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Agilent Technologies1
Uncertainty Propagation forMeasurements with Multiple Output Quantities
Michael DobbertBart Schrijver
July 2014
Agilent Technologies2
GUM Uncertainty Propagation
July 2014
Scalar Quantities
Agilent Technologies3
Measurement of a Sine Wave
ܣ ௨௧ܣ
߶ ൌ��Ͳ ߶ ൌ��߶௨௧
Linear Amplifier Input Sine Wave Output Sine Wave
ݐ ՜ ݐ ՜
July 2014
Agilent Technologies4
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
Agilent Technologies5
Matrix Transpose
July 2014
Agilent Technologies6
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
Agilent Technologies7
Derivation of a GeneralUncertainty Propagation Equation
July 2014
Agilent Technologies8
Variance
July 2014
Agilent Technologies9
Variance
July 2014
Agilent Technologies10
Variance Propagation
July 2014
Agilent Technologies11
Variance Propagation
July 2014
Agilent Technologies12
Variance Propagation
July 2014
Agilent Technologies13
Variance Propagation
July 2014
Agilent Technologies14
Variance Propagation
July 2014
Agilent Technologies15
Variance Propagation
July 2014
Agilent Technologies16
GUM Uncertainty Propagation
July 2014
Agilent Technologies17
GUM Uncertainty Propagation
July 2014
Scalar Quantities
Agilent Technologies18
Multiple Output Quantities
July 2014
Agilent Technologies19
Multiple Output Quantities
July 2014
Agilent Technologies20
General Uncertainty Equation
July 2014
Agilent Technologies21
Presentation
July 2014
Part 1: Develop a general solution for propagating measurement uncertainty.
Part 2: Apply the general solution to a complex-valued measurement.
Agilent Technologies22
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
Agilent Technologies23
Uncertainty of Complex Numbers
July 2014
Agilent Technologies24
Measurement of a Complex Number
July 2014
𝑡→
𝑡→
Agilent Technologies25
Measurement of a Complex Number
July 2014
Agilent Technologies26
Uncertainty Propagation for Complex Numbers
July 2014
M. Wollensack, METAS UncLib, Online: http://www.metas.ch/unclib, 2009
𝑐𝑑 𝑦
Agilent Technologies27
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
Agilent Technologies28
Uncertainty Propagation for Complex Numbers
July 2014
Agilent Technologies29
Conclusion
July 2014
Developed a general equation for uncertainty propagation .Applied our general equation to a complex valued measurement problem.
Agilent Technologies30
Conclusion
Why is this important?
July 2014
Agilent Technologies31
Questions?
Michael DobbertBart Schrijver
July 2014
Agilent Technologies32
Conic Sections
July 2014