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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.

Execute the following command in Julia's REPL:

]add FourierAnalysis

Although fully functional, this package needs throughout testing. Independent reviewers are more then welcome.

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:).

first name dot last name at gmail dot com

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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