FFT Specifications and Conventions

Stability: Stable

What it is

Use this page as the quick reference for Fourier normalization, axis conventions, and the APIs that follow those conventions.

Representative Signatures

TimeSeries.fft(fftlength=None, overlap=0, window="hann", ...)
ScalarField.fft_time(nfft=None)

Minimal Example

from gwexpy.timeseries import TimeSeries

# Use this page together with TimeSeries.fft / FrequencySeries.ifft / ScalarField.fft_time
# when you need the exact normalization and axis conventions.

Related Theory

• Physics Models

• Validated Algorithms

• FFT_Conventions

Related Tutorials

• Tutorial Index

• Getting Started

API Reference

The detailed generated API continues below on this page.

This document details the mathematical conventions, normalization, and sign definitions used for Fast Fourier Transforms (FFT) in gwexpy.

1. Temporal FFT (fft_time)

The temporal FFT is applied to the time axis (axis 0). It is designed for compatibility with gwpy and standard gravitational-wave data analysis practices.

Normalization

\(1/N\) normalization is applied to the forward transform. This means the amplitude of a sine wave \(A \sin(\omega t)\) appears as \(A\) in the resulting frequency spectrum (after accounting for the one-sided factor). The inverse transform (ifft_time) undoes this normalization.

Spectral Definition

• Type: One-sided (rfft). Only non-negative frequencies are returned.

• Correction: To preserve total power, all non-DC and non-Nyquist bins are multiplied by 2.

• Frequency Axis: \(f = [0, 1/2dt]\).

Sign Convention

• Forward: \(X[k] = \frac{1}{N} \sum_{n=0}^{N-1} x[n] e^{-i 2\pi k n / N}\)

• Inverse: \(x[n] = \sum_{k=0}^{N/2} X[k] e^{i 2\pi k n / N}\) (with appropriate bin weighting)

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2. Spatial FFT (fft_space)

The spatial FFT is applied to spatial axes (axes 1, 2, 3). It is designed for physical field analysis across multi-dimensional grids.

Normalization

Unnormalized. No factor is applied to the forward transform, following standard numpy.fft.fftn behavior. The inverse transform (ifft_space) applies \(1/N\) normalization.

Spectral Definition

• Type: Two-sided. Returns both positive and negative wavenumbers.

• Wavenumber Axis: By default, fft_space produces angular wavenumber \(k = 2\pi / \lambda\).

• Ordering: The zero-frequency (DC) component is at index 0 (not shifted). Use numpy.fft.fftshift externally if a centered spectrum is desired for visualization.

Sign Convention

• Forward: \(X[\mathbf{k}] = \sum_{\mathbf{n}} x[\mathbf{n}] e^{-i 2\pi \sum k_j n_j / N_j}\)

• Inverse: \(x[\mathbf{n}] = \frac{1}{\prod N_j} \sum_{\mathbf{k}} X[\mathbf{k}] e^{i 2\pi \sum k_j n_j / N_j}\)

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3. Spectral Density (spectral_density, compute_psd)

The spectral_density method provides power spectral density (PSD) estimation using the Welch method (averaged periodograms).

Scaling

• Density: Returns power per unit frequency/wavenumber resolution (\(V^2 / \text{Hz}\) or \(V^2 / [\text{unit}^{-1}]\)).

• Spectrum: Returns total power in each bin (\(V^2\)).

Wavenumber Convention in Signal Utils

Note that while fft_space uses angular wavenumber (\(k = 2\pi/\lambda\)), spectral_density(axis='x') produces non-angular wavenumber (\(f_k = 1/\lambda\)) by default, consistent with scipy.signal.welch. To convert to angular wavenumber, multiply the resulting axis by \(2\pi\).

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4. Normalization Summary Table

Method	Normalization (Forward)	Domain	Sign (\(ikx\))
fft_time	\(1/N\)	One-sided	Negative
ifft_time	\(N\)	One-sided	Positive
fft_space	\(1\)	Two-sided	Negative
ifft_space	\(1/N\)	Two-sided	Positive
spectral_density	Welch / \(1/N^2\)	One-sided	N/A (Magnitude sq.)