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def welch(x, fs=1.0, window='hann', 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 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'.
.. versionadded:: 1.2.0
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 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]_.
Consult the :ref:`tutorial_SpectralAnalysis` section of the :ref:`user_guide`
for a discussion of the scalings of the power spectral density and
the (squared) magnitude spectrum.
.. versionadded:: 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()
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