Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart...
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![Page 1: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/1.jpg)
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Uncertainty Propagation forMeasurements with Multiple Output Quantities
Michael DobbertBart Schrijver
July 2014
![Page 2: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/2.jpg)
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GUM Uncertainty Propagation
July 2014
Scalar Quantities
![Page 3: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/3.jpg)
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Measurement of a Sine Wave
ܣ ௨௧ܣ
߶ ൌ��Ͳ ߶ ൌ��߶௨௧
Linear Amplifier Input Sine Wave Output Sine Wave
ݐ ՜ ݐ ՜
July 2014
![Page 4: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/4.jpg)
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Presentation
July 2014
Part 1: Develop a general solution for propagating measurement uncertainty.
Part 2: Apply the general solution to a complex-valued measurement.
Rely on matrix equations
- matrix transpose
- matrix multiplication
![Page 5: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/5.jpg)
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Matrix Transpose
July 2014
![Page 6: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/6.jpg)
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Matrix Multiplication
July 2014
ܣ ൌ�� �ଵଵ �ଵଶ�ଶଵ �ଶଶ�ଷଵ �ଷଶ
൩
ܤ ൌ�� �ଵଵ �ଵଶ �ଵଷ�ଶଵ �ଶଶ �ଶଷ
൨
ቈל ל לל ל לל ל ל
�ଷଵ �ଵଷ �ଷଶ�ଶଷ
ܣܤ ൌ�� �ଵଵ�ଵଵ �ଵଶ�ଶଵ �ଵଵ �ଵଶ �ଵଶ�ଶଶ �ଵଵ �ଵଷ �ଵଶ�ଶଷ�ଶଵ �ଵଵ �ଶଶ�ଶଵ �ଶଵ �ଵଶ �ଶଶ�ଶଶ �ଶଵ �ଵଷ �ଶଶ�ଶଷ�ଷଵ �ଵଵ �ଷଶ�ଶଵ �ଷଵ �ଵଶ �ଷଶ�ଶଶ �ଷଵ �ଵଷ �ଷଶ�ଶଷ
൩
![Page 7: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/7.jpg)
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Derivation of a GeneralUncertainty Propagation Equation
July 2014
![Page 8: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/8.jpg)
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Variance
July 2014
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Variance
July 2014
![Page 10: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/10.jpg)
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Variance Propagation
July 2014
![Page 11: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/11.jpg)
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Variance Propagation
July 2014
![Page 12: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/12.jpg)
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Variance Propagation
July 2014
![Page 13: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/13.jpg)
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Variance Propagation
July 2014
![Page 14: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/14.jpg)
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Variance Propagation
July 2014
![Page 15: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/15.jpg)
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Variance Propagation
July 2014
![Page 16: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/16.jpg)
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GUM Uncertainty Propagation
July 2014
![Page 17: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/17.jpg)
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GUM Uncertainty Propagation
July 2014
Scalar Quantities
![Page 18: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/18.jpg)
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Multiple Output Quantities
July 2014
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Multiple Output Quantities
July 2014
![Page 20: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/20.jpg)
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General Uncertainty Equation
July 2014
![Page 21: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/21.jpg)
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Presentation
July 2014
Part 1: Develop a general solution for propagating measurement uncertainty.
Part 2: Apply the general solution to a complex-valued measurement.
![Page 22: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/22.jpg)
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Uncertainty of Complex Numbers
ܣ ௨௧ܣ
߶ ൌ��Ͳ ߶ ൌ��߶௨௧
Linear Amplifier Input Sine Wave Output Sine Wave
ݐ ՜ ݐ ՜
July 2014
𝜙𝑜𝑢𝑡
𝐴𝑜𝑢𝑡𝑎+𝑏𝑖
𝑅𝑒
𝐼𝑚
![Page 23: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/23.jpg)
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Uncertainty of Complex Numbers
July 2014
![Page 24: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/24.jpg)
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Measurement of a Complex Number
July 2014
𝑡→
𝑡→
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Measurement of a Complex Number
July 2014
![Page 26: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/26.jpg)
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Uncertainty Propagation for Complex Numbers
July 2014
M. Wollensack, METAS UncLib, Online: http://www.metas.ch/unclib, 2009
𝑐𝑑 𝑦
![Page 27: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/27.jpg)
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Uncertainty Propagation for Complex Numbers
>> vc = [ 0.0001 0; 0 0.0001 ];
>> vd = [ 2.8624e-7 2.4261e-7; 2.4261e-7 4.6598e-7 ];
>> c = LinProp( sqrt( 1/2 ) + 1i*sqrt( 1/2 ), vc );
>> d = LinProp( 0.01131 + 0.02746i, vd );
>> y = c * d;
>> get_covariance( y )
ans =
1.0e-06 *
0.2217 -0.0899
-0.0899 0.7069
July 2014
(Type B)
(Type A)
directivity
measurement equation
multiplier
![Page 28: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/28.jpg)
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Uncertainty Propagation for Complex Numbers
July 2014
![Page 29: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/29.jpg)
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Conclusion
July 2014
Developed a general equation for uncertainty propagation .Applied our general equation to a complex valued measurement problem.
![Page 30: Uncertainty Propagation for Measurements with Multiple Output Quantities Michael Dobbert Bart Schrijver July 2014 1 Agilent Technologies.](https://reader030.fdocuments.in/reader030/viewer/2022032414/56649efa5503460f94c0ba71/html5/thumbnails/30.jpg)
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Conclusion
Why is this important?
July 2014
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Questions?
Michael DobbertBart Schrijver
July 2014
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Conic Sections
July 2014