Robust Fourier Transform Algorithm Using Inversion Tools

In order to make Fourier transformation more robust and noise resistant, the tools of inverse problem theory are used. The unknown frequency spectra are assumed to be expanded by orthonormal square-integrable basis functions, and the expansion coefficients are determined by solving an over-determined inverse problem. It is proved that noise sensitivity can be appreciably be reduced by using the proposed LSQ Fourier Transform (LSQ-FT) method compared to Discrete Fourier Transform (DFT). It is shown the Iteratively Reweighted Least Squares algorithm using Cauchy weights (Cauchy-IRLS-FT) gives highly acceptable results in the case when the input data set contains outliers.