Probability distributions of random errors in frequency-domain measurements

In this paper, we investigate random errors in a frequency-domain measurement system, and we perform experiments at frequencies up to 20 GHz where the random errors are pronounced. We focus in obtaining a probability distribution of the S-parameter errors in magnitude (dB) and phase. Experiment results show that S-parameters uncertainty display a quasi-normal distribution at different spot frequencies. It is normally recommended that treatment of uncertainty should be done in terms of real and imaginary components of the S-parameters. However, experiments presented here demonstrate that the uncertainty can directly be treated in magnitude (dB) and phase, which is more convenient for an RF engineer. We also investigate the auto-correlation of the experimental data in real-imaginary as well as in the magnitude-phase planes. In this regard, measured data is treated as stochastic processes coming from a sample space that is wide sense stationary with zero mean. It is widely known that the real and imaginary parts of the S-parameter data are related by the Hilbert Transform and therefore their auto-correlations should also be similar. The auto-correlation analysis could be used in obtaining the imaginary (phase) uncertainty from the real (magnitude dB) part uncertainty. In this paper, indeed, we demonstrate that auto-correlation, across the whole frequency range, of the real part and imaginary are similar. Comparable results are shown for magnitude and phase. These results can be used for modeling purposes and obtain variability bounds in VNA measurements.