Fourier Transform as a Robust Inverse Problem

In this study a new method is proposed for the determination of the Fourier transform by using the tools of inverse problem theory. In this approach the Fourier spectrum is calculated after solving a (highly) over-determined inverse problem. This makes the proposed procedure more robust than the DFT or FFT. It is proved in two synthetic examples that – compared to DFT- the noise sensitivity of the IFT method (applying LSQ in the inversion) is appreciably reduced. Using an Iteratively Re-weighted Least Squares algorithm (with Cauchy weights) a robust and resistant method (W-IFT) is developed and tested. It can be expected, that the newly developed methods can serve as robust and noise-resistant alternatives of the common Fourier transforms in various fields of practical application.