Uncertainty Propagation Through Fourier Transforms

Real measurements come with uncertainties. Fourier transforms of real measurements should too.

fourierror propagates uncertainty through the discrete Fourier transform, giving you the full covariance of the real and imaginary components of the result rather than a bare complex spectrum. Two routes are available: an analytical propagation and a numerical (Monte Carlo) sampling approach, each with a DFT and an FFT implementation.

Function	Method
fourierror.analytical.dft	exact linear propagation through the DFT matrix
fourierror.analytical.fft	the same result, computed with FFTs
fourierror.numerical.dft	Monte Carlo sampling of the input distribution
fourierror.numerical.fft	Monte Carlo sampling, using the FFT for each sample

The package is built on scipp, so units and coordinates are carried through the transform — the correct “frequency” axis is derived for you from the coordinate you transform over, rather than left as bare array indices.

Installation

pip install fourierror

Usage

import numpy as np
import scipp as sc
import fourierror

t = sc.arange("time", 0.0, 6.4, 0.05, unit="s")
y = sc.sin(sc.to_unit(2 * np.pi * 1.25 * t * sc.scalar(1.0, unit="rad/s"), "rad"))
y.variances = np.full(t.shape, 0.01)
data = sc.DataArray(data=y, coords={"time": t})

result = fourierror.analytical.dft(data, coord="time")
result["real"], result["imag"]   # values with (co)variances on an "omega" axis

Input. A one-dimensional scipp.DataArray with variances, and the name of an evenly spaced coordinate to transform over (unevenly spaced coordinates raise a TypeError). The variances are treated as uncorrelated; to supply correlated uncertainties, pass a fourierror.data_array.CovDataArray, whose full covariance matrix is propagated.

Output. A scipp.DataGroup with "real" and "imag" entries:

• Values are (2 / n) * numpy.fft.fft(values) (for the numerical methods, the sample mean of this), so a cosine of amplitude A whose frequency lies on the grid appears in "real" with height A at its positive and negative frequencies (a sine appears in "imag" as -A and +A).

• The coordinate omega is the angular frequency, 2π * numpy.fft.fftfreq(n, d), in the reciprocal unit of the input coordinate, in NumPy’s order (zero first, negative frequencies in the second half).

• Transforming correlates the output points, so each entry is usually a CovDataArray carrying the full covariance matrix (via .covariance); .variances gives its diagonal. If a covariance is diagonal to within diag_tol, a plain scipp.DataArray with variances is returned instead.

If you’re new to scipp, the getting started guide is a good first stop. See Usage for a worked example and the API reference for every function.

Citing

The propagation scheme, and guidance on when to prefer the numerical or analytical method, is described in the accompanying paper. If you use fourierror in your work, please cite:

T. Zolzaya and A. R. McCluskey, “Numerical and analytical strategies to propagate correlated uncertainties through the discrete Fourier transform”, arXiv:2611.12359 (2026).

@misc{zolzaya2026numerical,
  title         = {Numerical and analytical strategies to propagate correlated uncertainties through the discrete {F}ourier transform},
  author        = {Zolzaya, Tuguldur and McCluskey, Andrew R.},
  year          = {2026},
  eprint        = {2611.12359},
  archivePrefix = {arXiv},
  url           = {https://arxiv.org/abs/2611.12359}
}

Contributors