Uncertainty Propogation for Measurements with Multiple Output Quantities

The ISO Guide to the Expression of Uncertainty in Measurement [1] (GUM) limits the description of the law of propagation of uncertainty to real input quantities and a single real output quantity. The GUM provides little guidance for uncertainty analysis of measurements with multiple output quantities, such as complex valued S-Parameter measurements that have both real and imaginary components. Complex measurement quantities are common in RF and microwave measurements. Likewise, measurements with multiple output quantities exist in many disciplines. Supplement 2 [2] to the GUM extends the law of propagation of uncertainty to an arbitrary number of output quantities, which is a more general solution. This paper discusses this more general solution clearly and concisely using matrix notation. It demonstrates that the GUM expressions for uncertainty propagation are a specific case of this more general solution. This method is then applied to a practical measurement uncertainty example involving complex quantities.