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.
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