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scipy.signal.

# welch

scipy.signal.welch(x, fs=1.0, window='hann_periodic', nperseg=None, noverlap=None, nfft=None, detrend='constant', return_onesided=True, scaling='density', axis=-1, average='mean')

  Estimate power spectral density using Welch’s method.

  Welch’s method [1] computes an estimate of the power spectral density by dividing the data into overlapping segments, computing a modified periodogram for each segment and averaging the periodograms.

  Parameters:
    x array_like

      Time series of measurement values

    fs float, optional

      Sampling frequency of the x time series. Defaults to 1.0.

    window str or tuple or array_like, optional

      Desired window to use. If window is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. See `get_window` for a list of windows and required parameters. If window is array_like it will be used directly as the window and its length must be nperseg. Defaults to a periodic Hann window.

    nperseg int, optional

      Length of each segment. Defaults to None, but if window is str or tuple, is set to 256, and if window is array_like, is set to the length of the window.

    noverlap int, optional

      Number of points to overlap between segments. If None, `noverlap = nperseg // 2`. Defaults to None.

    nfft int, optional

      Length of the FFT used, if a zero padded FFT is desired. If None, the FFT length is nperseg. Defaults to None.

    detrend str or function or False, optional

      Specifies how to detrend each segment. If `detrend` is a string, it is passed as the type argument to the `detrend` function. If it is a function, it takes a segment and returns a detrended segment. If `detrend` is False, no detrending is done. Defaults to ‘constant’.

    return_onesided bool, optional

      If True, return a one-sided spectrum for real data. If False return a two-sided spectrum. Defaults to True, but for complex data, a two-sided spectrum is always returned.

    scaling{ ‘density’, ‘spectrum’ }, optional

      Selects between computing the power spectral density (‘density’) where Pxx has units of V**2/Hz and computing the squared magnitude spectrum (‘spectrum’) where Pxx has units of V**2, if x is measured in V and fs is measured in Hz. Defaults to ‘density’

    axis int, optional

      Axis along which the periodogram is computed; the default is over the last axis (i.e. `axis=-1`).

    average{ ‘mean’, ‘median’ }, optional

      Method to use when averaging periodograms. Defaults to ‘mean’.

      Added in version 1.2.0.

  Returns:
    f ndarray

      Array of sample frequencies.

    Pxx ndarray

      Power spectral density or power spectrum of x.

  See also

  `csd`

    Cross power spectral density using Welch’s method

  `periodogram`

    Simple, optionally modified periodogram

  `lombscargle`

    Lomb-Scargle periodogram for unevenly sampled data

  Notes

  An appropriate amount of overlap will depend on the choice of window and on your requirements. For the default Hann window an overlap of 50% is a reasonable trade-off between accurately estimating the signal power, while not over counting any of the data. Narrower windows may require a larger overlap. If noverlap is 0, this method is equivalent to Bartlett’s method [2].

  The ratio of the squared magnitude (`scaling='spectrum'`) divided by the spectral power density (`scaling='density'`) is the constant factor of `sum(abs(window)**2)*fs / abs(sum(window))**2`. If return_onesided is `True`, the values of the negative frequencies are added to values of the corresponding positive ones.

  Consult the Spectral Analysis section of the SciPy User Guide for a discussion of the scalings of the power spectral density and the (squared) magnitude spectrum.

  Added in version 0.12.0.

  References

  [1]

  P. Welch, “The use of the fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms”, IEEE Trans. Audio Electroacoust. vol. 15, pp. 70-73, 1967.

  [2]

  M.S. Bartlett, “Periodogram Analysis and Continuous Spectra”, Biometrika, vol. 37, pp. 1-16, 1950.

  Examples

      >>> import numpy as np
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
      >>> rng = np.random.default_rng()

  Generate a test signal, a 2 Vrms sine wave at 1234 Hz, corrupted by 0.001 V**2/Hz of white noise sampled at 10 kHz.

      >>> fs = 10e3
      >>> N = 1e5
      >>> amp = 2*np.sqrt(2)
      >>> freq = 1234.0
      >>> noise_power = 0.001 * fs / 2
      >>> time = np.arange(N) / fs
      >>> x = amp*np.sin(2*np.pi*freq*time)
      >>> x += rng.normal(scale=np.sqrt(noise_power), size=time.shape)

  Compute and plot the power spectral density.

      >>> f, Pxx_den = signal.welch(x, fs, nperseg=1024)
      >>> plt.semilogy(f, Pxx_den)
      >>> plt.ylim([0.5e-3, 1])
      >>> plt.xlabel('frequency [Hz]')
      >>> plt.ylabel('PSD [V**2/Hz]')
      >>> plt.show()

  If we average the last half of the spectral density, to exclude the peak, we can recover the noise power on the signal.

      >>> np.mean(Pxx_den[256:])
      0.0009924865443739191

  Now compute and plot the power spectrum.

      >>> f, Pxx_spec = signal.welch(x, fs, 'flattop', 1024, scaling='spectrum')
      >>> plt.figure()
      >>> plt.semilogy(f, np.sqrt(Pxx_spec))
      >>> plt.xlabel('frequency [Hz]')
      >>> plt.ylabel('Linear spectrum [V RMS]')
      >>> plt.show()

  The peak height in the power spectrum is an estimate of the RMS amplitude.

      >>> np.sqrt(Pxx_spec.max())
      2.0077340678640727

  If we now introduce a discontinuity in the signal, by increasing the amplitude of a small portion of the signal by 50, we can see the corruption of the mean average power spectral density, but using a median average better estimates the normal behaviour.

      >>> x[int(N//2):int(N//2)+10] *= 50.
      >>> f, Pxx_den = signal.welch(x, fs, nperseg=1024)
      >>> f_med, Pxx_den_med = signal.welch(x, fs, nperseg=1024, average='median')
      >>> plt.semilogy(f, Pxx_den, label='mean')
      >>> plt.semilogy(f_med, Pxx_den_med, label='median')
      >>> plt.ylim([0.5e-3, 1])
      >>> plt.xlabel('frequency [Hz]')
      >>> plt.ylabel('PSD [V**2/Hz]')
      >>> plt.legend()
      >>> plt.show()