Measurement Uncertainty of Complex-Valued Microwave Quantities

Abstract

Abstract-This paper presents an evaluation of measurement uncertainty for complex-valued quantities in microwave applications, mainly focusing on the non-linear transformation of measurement uncertainty from rectangular coordinate to polar coordinate. Based on the law of propagation of uncertainty in matrix form, general expressions of the covariance matrix for the magnitude and phase uncertainties in polar coordinate have been derived, and several different application scenarios have been analyzed and evaluated with numerical simulations. This is followed by some recommendations on the coordinate transformations in practical microwave measurements.

Comparative Evaluation of Uncertainty Transformation for Measuring Complex-Valued Quantities

“…Both the equations are the same as the results derived using the separated LPU transformations for |S| and φ in [11]. It is also noted that if u(φ) > π 2 (happened when the uncertainty region encompasses the origin), u(φ) = π 2 [6].…”

“…The law of propagation of uncertainty (LPU) in [5] is then a possible solution to propagate the uncertainty in rectangular coordinate to polar coordinate, and has been preliminarily discussed in [11]. As a continued work of [11], we will compare its performance for the uncertainty coordinate transformation with another method using the coordinate rotation [10] in this paper. In the following, both the methods are described in details in Section 2.…”

Practical evaluation of phase uncertainty in scattering parameter measurements

“…As reported in [5], [6], uncertainty evaluated in a rectangular coordinate could be transformed to |S| and φ in a polar coordinate. Law of Propagation of Uncertainty in [2] and Coordinate Rotation [5] have found to be possible solutions for uncertainty transformation.…”

Direct Comparison Transfer of Microwave Power Sensor Calibration With an Adaptor: Modeling and Evaluation

“…For the Type-A method, the standard uncertainty is evaluated by the statistical analysis of series of observations, while for the Type-B method, the standard uncertainty is obtained from other information including previous measurement data, specifications from manufacturers, data provided in calibration and other certificates, and uncertainties assigned to reference data taken from handbooks etc.. It is noted that in this study for the complex-valued microwave quantities such as S-parameter and reflection coefficient, their standard uncertainties are evaluated with the assumption of zero correlation between their real and imaginary parts [12,13].…”