Circular Statistics for Antenna Measurements

Statistics is a necessary tool to assess the uncertainties in a measurement technique. In this way, we can determine the reliability of the technique and can aim to improve this. In measurement techniques where both the amplitude and phase are important, statistics should be applied not only to the magnitude, but also to the phase. Linear statistics has several disadvantages when applied to the phase. Most importantly, the computed mean is dependent on the chosen branch cut position and the computed confidence interval is not completely accurate, even if the computed mean is correct. Although circular statistics does not have these issues, it is, to the best of our knowledge, not yet used in antenna measurements, except for channel sounding. In this article, methods known in circular statistics to compute the mean and deviation are explained, along with some circular distribution functions. The effects are shown in two case studies for a planar near-field (NF) measurement and a phase pattern measurement. Furthermore, a methodology is constructed by the steps that should be taken to compute the expanded uncertainty of the phase. Using circular distribution functions, this can be computed more accurately at the cost of a higher computation time.