Integral Transforms from Finite Data: An Application of Gaussian Process\n Regression to Fourier Analysis

Computing accurate estimates of the Fourier transform of analog signals from\ndiscrete data points is important in many fields of science and engineering.\nThe conventional approach of performing the discrete Fourier transform of the\ndata implicitly assumes periodicity and bandlimitedness of the signal. In this\npaper, we use Gaussian process regression to estimate the Fourier transform (or\nany other integral transform) without making these assumptions. This is\npossible because the posterior expectation of Gaussian process regression maps\na finite set of samples to a function defined on the whole real line, expressed\nas a linear combination of covariance functions. We estimate the covariance\nfunction from the data using an appropriately designed gradient ascent method\nthat constrains the solution to a linear combination of tractable kernel\nfunctions. This procedure results in a posterior expectation of the analog\nsignal whose Fourier transform can be obtained analytically by exploiting\nlinearity. Our simulations show that the new method leads to sharper and more\nprecise estimation of the spectral density both in noise-free and\nnoise-corrupted signals. We further validate the method in two real-world\napplications: the analysis of the yearly fluctuation in atmospheric CO2 level\nand the analysis of the spectral content of brain signals.\n