Expanded uncertainty regions for complex quantities in polar coordinates

Hall, B. D.

doi:10.1088/0026-1394/52/4/486

Cited by 7 publications

(4 citation statements)

References 13 publications

(25 reference statements)

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“…The shape of that region is arbitrary, so this article considers using ellipses, circles, rectangles and two forms of parallelogram to report the expanded uncertainty of measurement results expressed in rectangular coordinates 1 . The performance of these uncertainty regions is compared in terms of the coverage probability, or level of confidence, achieved under different measurement error conditions.…”

Section: Introductionmentioning

confidence: 99%

“…Unfortunately, the extent of an elliptical region is difficult to report in a simple way. Alternatively, circular and rectangular regions are easy to describe, but do not use in their construction all the uncertainty 1 Uncertainty regions for complex quantities expressed in polar coordinates are the subject of another paper [1]. information available.…”

Section: Introductionmentioning

confidence: 99%

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Expanded uncertainty regions for complex quantities

Hall¹

2013

Metrologia

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The expanded measurement uncertainty of a complex quantity is a region in the complex plane surrounding the measured value. This paper considers different shaped uncertainty regions in the form of ellipses, circles, rectangles and parallelograms. The different types of region are compared, under a variety of measurement error conditions, with regard to coverage probability and relative area. Elliptical confidence regions are commonly used in multivariate statistics. However, this shape has not been adopted widely in metrology, perhaps because there is no simple way to report the extent of an elliptical region. The other shapes considered are easier to use. Unfortunately, the coverage probability of circular uncertainty regions is found to be sensitive to both the form of the distribution of measurement errors and to the number of degrees of freedom, making this shape a poor choice. Parallelograms and rectangles both performed well, with parallelograms giving the best results overall.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Expanded uncertainty regions for complex quantities

Hall¹

2013

Metrologia

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“…Current practice is often to report results as independent uncertainty intervals for real-valued magnitude and phase 7 , rather than an annular sector (simultaneous uncertainty intervals for magnitude and phase, forming a region of uncertainty). The coverage probability is, of course, overstated if independent intervals are misinterpreted as describing a region for the complex quantity [23]. For example, to construct a region with approximately 95% coverage, ( ) ∞ = k 2.24 GUM,0.975 should be used when calculating the polar coordinate intervals.…”

Section: Polar Coordinatesmentioning

confidence: 99%

Evaluating the measurement uncertainty of complex quantities: a selective review

Hall

2015

Metrologia

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“…However, for assessing the complex uncertainty of S-parameters involving multiple frequency points and variables, the covariance matrix method is preferred. The covariance matrix is considered an extension of linear propagation, which can effectively address the uncertainty evaluation and propagation of such complex objects [ 3 , 4 , 5 , 6 ].…”

Section: Introductionmentioning

confidence: 99%

Assessment of Measurement Uncertainty for S-Parameter Measurement Based on Covariance Matrix

Zhu,

Wang,

Zhao

et al. 2024

Sensors

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S-parameters are widely used to detail the scattering parameters of radio frequency (RF) components and microwave circuit modules. The vector network analyzer (VNA) is the most commonly used device for measuring S-parameters. Given the multiple frequency points, complex values, and intricate uncertainty propagation involved, accurately assessing the uncertainty of S-parameter measurements is difficult. In this study, we proposed a new method for assessing S-parameter uncertainty based on the covariance matrices, tracing back to the nominal uncertainty of calibration standards. First, we analyzed the relevant theory of uncertainty assessment using covariance matrices and subsequently deduced the mechanism of Type B uncertainty propagating from calibration standards to error model coefficients and S-parameter measurements to evaluate Type B measurement uncertainty. In this study, a novel measurement system was constructed for measuring grounded coplanar waveguides by using a VNA and calibration standards with 8- and 12-error models. Initially, the model assessed the Type B uncertainty of measuring four S-parameters of a grounded coplanar waveguide. Next, the VNA calibrated with the 12-error model was used to conduct multiple repeated measurements to assess the Type A uncertainty of the grounded coplanar waveguide. Finally, the composite uncertainty was constructed, which demonstrated that the proposed method can be used for assessing the uncertainty of S-parameters.

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Expanded uncertainty regions for complex quantities in polar coordinates

Cited by 7 publications

References 13 publications

Expanded uncertainty regions for complex quantities

Expanded uncertainty regions for complex quantities

Evaluating the measurement uncertainty of complex quantities: a selective review

Assessment of Measurement Uncertainty for S-Parameter Measurement Based on Covariance Matrix

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