A Closed-Form Approach to the Inverse Fourier Transform and Its Applications

The inverse Fourier transform of a complex-valued weighted exponential frequency-domain function is derived. A bounded causal frequency-domain transfer function can be approximated by a sum of exponential functions using numerical techniques such as the Prony's method or the matrix pencil method, which leads to the corresponding time-domain response through an exact inverse Fourier transform. The method is demonstrated in examples with given tabulated frequency-domain data for function interpolation and extrapolation, direct generation of time-domain pulse responses, and deriving time-domain reflectometry waveform from measured S-parameters.