scope.fourier module

scope.fourier.fit_fourier.broken_power_law(freq, freq0, N_c, pl_index, N_w)[source]

Model of the broken power law.

A piecewise function joining two models of power law: COLOURED NOISE and WHITE NOISE.

Parameters:

  • freq (numpy array) – Fourier frequencies

  • freq0 (float) – X-coordinate of the knot

  • N_c (float) – Energy (coefficient) of the coloured-noise model

  • pl_index (float) – Index of the coloured-noise model

  • N_w (float) – Energy (coefficient) of the white-noise model

Returns:

The function

Return type:

numpy array

scope.fourier.fit_fourier.continuous_power_law(freq, N_c, pl_index, N_w)[source]

Model of the continuous power law.

Superposition of two models of power law: COLOURED NOISE + WHITE NOISE.

Parameters:

  • freq (numpy array) – Fourier frequencies

  • N_c (float) – Energy (coefficient) of the coloured-noise model

  • pl_index (float) – Index of the coloured-noise model

  • N_w (float) – Energy (coefficient) of the white-noise model

Returns:

The function

Return type:

numpy array

scope.fourier.fit_fourier.fit_fourier(x, dt, fap, plot_spectrum=False)[source]

A debiased least squares fitting algorithm.

This function fits the input signal with the continuous power law using debiased least squares fitting, and calculates the frequency-dependent confidence limit based on a user-defined false alarm probability.

The method of debiased least squares fitting is based on [1].

[1] ‘https://www.aanda.org/articles/aa/abs/2005/07/aa1453/aa1453.html’

Parameters:

  • x (numpy array) – Noise signal

  • dt (float) – Interval of the time series

  • fap (float) – False alarm probability

  • plot_spectrum (bool, optional) – Plot the power spectrum, with fittings by the broken power law and continuous power law. The confidence limits of the continous power law fit are also plotted. The default is False.

Returns:

fit_fourier_result –

scope.fourier.fit_fourier.power

FFT power

Type:

numpy array

scope.fourier.fit_fourier.frequency

Fourier frequencies

Type:

numpy array

scope.fourier.fit_fourier.frequency0

Frequency at which the signal varies from coloured noise to white noise

Type:

float

scope.fourier.fit_fourier.expectation_broken

Frequency dependent expectation of the FFT power calculated from the broken power law model

Type:

numpy array

scope.fourier.fit_fourier.expectation_continuous

Frequency dependent expectation of the FFT power calculated from the continuous power law model

Type:

numpy array

scope.fourier.fit_fourier.white_energy

Energy of the white-noise component

Type:

float

scope.fourier.fit_fourier.color_energy

Energy of the coloured-noise component

Type:

float

scope.fourier.fit_fourier.pl_index

Index of the coloured-noise component

Type:

float

scope.fourier.fit_fourier.confidence_limit

Frequency dependent confidence limit based on the provided false alarm probability

Type:

numpy array

Return type:

dict

scope.fourier.fit_fourier.log_power(freq, N_c, pl_index, N_w)[source]

Function describing the logarithm (base 10) of the power spectrum.

Calculates the logarithm (base 10) of the continuous power law + bias. The bias term originates from the fact that the power spectrum is scattered around the true spectrum with a chi-squared distribution with 2 degrees of freedom.

Parameters:

  • freq (numpy array) – Fourier frequencies

  • N_c (float) – Energy (coefficient) of the coloured-noise model

  • pl_index (float) – Index of the coloured-noise model

  • N_w (float) – Energy (coefficient) of the white-noise model

Returns:

The function

Return type:

numpy array

scope.fourier.fit_fourier.piecewise_linear(x, x0, y0, k1)[source]

A piecewise linear function to be fitted.

The piecewise linear function consists of the function of a straight line and a flat line.

Parameters:

  • x (numpy array) – Fourier frequency

  • x0 (float) – X-coordinate of the knot

  • y0 (float) – Y-coordinate of the knot

  • k1 (float) – Gradient of the straight line

Returns:

The function

Return type:

numpy array