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scipy.signal.

# csd

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

  Estimate the cross power spectral density, Pxy, using Welch’s method.

  Parameters:
    x array_like

      Time series of measurement values

    y array_like

      Time series of measurement values

    fs float, optional

      Sampling frequency of the x and y 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 and may not be greater than nperseg.

    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 cross spectral density (‘density’) where Pxy has units of V**2/Hz and computing the cross spectrum (‘spectrum’) where Pxy has units of V**2, if x and y are measured in V and fs is measured in Hz. Defaults to ‘density’

    axis int, optional

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

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

      Method to use when averaging periodograms. If the spectrum is complex, the average is computed separately for the real and imaginary parts. Defaults to ‘mean’.

      Added in version 1.2.0.

  Returns:
    f ndarray

      Array of sample frequencies.

    Pxy ndarray

      Cross spectral density or cross power spectrum of x,y.

  See also

  `periodogram`

    Simple, optionally modified periodogram

  `lombscargle`

    Lomb-Scargle periodogram for unevenly sampled data

  `welch`

    Power spectral density by Welch’s method. [Equivalent to csd(x,x)]

  `coherence`

    Magnitude squared coherence by Welch’s method.

  Notes

  By convention, Pxy is computed with the conjugate FFT of X multiplied by the FFT of Y.

  If the input series differ in length, the shorter series will be zero-padded to match.

  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.

  The ratio of the cross spectrum (`scaling='spectrum'`) divided by the cross spectral 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 a spectral density and an (amplitude) spectrum.

  Welch’s method may be interpreted as taking the average over the time slices of a (cross-) spectrogram. Internally, this function utilizes the `ShortTimeFFT` to determine the required (cross-) spectrogram. An example below illustrates that it is straightforward to calculate Pxy directly with the `ShortTimeFFT`. However, there are some notable differences in the behavior of the `ShortTimeFFT`:

  - There is no direct `ShortTimeFFT` equivalent for the `csd` parameter combination `return_onesided=True, scaling='density'`, since `fft_mode='onesided2X'` requires `'psd'` scaling. The is due to `csd` returning the doubled squared magnitude in this case, which does not have a sensible interpretation.

  - `ShortTimeFFT` uses float64 / complex128 internally, which is due to the behavior of the utilized `fft` module. Thus, those are the dtypes being returned. The `csd` function casts the return values to float32 / complex64 if the input is float32 / complex64 as well.

  - The `csd` function calculates `np.conj(Sx[q,p]) * Sy[q,p]`, whereas `spectrogram` calculates `Sx[q,p] * np.conj(Sy[q,p])` where `Sx[q,p]`, `Sy[q,p]` are the STFTs of x and y. Also, the window positioning is different.

  Added in version 0.16.0.

  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]

  Rabiner, Lawrence R., and B. Gold. “Theory and Application of Digital Signal Processing” Prentice-Hall, pp. 414-419, 1975

  Examples

  The following example plots the cross power spectral density of two signals with some common features:

      >>> import numpy as np
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
      >>> rng = np.random.default_rng()
      ...
      ... # Generate two test signals with some common features:
      >>> N, fs = 100_000, 10e3  # number of samples and sampling frequency
      >>> amp, freq = 20, 1234.0  # amplitude and frequency of utilized sine signal
      >>> noise_power = 0.001 * fs / 2
      >>> time = np.arange(N) / fs
      >>> b, a = signal.butter(2, 0.25, 'low')
      >>> x = rng.normal(scale=np.sqrt(noise_power), size=time.shape)
      >>> y = signal.lfilter(b, a, x)
      >>> x += amp*np.sin(2*np.pi*freq*time)
      >>> y += rng.normal(scale=0.1*np.sqrt(noise_power), size=time.shape)
      ...
      ... # Compute and plot the magnitude of the cross spectral density:
      >>> nperseg, noverlap, win = 1024, 512, 'hann'
      >>> f, Pxy = signal.csd(x, y, fs, win, nperseg, noverlap)
      >>> fig0, ax0 = plt.subplots(tight_layout=True)
      >>> ax0.set_title(f"CSD ({win.title()}-window, {nperseg=}, {noverlap=})")
      >>> ax0.set(xlabel="Frequency $f$ in kHz", ylabel="CSD Magnitude in V²/Hz")
      >>> ax0.semilogy(f/1e3, np.abs(Pxy))
      >>> ax0.grid()
      >>> plt.show()

  The cross spectral density is calculated by taking the average over the time slices of a spectrogram:

      >>> SFT = signal.ShortTimeFFT.from_window('hann', fs, nperseg, noverlap,
      ...                                       scale_to='psd', fft_mode='onesided2X',
      ...                                       phase_shift=None)
      >>> Sxy1 = SFT.spectrogram(y, x, detr='constant', k_offset=nperseg//2,
      ...                        p0=0, p1=(N-noverlap) // SFT.hop)
      >>> Pxy1 = Sxy1.mean(axis=-1)
      >>> np.allclose(Pxy, Pxy1)  # same result as with csd()
      True

  As discussed in the Notes section, the results of using an approach analogous to the code snippet above and the `csd` function may deviate due to implementation details.

  Note that the code snippet above can be easily adapted to determine other statistical properties than the mean value.