On the Statistical Characterization of the Discrete Fourier Transform of Noisy Space Vectors

Abstract

This paper investigates the statistical effects of additive noise on the magnitude of the space-vector spectral lines estimated through a discrete Fourier transform (DFT). In fact, the space vector is a well-known and effective tool for monitoring power quality issues in modern power systems. In many practical cases waveform distortion requires a DFT in order to extract the fundamental component and the harmonic/interharmonic content. In particular, the space vector shape (on the complex plane) of the fundamental component provides information such as voltage dips, whereas space-vector spectral lines provide information related to waveform distortion. Additive noise mainly impacts on low-magnitude spectral lines which require a statistical characterization. Conventional results available in the literature cannot be used in a straightforward way because a space vector is a complex-valued function, therefore special care is needed for proper interpretation and use of its properties in the frequency domain. In the paper, the probability density function, the mean value and the variance of the magnitude of space-vector DFT spectral lines are derived in closed form. Analytical results are validated by means of numerical simulations.