An ultra-precise fast fourier transform

Abstract

<h2>Abstract</h2> The Fast Fourier Transform (FFT) is a cornerstone of digital signal processing, generating a computationally efficient estimate of the frequency content of a time series. Its limitations include: (1) information is only provided at discrete frequency steps, so further calculation, for example interpolation, is often used to obtain improved estimates of peak frequencies and amplitudes; (2) ‘energy' from spectral peaks may ‘leak' into adjacent frequencies, potentially causing lower amplitude peaks to be distorted or hidden; (3) the FFT, like many other DSP algorithms, is a discrete time approximation of continuous time mathematics. This paper describes a new FFT calculation which uses two windowing functions, derived from Prism Signal Processing. Separate FFT results are obtained from each windowing function applied to the data set. Calculations based on the two FFT results yields high precision estimates of spectral peak location (frequency), amplitude and phase. This technique addresses FFT limitations as follows: (1) spectral peak parameters are calculated directly, unrestricted by FFT frequency step discretization; (2) the windowing functions have narrowband characteristics which attenuate and localize spectral leakage; (3) the windowing functions incorporate a Romberg Integration mechanism to overcome the discrete/continuous time approximation.