Projecting interval uncertainty through the discrete Fourier transform: an application to time signals with poor precision

The discrete Fourier transform (DFT) is often used to decompose a signal into a finite number of harmonic components. The efficient and rigorous propagation of the error present in a signal through the transform can be computationally challenging. Real data is always subject to imprecision because of measurement uncertainty. For example, such uncertainty may come from sensors whose precision is affected by degradation, or simply from digitisation. On many occasions, only error bounds on the signal may be known, thus it may be necessary to automatically propagate the error bounds without making additional artificial assumptions. This paper presents a method that can automatically propagate interval uncertainty through the DFT while yielding the exact bounds on the Fourier amplitude and on an estimation of the Power Spectral Density (PSD) function. The method allows technical analysts to project interval uncertainty – present in the time signals – to the Fourier amplitude and PSD function without making assumptions about the dependence and the distribution of the error over the time steps. Thus, it is possible to calculate and analyse system responses in the frequency domain without conducting extensive Monte Carlo simulations nor running expensive optimisations in the time domain. The applicability of this method in practice is demonstrated by a technical application. It is also shown that conventional Monte Carlo methods severely underestimate the uncertainty.