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# scipy.signal.welch

`scipy.signal.``welch`(x, fs=1.0, window='hanning', nperseg=256, noverlap=None, nfft=None, detrend='constant', return_onesided=True, scaling='density', axis=-1)

  Estimate power spectral density using Welch’s method.

  Welch’s method [R215] 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. 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 will be used for nperseg. Defaults to ‘hanning’.

  nperseg : int, optional

  > Length of each segment. Defaults to 256.

  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 `detrend`. 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. Note that 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 power 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`).

  Returns:

  f : ndarray

  > Array of sample frequencies.

  Pxx : ndarray

  > Power spectral density or power spectrum of x.

  See also

  `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 ‘hanning’ 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 [R216].

  New in version 0.12.0.

  References

  [R215] | (1, 2) 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.

  [R216] | (1, 2) M.S. Bartlett, “Periodogram Analysis and Continuous Spectra”, Biometrika, vol. 37, pp. 1-16, 1950.

  Examples

      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
      >>> np.random.seed(1234)

  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 += np.random.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()

  (Source code)

  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