In recent years there has been a renewed interest in finding fast algorithms to compute accurately the linear canonical transform (LCT) of a given function. This is driven by the large number of applications of the LCT in optics and signal processing. The well-known integral transforms: Fourier, fractional Fourier, bilateral Laplace and Fresnel transforms are special cases of the LCT. In this paper we obtain an O(N log N) algorithm to compute the LCT by using a chirp-FFT-chirp transformation yielded by a convergent quadrature formula for the fractional Fourier transform. This formula gives a unitary discrete LCT in closed form. In the case of the fractional Fourier transform the algorithm computes this transform for arbitrary complex values inside the unitary circle and not only at the boundary. In the case of the ordinary Fourier transform the algorithm improves the output of the FFT.
Background: Fourier transform is a basic tool for analyzing biological signals and is computed for a finite sequence of data sample. The electroencephalographic (EEG) signals analyzed with this method provide only information based on the frequency range, for short periods. In some cases, for long periods it can be useful to know whether EEG signals coincide or have a relative phase between them or with other biological signals. Some studies have evidenced that sex hormones and EEG signals show oscillations in their frequencies across a period of 28 days; so it seems of relevance to seek after possible patterns relating EEG signals and endogenous sex hormones, assumed as long time-periodic functions to determine their typical periods, frequencies and relative phases.
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