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**FourierAnalysis** is a signal-processing **Julia** package for
performing the analysis of *real multivariate data* (e.g., multivariate time series)
in the *frequency* domain and in the *time-frequency* domain. It is based upon the
DSP.jl, FFTW.jl and AbstractFFTs.jl packages.

In the frequency domain *FourierAnalysis* computes **spectra**, *linear* and
*non-linear* **cross-spectral matrices** and several *linear* and *non-linear* **coherence matrices** using the sliding-windows Welch method.

Time-frequency representations are obtained applying a
filter-bank and the
Hilbert transform.
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

**mean amplitude****mean direction****phase concentration**(the non-linear version of which is the directional statistic*circular mean resultant length*)**comodulation****coherence**(the non-linear version of which is known as*phase-locking values*or*phase coherence*).

All these measures are provided in a simple and unified fashion, following the conceptual approach previously illustrated in the context of electroencephalography (Congedo, 2018), 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.

## Installation

Execute the following command in Julia's REPL:

```
]add FourierAnalysis
```

## Disclaimer

Although fully functional, this package is still in a pre-release stage. It needs throughout testing. Independent reviewers are more then welcome.

## About the Author

Marco Congedo is a Research Director of CNRS (Centre National de la Recherche Scientifique), working at UGA (University of Grenoble Alpes), in Grenoble (France), the city where Jean-Baptiste Joseph Fourier has served as a Prefect:).

## Contact

first name *dot* last name *at* gmail *dot* com

## Examples

```
using FourierAnalysis, Plots
# Set sampling rate (sr) and FFT window length (wl):
sr, wl = 128, 128
# Generate a sinusoidal wave at 10Hz with peak amplitude 0.5 and add some white noise:
v = sinusoidal(0.5, 10, sr, wl*16) + randn(wl*16)
# Get the power spectrum with a rectangular tapering window:
S = spectra(v, sr, wl; tapering=rectangular)
# Plot the power spectrum:
plot(S; fmax=24)
# The same syntax applies in the case of multivariate data (e.g., 4 time-series):
V = randn(wl*16, 4)
S = spectra(V, sr, wl; tapering=hamming)
plot(S)
# Get the analytic amplitude in the time-Frequency domain:
A = TFamplitude(v, sr, wl; fmax=24)
# plot the analytic amplitude:
heatmap(A.y)
```

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