Predicting the variance of a measurement with 1/f noise

Abstract

Measurement devices always add noise to the signal of interest and it is necessary to evaluate the variance of the results. This article focuses on stationary random processes whose Power Spectrum Density is a power law of frequency. For flicker noise, behaving as $1/f$ and which is present in many different phenomena, the usual way to compute the variance leads to infinite values. This article proposes an alternative definition of the variance which takes into account the fact that measurement devises need to be calibrated. This new variance, which depends on the calibration duration, the measurement duration and the duration between the calibration and the measurement, allows avoiding infinite values when computing the variance of a measurement.