what's your favorite application of the Fourier transform? source: 3Blue1Brown But what is the Fourier Transform? A visual introduction, The Powerful Fourier Transform Quanta Science Channel The Fourier transform is a mapping from time domain to the frequency domain, given by X(ω)=∫−∞∞x(t)e−iωtdt, where x(t) is the time domain signal, X(ω) is its Fourier transform, and ω is the angular frequency. Although the properties of the Fourier transform are well known (e.g., Oppenheim and Schafer, 2010), we address two points of special relevance to connectivity estimation. First, the Fourier transform is linear. This has an important implication for network identification. For linear time-invariant systems, such as those described by Eq. (1), there can be no cross-frequency interactions. For example, if there is 10 Hz activity in the input, then, in the ideal case, all of that will be mapped to 10 Hz activity in the output (although power and phase may vary). Cross-spectral interactions are therefore a signature of non-linear interactions. Second, the practical application of Fourier transform methods entail using a discrete (rather than continuous) transform, combined with a windowing function to limit the transform to finite bounds and minimize the leakage of the high frequency components due to the finite bounds. Since the window width is typically fixed, the Fourier transform does not provide time domain resolution at a scale less than the window width. It is therefore of limited use for time–frequency analysis. Greenblatt, Pflieger, Ossadtchi. (2012). Connectivity measures applied to human brain electrophysiological data. Journal of Neuroscience Methods. #fourier #fouriertransform #math #calculus #apcalculus - @isomorphc