Source code for

A number of utility functions, for example for dealing with
autocorrelation functions, Fourier transforms, and smoothing.

import numpy as np
from scipy.signal import correlate
from numpy.typing import NDArray
import pandas as pd

[docs]def compute_acf(Z: NDArray[float], delta_t: float = 1.0, method='scipy'): r""" Computes the autocorrelation function (ACF) for a one-dimensional signal :math:`Z` in time as .. math:: ACF(\tau) = \frac{\left < Z(t) Z^*(t+\tau) \right >}{\left < Z(t) Z^*(t) \right >} Here, only the real part of the ACF is returned since if :math:`Z` is complex the imaginary part should average out to zero for any stationary signal. Parameters ---------- Z complex time signal delta_t spacing in time between two consecutive values in :math:`Z` method implementation to use; possible values: `numpy` and `scipy` (default and usually faster) """ # keep only real part and normalize acf = _compute_correlation_function(Z, Z, method) acf = np.real(acf) acf /= acf[0] time_lags = delta_t * np.arange(0, len(acf), 1) return time_lags, acf
def _compute_correlation_function(Z1, Z2, method='scipy'): N = len(Z1) assert len(Z1) == len(Z2) if method == 'scipy': cf = correlate(Z1, Z2, mode='full')[N - 1:] / np.arange(N, 0, -1) elif method == 'numpy': cf = np.correlate(Z1, Z2, mode='full')[N - 1:] / np.arange(N, 0, -1) else: raise ValueError('method must be either numpy or scipy') return cf # smoothing functions / FFT filters # -------------------------------------
[docs]def gaussian_decay(t: NDArray[float], t_sigma: float): r""" Evaluates a gaussian distribution in time :math:`f(t)`, which can be applied to an ACF in time to artificially damp it, i.e., forcing it to go to zero for long times. .. math:: f(t) = \exp{\left [-\frac{1}{2} \left (\frac{t}{t_\mathrm{sigma}}\right )^2 \right ] } Parameters ---------- t time array t_sigma width (standard deviation of the gaussian) of the decay """ return np.exp(- 1 / 2 * (t / t_sigma) ** 2)
[docs]def fermi_dirac(t: NDArray[float], t_0: float, t_width: float): r""" Evaluates a Fermi-Dirac-like function in time :math:`f(t)`, which can be applied to an ACF in time to artificially damp it, i.e., forcing it to go to zero for long times without affecting the short-time correlations too much. .. math:: f(t) = \frac{1}{\exp{[(t-t_0)/t_\mathrm{width}}] + 1} Parameters ---------- t time array t_0 starting time for decay t_width width of the decay """ return 1.0 / (np.exp((t - t_0) / t_width) + 1)
[docs]def smoothing_function(data: NDArray[float], window_size: int, window_type: str = 'hamming'): """ Smoothing function for 1D arrays. This functions employs pandas rolling window average Parameters ---------- data 1D data array window_size The size of smoothing/smearing window window_type What type of window-shape to use, e.g. ``'blackman'``, ``'hamming'``, ``'boxcar'`` (see pandas and scipy documentaiton for more details) """ series = pd.Series(data) new_data = series.rolling(window_size, win_type=window_type, center=True, min_periods=1).mean() return np.array(new_data)