = fig.add_subplot(2, 3, 6, sharex=ax2, sharey=ax2)
    ax4.psd(y, NFFT=len(t)//2, pad_to=len(t), noverlap=0, Fs=fs)
    ax4.psd(y, NFFT=len(t)//2, pad_to=len(t), noverlap=int(0.05*len(t)/2.), Fs=fs)
    ax4.psd(y, NFFT=len(t)//2, pad_to=len(t), noverlap=int(0.2*len(t)/2.), Fs=fs)
    ax4.set_ylabel('')
    plt.title('overlap')

    plt.show()





.. image:: /gallery/pylab_examples/images/sphx_glr_psd_demo_002.png
    :align: center




This is a ported version of a MATLAB example from the signal
processing toolbox that showed some difference at one time between
Matplotlib's and MATLAB's scaling of the PSD.



.. code-block:: python


    fs = 1000
    t = np.linspace(0, 0.3, 301)
    A = np.array([2, 8]).reshape(-1, 1)
    f = np.array([150, 140]).reshape(-1, 1)
    xn = (A * np.sin(2 * np.pi * f * t)).sum(axis=0)
    xn += 5 * np.random.randn(*t.shape)

    fig, (ax0, ax1) = plt.subplots(ncols=2)

    fig.subplots_adjust(hspace=0.45, wspace=0.3)
    yticks = np.arange(-50, 30, 10)
    yrange = (yticks[0], yticks[-1])
    xticks = np.arange(0, 550, 100)

    ax0.psd(xn, NFFT=301, Fs=fs, window=mlab.window_none, pad_to=1024,
            scale_by_freq=True)
    ax0.set_title('Periodogram')
    ax0.set_yticks(yticks)
    ax0.set_xticks(xticks)
    ax0.grid(True)
    ax0.set_ylim(yrange)

    ax1.psd(xn, NFFT=150, Fs=fs, window=mlab.window_none, pad_to=512, noverlap=75,
            scale_by_freq=True)
    ax1.set_title('Welch')
    ax1.set_xticks(xticks)
    ax1.set_yticks(yticks)
    ax1.set_ylabel('')  # overwrite the y-label added by `psd`
    ax1.grid(True)
    ax1.set_ylim(yrange)

    plt.show()




.. image:: /gallery/pylab_examples/images/sphx_glr_psd_demo_003.png
    :align: center




This is a ported version of a MATLAB example from the signal
processing toolbox that showed some difference at one time between
Matplotlib's and MATLAB's scaling of the PSD.

It uses a complex signal so we can see that complex PSD's work properly.



.. code-block:: python


    prng = np.random.RandomState(19680801)  # to ensure reproducibility

    fs = 1000
    t = np.linspace(0, 0.3, 301)
    A = np.array([2, 8]).reshape(-1, 1)
    f = np.array([150, 140]).reshape(-1, 1)
    xn = (A * np.exp(2j * np.pi * f * t)).sum(axis=0) + 5 * prng.randn(*t.shape)

    fig, (ax0, ax1) = plt.subplots(ncols=2)

    fig.subplots_adjust(hspace=0.45, wspace=0.3)
    yticks = np.arange(-50, 30, 10)
    yrange = (yticks[0], yticks[-1])
    xticks = np.arange(-500, 550, 200)

    ax0.psd(xn, NFFT=301, Fs=fs, window=mlab.window_none, pad_to=1024,
            scale_by_freq=True)
    ax0.set_title('Periodogram')
    ax0.set_yticks(yticks)
    ax0.set_xticks(xticks)
    ax0.grid(True)
    ax0.set_ylim(yrange)

    ax1.psd(xn, NFFT=150, Fs=fs, window=mlab.window_none, pad_to=512, noverlap=75,
            scale_by_freq=True)
    ax1.set_title('Welch')
    ax1.set_xticks(xticks)
    ax1.set_yticks(yticks)
    ax1.set_ylabel('')  # overwrite the y-label added by `psd`
    ax1.grid(True)
    ax1.set_ylim(yrange)

    plt.show()



.. image:: /gallery/pylab_examples/images/sphx_glr_psd_demo_004.png
    :align: center




**Total running time of the script:** ( 0 minutes  0.282 seconds)



.. container:: sphx-glr-footer


  .. container:: sphx-glr-download

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



  .. container:: sphx-glr-download

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

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