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‎.github/dependabot.yml‎

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# https://docs.github.com/github/administering-a-repository/configuration-options-for-dependency-updates
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version: 2
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updates:
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- package-ecosystem: "github-actions"
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directory: "/" # Location of package manifests
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schedule:
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interval: "weekly"

‎.github/workflows/CI.yml‎

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name: CI
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on:
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push:
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branches:
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- master # update to match your development branch (master, main, dev, trunk, ...)
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tags: '*'
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pull_request:
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jobs:
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build:
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permissions:
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contents: write
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pull-requests: read
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statuses: write
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runs-on: ubuntu-latest
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steps:
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- uses: actions/checkout@v4
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- uses: julia-actions/setup-julia@v2
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with:
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version: '1.11'
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- uses: julia-actions/cache@v1
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- name: Install dependencies
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run: julia --project=docs/ -e ' using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.add("DocumenterTools"); Pkg.instantiate()'
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- name: Build and deploy
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env:
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GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} # If authenticating with GitHub Actions token
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DOCUMENTER_KEY: ${{ secrets.DOCUMENTER_KEY }} # If authenticating with SSH deploy key
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run: julia --project=docs/ docs/make.jl

‎.github/workflows/CompatHelper.yml‎

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name: CompatHelper
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on:
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schedule:
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- cron: '00 00 * * *'
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jobs:
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CompatHelper:
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runs-on: ubuntu-latest
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steps:
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- uses: julia-actions/setup-julia@latest
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with:
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version: 1.3
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- name: Pkg.add("CompatHelper")
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run: julia -e 'using Pkg; Pkg.add("CompatHelper")'
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- name: CompatHelper.main()
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env:
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GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
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run: julia -e 'using CompatHelper; CompatHelper.main()'

‎.github/workflows/TagBot.yml‎

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name: TagBot
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on:
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issue_comment:
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types:
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- created
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workflow_dispatch:
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inputs:
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lookback:
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default: "3"
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permissions:
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actions: read
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checks: read
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contents: write
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deployments: read
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issues: read
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discussions: read
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packages: read
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pages: read
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pull-requests: read
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repository-projects: read
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security-events: read
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statuses: read
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jobs:
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TagBot:
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if: github.event_name == 'workflow_dispatch' || github.actor == 'JuliaTagBot'
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runs-on: ubuntu-latest
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steps:
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- uses: JuliaRegistries/TagBot@v1
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with:
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token: ${{ secrets.GITHUB_TOKEN }}
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# Edit the following line to reflect the actual name of the GitHub Secret containing your private key
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ssh: ${{ secrets.DOCUMENTER_KEY }}
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# ssh: ${{ secrets.NAME_OF_MY_SSH_PRIVATE_KEY_SECRET }}

‎.gitignore‎

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#Emacs temp files
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*~
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[#]*[#]
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.\#*
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# MacOS files
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.DS_Store
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# VS code files
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# *.vscode
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*.code-workspace
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# Files generated by invoking Julia with --code-coverage
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*.jl.cov
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*.jl.*.cov
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# Files generated by invoking Julia with --track-allocation
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*.jl.mem
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# System-specific files and directories generated by the BinaryProvider and BinDeps packages
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# They contain absolute paths specific to the host computer, and so should not be committed
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deps/deps.jl
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deps/build.log
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deps/downloads/
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deps/usr/
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deps/src/
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# Build artifacts for creating documentation generated by the Documenter package
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docs/build/
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docs/site/
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# File generated by Pkg, the package manager, based on a corresponding Project.toml
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# It records a fixed state of all packages used by the project. As such, it should not be
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# committed for packages, but should be committed for applications that require a static
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# environment.
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Manifest.toml
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# Figures
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# Add here folders where figures are generated during development
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# to check what is ignored use 'git status' in git Bash

‎LICENSE‎

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MIT License
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Copyright (c) 2019-2025, Marco Congedo, CNRS, Grenobe, France:
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https://sites.google.com/site/marcocongedo/home
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Permission is hereby granted, free of charge, to any person obtaining a copy
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of this software and associated documentation files (the "Software"),
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to deal in the Software without restriction, including without limitation
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the rights to use, copy, modify, merge, publish, distribute, sublicense,
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and/or sell copies of the Software, and to permit persons to whom the
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Software is furnished to do so, subject to the following conditions:
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The above copyright notice and this permission notice shall be included in
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all copies or substantial portions of the Software.
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THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
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MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
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IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM,
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DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
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TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
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WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

‎Project.toml‎

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name = "FourierAnalysis"
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uuid = "e7e9c730-dc46-11e9-3633-f1ab55cc17e1"
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authors = ["Marco-Congedo <marco.congedo@gmail.com>"]
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version = "1.2.4"
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[deps]
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AbstractFFTs = "621f4979-c628-5d54-868e-fcf4e3e8185c"
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DSP = "717857b8-e6f2-59f4-9121-6e50c889abd2"
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FFTW = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
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LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
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PosDefManifold = "f45a3650-5c51-11e9-1e9a-133aa5e309cf"
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RecipesBase = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
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Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
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[compat]
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AbstractFFTs = "1.0, 1.2, 1.3, 1.4, 1.5"
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DSP = "0.7, 0.8"
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FFTW = "1.3, 1.4, 1.5, 1.6, 1.7, 1.8"
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PosDefManifold = "0.5"
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RecipesBase = "1.1, 1.2, 1.3"
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Statistics = "1.11"
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LinearAlgebra = "1.11"
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julia = "1"
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[extras]
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Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
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[targets]
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test = ["Test"]

‎README.md‎

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| **Documentation** |
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|:---------------------------------------:|
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| [![](https://img.shields.io/badge/docs-dev-blue.svg)](https://Marco-Congedo.github.io/FourierAnalysis.jl/dev) |
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![](/docs/src/assets/Fig1.jpg)
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**FourierAnalysis** is a signal-processing [**Julia**](https://julialang.org/) package for
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performing the analysis of *real multivariate data* (e.g., multivariate time series)
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in the *frequency* domain and in the *time-frequency* domain. It is based upon the
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[DSP.jl](https://github.com/JuliaDSP/DSP.jl), [FFTW.jl](https://github.com/JuliaMath/FFTW.jl) and [AbstractFFTs.jl](https://github.com/JuliaMath/AbstractFFTs.jl) packages.
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In the frequency domain *FourierAnalysis* computes **spectra**, *linear* and
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*non-linear* **cross-spectral matrices** and several *linear* and *non-linear* **coherence matrices** using the sliding-windows [Welch method](https://en.wikipedia.org/wiki/Welch%27s_method).
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Time-frequency representations are obtained applying a
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[filter-bank](https://en.wikipedia.org/wiki/Filter_bank) and the
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[Hilbert transform](https://en.wikipedia.org/wiki/Hilbert_transform).
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This way *FourierAnalysis* computes the **analytic signal**, from which the **instantaneous amplitude** (envelope) and **instantaneous phase** are obtained, along with several popular *linear* and *non-linear*, *weighted*, *univariate* and *bivariate* statistics, such as
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- **mean amplitude**
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- **mean direction**
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- **phase concentration** (the non-linear version of which is the directional statistic *circular mean resultant length*)
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- **comodulation**
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- **coherence** (the non-linear version of which is known as *phase-locking values* or *phase coherence*).
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All these measures are provided in a simple and unified fashion, following the conceptual approach previously illustrated in
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in the context of electroencephalography ([Congedo, 2018](https://hal.archives-ouvertes.fr/hal-01868538v2/document)), for which all default settings have been tailored. The package has been written with the *do-it-with-one-line* spirit, but without sacrificing full control over relevant options.
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## Installation
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Execute the following command in Julia's REPL:
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]add FourierAnalysis
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## Disclaimer
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Although fully functional, this package is still in a pre-release stage. It needs throughout testing.
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Independent reviewers are more then welcome.
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## About the Author
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[Marco Congedo](https://sites.google.com/site/marcocongedo) is
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a Research Director of [CNRS](http://www.cnrs.fr/en) (Centre National de la Recherche Scientifique), working at [UGA](https://www.univ-grenoble-alpes.fr/english/) (University of Grenoble Alpes), in Grenoble (France), the city where Jean-Baptiste Joseph Fourier has served as a Prefect:).
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## Contact
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first name *dot* last name *at* gmail *dot* com
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## Examples
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```
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using FourierAnalysis, Plots
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# Set sampling rate (sr) and FFT window length (wl):
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sr, wl = 128, 128
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# Generate a sinusoidal wave at 10Hz with peak amplitude 0.5 and add some white noise:
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v = sinusoidal(0.5, 10, sr, wl*16) + randn(wl*16)
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# Get the power spectrum with a rectangular tapering window:
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S = spectra(v, sr, wl; tapering=rectangular)
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# Plot the power spectrum:
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plot(S; fmax=24)
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# The same syntax applies in the case of multivariate data (e.g., 4 time-series):
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V = randn(wl*16, 4)
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S = spectra(V, sr, wl; tapering=hamming)
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plot(S)
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# Get the analytic amplitude in the time-Frequency domain:
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A = TFamplitude(v, sr, wl; fmax=24)
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# plot the analytic amplitude:
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heatmap(A.y)
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```
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| **Documentation** |
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|:---------------------------------------:|
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| [![](https://img.shields.io/badge/docs-dev-blue.svg)](https://Marco-Congedo.github.io/FourierAnalysis.jl/dev) |

‎docs/Project.toml‎

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authors = ["Marco Congedo"]
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version = "1.2.4"
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[deps]
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Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
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DocumenterCitations = "daee34ce-89f3-4625-b898-19384cb65244"
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DocumenterInterLinks = "d12716ef-a0f6-4df4-a9f1-a5a34e75c656"
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DocumenterTools = "35a29f4d-8980-5a13-9543-d66fff28ecb8"
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LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
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Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
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FourierAnalysis = "e7e9c730-dc46-11e9-3633-f1ab55cc17e1"
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[compat]
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Documenter = "1.11"
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LinearAlgebra = "1.11"
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Statistics = "1.11"

‎docs/make.jl‎

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push!(LOAD_PATH,"../src/")
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push!(LOAD_PATH, @__DIR__)
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using Documenter, DocumenterTools, DocumenterCitations, DocumenterInterLinks
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using FourierAnalysis
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makedocs(
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sitename="FourierAnalysis",
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format = Documenter.HTML(),
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authors="Marco Congedo, CNRS, France",
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modules=[FourierAnalysis],
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pages = [
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"index.md",
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"Main Module" => "MainModule.md",
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"Tapering Windows" => "tapers.md",
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"frequency domain" => Any[
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"Spectral Estimations" => "spectra.md",
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"Cross-Spectral Matrices" => "crossspectra.md",
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"Coherence Matrices" => "coherence.md",
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"Goertzel's Algorithms" => "goertzel.md",
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],
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"time-frequency(TF) domain" => Any[
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"TF Representations" => "timefrequency.md",
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"TF Univariate Measures" => "timefrequencyuni.md",
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"TF Bivariate Measures " => "timefrequencybi.md",
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],
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"utilities" => Any[
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"Plots" => "recipes.md",
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"Tools" => "tools.md",
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"FFTW planners" => "fftw.md",
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"Filter Banks" => "filters.md",
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"Hilbert Transform" => "hilbert.md",
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]
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]
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)
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deploydocs(
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repo = "github.com/Marco-Congedo/FourierAnalysis.jl.git",
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target = "build",
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devurl = "dev",
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)

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