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{"@context":"https://schema.org","@type":"DiscussionForumPosting","headline":"Scaling of `mlab.magnitude_spectrum()` is inconsistent","articleBody":"Compared to `signal.welch()` and `mlab.psd()`, the scaling of `mlab.magnitude_spectrum()` is not consistent. The normalization of the window seems to be missing, as the following example shows:\r\n\r\n![spectrum_scaling](https://cloud.githubusercontent.com/assets/12721170/24580072/395e14c0-1702-11e7-97da-b79c32451873.png)\r\n\r\n    import numpy as np\r\n    import matplotlib.pyplot as plt\r\n    from matplotlib import mlab\r\n    from scipy import signal\r\n    \r\n    tau, n = 10, 2**10  # 10 second signal with 1024 points\r\n    T = tau/n  # sampling interval\r\n    \r\n    t = np.arange(n)*T\r\n    x = 4*np.sin(40*np.pi*t)  # 20 Hz sine\r\n    w = signal.hanning(n, False)\r\n    \r\n    f = np.fft.fftfreq(n, T)  # frequency slots\r\n    df = f[1] - f[0]\r\n    X = np.fft.fft(x*w)/w.sum()  # Windowed FFT scaled to amplitude\r\n    \r\n    f_Pxx, Pxx = signal.welch(x, fs=1/T, window='hanning', nperseg=n, noverlap=0,\r\n                              detrend=False, return_onesided=False,\r\n                              scaling='spectrum')\r\n    P_psd, f_psd = mlab.psd(x, NFFT=n, Fs=1/T,  window=w,\r\n                            sides='twosided', scale_by_freq=False)\r\n    \r\n    # shift for plotting:\r\n    f, X, f_Pxx, Pxx = (np.fft.fftshift(zz) for zz in (f, X, f_Pxx, Pxx))\r\n    \r\n    fg0, axx0 = plt.subplots(4, 1, sharex=True, num=1, figsize=(7, 7), clear=True)\r\n    \r\n    axx0[0].set_title(r\"FFT Magnitude Spectrum of $x(t)=4\\sin(40\\pi t)$\")\r\n    axx0[0].plot(f, np.abs(X), 'C0.-')\r\n    \r\n    axx0[1].set_title(\"Scipy's Welch Algorithm\")\r\n    axx0[1].plot(f_Pxx, np.sqrt(Pxx), \"C1.-\")\r\n    \r\n    axx0[2].set_title(\"psd()\")\r\n    axx0[2].plot(f_psd, np.sqrt(P_psd), \"C2.-\")\r\n    \r\n    axx0[3].set_title(\"magnitude_spectrum()\")\r\n    axx0[3].magnitude_spectrum(x, Fs=1/T, sides='twosided', color=\"C3\", marker='.',\r\n                               scale='linear')\r\n    # Correct result:\r\n    # axx0[3].magnitude_spectrum(x, Fs=1/T, sides='twosided', color=\"C3\", marker='.',\r\n    #                           scale='linear', window=w/w.sum())\r\n    \r\n    axx0[-1].set_xlabel(r\"Frequency $f$\")\r\n    axx0[0].set_xlim(19, 21)\r\n    for ax in axx0:\r\n        ax.grid(True)\r\n        ax.set_ylabel(r\"$|X(f)|$\")\r\n    fg0.tight_layout()\r\n    plt.show()\r\n\r\nI tried fixing it, but I could find the time to fix `mlab._spectral_helper()` to perform consistently with `mlab.magnitude_spectrum()` and `mlab.spegram()`.\r\n\r\nPS: Tested on Matplotlib 2.0.0 (Anaconda Python 3.6 on Debian).","author":{"url":"https://github.com/DietBru","@type":"Person","name":"DietBru"},"datePublished":"2017-04-01T15:51:12.000Z","interactionStatistic":{"@type":"InteractionCounter","interactionType":"https://schema.org/CommentAction","userInteractionCount":0},"url":"https://github.com/8417/matplotlib/issues/8417"}