A Closed-Form Representation of an Upper Limit Error Function and its Interpretation on Measurements with Noise

Any measurement of an electrical quantity, e.g. in network or spectrum analysis, is influenced by noise inducing a measurement uncertainty, the statistical quantification of which is rarely discussed in literature. A measurement uncertainty in such a context means a measurement error that is associated with a given probability, e.g. one standard deviation. The measurement uncertainty mainly depends on the signal-to-noise-ratio (SNR), but additionally can be influenced by the acquisition stage of the measurement setup. The analytical treatment of noise is hardly feasible as the physical nature of a noise vector needs to account for a certain magnitude and phase in a combined probability function. However, in a previous work a closed-form analytical solution for the uncertainties of amplitude and phase measurements depending on the SNR has been derived and validated. The derived formula turned out to be a good representation of the measured reality, though several approximations had to be made for the sake of an analytical expression. This contribution gives a physical interpretation on the approximations made and discusses the results in the context of the acquisition of measurement data.