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                       Uncertainty Propagation for
                Measurements with Multiple Output Quantities
                                        Michael Dobbert
                                [email protected]

                                         Bart Schrijver
                                  [email protected]

                                     Keysight Technologies
                                  1400 Fountaingrove Parkway
                                    Santa Rosa, CA, 95403


Abstract: The ISO Guide to the Expression of Uncertainty in Measurement (GUM) [1] 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.


1. Introduction
The GUM [1] assumes that a measurement system is modeled as a function of multiple real input
quantities and a single real output quantity. This is represented as

                                              (            ).                             (1)

   In this case, the measurand, , is a scalar quantity as are each                . There exist,
however, measurement problems where the measurand must be represented by more than one
quantity. To demonstrate this, consider the following example from electrical metrology.

   A common task in electrical metrology is the measurement of sine waves. Sine waves, of
course, are represented by the sine function

                                        ( )        (        ),                            (2)

where

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