A generalised approach to the propagation of uncertainty in complex S-parameter measurements

Measurement Uncertainty of Complex-Valued Microwave Quantities

“…From the literature, transformation of the uncertainty evaluated in rectangular coordinate to polar coordinate is suggested to be a good solution [10]. However the transformation has not yet been fully explored for realistic measurement scenarios.…”

“…This is because the representation in polar coordinate bears a direct relationship to some physical phenomena affecting the measurement process. For example in microwave applications, phase is directly related to the electrical path length of a signal, while magnitude is directly related to the amplified/attenuated level of a transmitted signal [10]. Therefore, it is often required to evaluate the measurement uncertainties of the magnitude and phase representations in polar coordinate.…”

“…It is noted that the estimated correlation coefficient in covariance matrix is rarely meaningful based on such relatively small sample sizes (other examples are 5 to 6 samples in [1,10]), and is inherently unreliable for practical measurements. Hence, the proposed method in [3] has limited applications without detailed knowledge of its dependence on the correlation coefficient in the covariance matrix.…”

“…For the often experienced complex-valued microwave quantities, the associated measurement uncertainties have been strongly recommended to be evaluated in rectangular coordinate [10], following the internationally recommended guidelines [5]. More efforts on the evaluations of measurement uncertainty for complex-valued quantities can be found in [11][12][13][14].…”

“…In most of works [10][11][12][13][14], covariance matrix which groups together information about the standard uncertainties and correlation between the real and imaginary components has been used to describe the complex-valued quantities. With the covariance matrix, the uncertainty evaluated in the rectangular coordinate could be transformed into the desired polar coordinate using the law of propagation of uncertainty in matrix form [10,13,14].…”