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Uncertainty Propagation forMeasurements with Multiple Output Quantities

Michael DobbertBart Schrijver

July 2014

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GUM Uncertainty Propagation

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Scalar Quantities

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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

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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

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Matrix Transpose

July 2014

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Matrix Multiplication

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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

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Derivation of a GeneralUncertainty Propagation Equation

July 2014

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Variance

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Variance

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Variance Propagation

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Variance Propagation

July 2014

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Variance Propagation

July 2014

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Variance Propagation

July 2014

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Variance Propagation

July 2014

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Variance Propagation

July 2014

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GUM Uncertainty Propagation

July 2014

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GUM Uncertainty Propagation

July 2014

Scalar Quantities

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Multiple Output Quantities

July 2014

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Multiple Output Quantities

July 2014

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General Uncertainty Equation

July 2014

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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.

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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

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Uncertainty of Complex Numbers

July 2014

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Measurement of a Complex Number

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𝑡→

𝑡→

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Measurement of a Complex Number

July 2014

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Uncertainty Propagation for Complex Numbers

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M. Wollensack, METAS UncLib, Online: http://www.metas.ch/unclib, 2009

𝑐𝑑 𝑦

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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

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(Type B)

(Type A)

directivity

measurement equation

multiplier

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Uncertainty Propagation for Complex Numbers

July 2014

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Conclusion

July 2014

Developed a general equation for uncertainty propagation .Applied our general equation to a complex valued measurement problem.

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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