Software Tools for Evaluation of Measurement Models for Complex-valued Quantities in Accordance with Supplement 2 to the GUM Speaker: C.M.Tsui The Government of the Hong Kong Special Administrative Region Standards and Calibration Laboratory 36/F

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Software Tools for Evaluation of Measurement
Models for Complex-valued Quantities in
Accordance with Supplement 2 to the GUMSpeaker
C.M.TsuiThe Government of the Hong Kong Special
Administrative RegionStandards and Calibration
Laboratory36/F Immigration Tower, 7 Gloucester
Road, Wanchai, Hong KongPhone (852) 2829 4850,
Fax [phone], Email cmtsui_at_itc.gov.hkAut
hors C.M.Tsui, Y.K.Yan, H.W. LiThe Government
of the Hong Kong Special Administrative
RegionStandards and Calibration Laboratory
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Case 1 A Simple Case First
?
What are the standard uncertainties of output
quantities Y1 and Y2 ? And what is the
correlation between them ?
X1
Multivariate Measurement Model
Y1

• Y1 X1 X3
• Y2 X2 X3

X2
Y2
X3
Input quantities X1, X2, X3 are independent and
have Gaussian distribution with mean 0 and
standard uncertainties of 1.
3
Case 2 A little bit more complicated
?
X1
Measurement Model
Y1

• Y1 X1 X3
• Y2 X2 X3

X2
Y2
X3
Input quantities X1, X2 remain the same. X3 now
has rectangular distribution with mean 0 and
standard uncertainty of 3.
3 v3
0
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Case 3 Complex Number Measurement Model

• A simplified measurement model for the effective
  output voltage reflection coefficient (?) of a
  power splitter.
• The 3-port S-parameter and ? are complex
  quantities.

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A Short Demonstration First Dont Worry If You
Dont Know What I am Doing I Will Explain Shortly
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Lets Try Some Variation

• What happens if we change the measurement model
  from this
• Y1 X1 X3
• Y2 X2 X3
• to that
• Y1 X1 X3
• Y2 X2 - X3

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GUM and the Supplements

• In 1993, the GUM was released. It defined the GUM
  Uncertainty Framework (GUF) for evaluation of
  measurement uncertainties.
• There are limitations in the application of GUF.
  In 2008, Supplement 1 was published. It concerned
  with the propagation of probability distributions
  through a measurement model.
• The conditions for valid application of the GUF
  are described in details in sections 5.7 and 5.8
  of Supplement 1. When these conditions cannot be
  met, a Monte Carlo method (MCM) should be used.

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GUM and the Supplements

• The GUM and Supplement 1 mainly deal with models
  having any number of input but only one output
  quantity.
• In many real world measurement systems,
  especially when complex numbers are involved,
  there are more than 1 output. The output may be
  correlated. The coverage region is
  multidimensional and much more complicated than
  the univariate cases.
• The GUM and Supplement 1 are inadequate for
  measurement models with multiple outputs. The new
  Supplement 2, deals with models having any number
  of input quantities and any number of output
  quantities.

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The SCL software tools (1)

• The SCL software tools support evaluation of
  complex valued measurement models in accordance
  with Supplement 2. Users only need to
• encode the measurement model as a Visual Basic
  subroutine
• specify parameters of uncertainty components,
  such as estimates, standard uncertainties and
  probability distribution function (PDF)
• The software tools, written in Visual C and
  Visual Basic, are tightly integrated with
  Microsoft Excel which serves as front-end user
  interface.
• Computational intensive routines that require
  faster execution speed were developed in Visual
  C and compiled into a Dynamic Link Library
  (DLL).

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The SCL software tools (2)

• The SCL software tools include two parts that can
  be used separately.
• Simulator
• User Defined Function
• The simulator enable the users to vary the
  parameters of input quantities and the
  measurement model interactively. Users can
  experiment with different configurations of the
  measurement system and see the effects on the
  measurement uncertainties.
• The user-defined function can be embedded in any
  Microsoft Excel worksheet like an ordinary
  function for GUF or MCM computation.

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The 3 Stages of Uncertainty Evaluation

• Formulation
• Define input X and output Y.
• Establish mathematical relationship between X and
  Y (i.e. the measurement model)
• Propagation
• Obtain PDF of Y from X through the measurement
  model.
• Summarizing
• The expectation, covariance matrix and coverage
  region of Y are obtained from PDF of Y.

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Formulation (setting up measurement model)

• Section 9.2 of Supplement 2 describes the
  following additive measurement model.
• Y1 X1 X3
• Y2 X2 X3
• Public Static Sub model(x() As Double, y() As
  Double)
• y(1) x(1) x(3)
• y(2) x(2) x(3)
• End Sub

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Formulation (Setting up Input Quantities and
Assigning PDF)
The following PDF types are supported CTP -
curvilinear trapezoid E - exponential G -
Gaussian R - rectangular TP - trapezoidal TR -
triangular T - student-t U - arc sine.
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Propagation

• 3 ways to propagate the distributions.
• Analytical method (impractical for most real
  world measurement systems)
• First order Taylor series approximation of the
  measurement model (GUF)
• Numerical method (MCM)
• The SCL software tools support the second and
  third methods.

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Propagation (First order Taylor series
approximation)

• the input quantities are characterized by
• a vector representing the estimates of the input
  quantities x (x1, xN)T
• a covariance matrix Ux containing the covariance
  of the input quantities.
• Measurement model denoted by Y f(X),
• Estimate y f(x).
• Covariance matrix Uy CxUxCxT
• Cx is the sensitivity matrix. The entry at row i
  and column j of Cx is given by the partial
  derivative ?fi/?xj.

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Propagation (MCM)

• The idea of MCM is to make large number of draws
  from the PDF of the input quantities and to
  derive the output quantities for each draw. The
  larger is the number of draws, the more reliable
  are the results. The trade-off is a longer
  computation time.
• There are two ways to select the number of Monte
  Carlo trials. A fixed number can be chosen
  beforehand. Alternatively an adaptive algorithm
  may be used to determine the number of trials
  on-the-fly based on the stability of the
  simulated output.

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Summarizing
Histogram of PDF for an output quantity.
Contour of the joint PDF for the output
quantities
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Summarizing (coverage region)

• It is not easy to determine the multidimensional
  coverage regions for vector output quantities.
• Supplement 2 considers only two types of coverage
  regions for multivariate cases hyper-ellipse and
  hyper-rectangle.
• They are characterized by two sets of quantities
  the covariance matrix for the output quantities
  and a scalar parameter (kp for hyper-ellipse and
  kq for hyper-rectangle) which determines the
  volume under the PDF corresponding to the
  coverage probability.

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Case 3 Complex Number Measurement Model

• A simplified measurement model for the effective
  output voltage reflection coefficient (?) of a
  power splitter.
• The 3-port S-parameter and ? are complex
  quantities.

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Visual Basic User Defined Data Type

• It is quite easy to represent complex numbers in
  Visual Basic by means of the user defined data
  type.
• The following declares a user defined data type
  called Complex (actually you can give it any
  name you like), variable r is the real part and i
  is the imaginary part
• Type Complex
• r As Double
• i As Double
• End Type

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Declare a Variable as Complex Quantity

• Suppose the variable x(1) stores the magnitude of
  a complex number and x(2) stores the phase.
• We can declare a variable S22 as complex data
  type and initialize its real and imaginary parts
  as follows
• Dim S22 As Complex
• S22.r x(1) Cos(x(2))
• S22.i x(1) Sin(x(2))
• From now on you can treat S22 as a single item.

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Function to add complex numbers

• Next, we can define function to manipulate
  complex numbers. The following is a function to
  add two complex numbers
• Public Function cadd(a As Complex, b As Complex)
  As Complex
• cadd.r a.r b.r
• cadd.i a.i b.i
• End Function
• The following is an example of using the cadd
  function.
• R PQ (1 i)(3 2i) 4 I
• Dim P As Complex, Q As Complex, R As Complex
• P.r 1 P.i -1
• Q.r 3 Q.i 2
• R cadd(P, Q)

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Function to multiply complex numbers

• From high school algebra, for complex number
  multiplication, we have
• (p qi) (x yi) (px-qy) (qxpy)i
• Public Function cmul(a As Complex, b As Complex)
  As Complex
• cmul.r a.r b.r - a.i b.i
• cmul.i a.r b.i a.i b.r
• End Function
•  

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Function to divide complex numbers

•  
• For complex number division, we have
• Public Function cdiv(a As Complex, b As Complex)
  As Complex
• Dim D As Double
• D b.r b.r b.i b.i
• cdiv.r (a.r b.r a.i b.i) / D
• cdiv.i (a.i b.r - a.r b.i) / D
• End Function

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Representing Complex Number Equation

• Using the above functions, the following complex
  number equation can be represented by a single
  Visual Basic statement
• VRC cminus(S22, cdiv(cmul(S12, S23), S13))

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Putting All Together
Type Complex r As Double i As
Double End Type Public Static Sub model(x() As
Double, y() As Double) ' X(1) s22 magnitude
X(2) s22 phase ' X(3) s12 magnitude
X(4) s12 phase ' X(5) s23 magnitude
X(6) s23 phase ' X(7) s13 magnitude
X(8) s13 phase Dim S22 As Complex, S12 As
Complex, S23 As Complex, S13 As Complex Dim VRC
As Complex S22.r x(1) Cos(x(2)) S22.i x(1)
Sin(x(2)) S12.r x(3) Cos(x(4)) S12.i
x(3) Sin(x(4)) S23.r x(5) Cos(x(6)) S23.i
x(5) Sin(x(6)) S13.r x(7) Cos(x(8))
S13.i x(7) Sin(x(8)) VRC cminus(S22,
cdiv(cmul(S12, S23), S13)) y(1) VRC.r y(2)
VRC.i End Sub
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The Input Quantities for Case 3

• Many people think it will be more appropriate to
  treat the magnitude and phase, rather than the
  real and imaginary parts, of S-parameter measured
  by a network analyzer as independent and use them
  as input quantities to a measurement model.

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Some Considerations in Encoding Complex Quantity
Models (1)

• There are commonly two ways to represent a
  complex quantity either in terms of its real and
  imaginary parts or in polar form by its magnitude
  and phase.
• It is recommended that the output quantities of
  the model should be in terms of real and
  imaginary parts.
• The reason is that in the summarizing stage of
  MCM, statistical analysis is applied to the MCM
  trials.
• It is known that statistical analysis on complex
  quantities will produce different results
  depending on whether the inputs to the
  statistical analysis are represented in polar
  form or not.

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Some Considerations in Encoding Complex Quantity
Models (2)

• Potential problems in statistical analysis on
  polar form
• The transformation between rectangular and polar
  form is non-linear.
• The real and imaginary axes extend to plus and
  minus infinity while the magnitude is always
  non-negative.
• The phase is cyclical in nature.
• To avoid these problems, the output quantities of
  the measurement model should be in terms of real
  and imaginary parts.
• Results should only be converted into polar form
  after statistical analysis.

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User Defined Function

• The second part of the SCL software tools is an
  Excel user-defined function (UDF) that can be
  embedded in any Excel worksheet for GUF or MCM
  computation.
• An Excel UDF behaves just like other Excel
  built-in functions. It takes arguments and
  returns a value. An Excel UDF will be executed
  whenever any value in its argument list changes.
  Re-calculation is automatic.
• A drawback of Excel UDF is that if it takes a
  long time to execute, such as when running MCM
  for a large number of trials on a complicated
  measurement model, Microsoft Excel will appear
  frozen.

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User Defined Function
The syntax of the UDF is gum2(range, index,
sim_mode, sim_par, conf_type, model_index) range
To point to the table of input
quantities. index To specify the return
parameter of the function sim_mode Enter 1 to
select adaptive simulation mode. Enter 2 to
select fixed sample size simulation mode sim_par
Enter number of significant digits (1 or 2)
for adaptive simulation mode. Enter the number of
trials for fixed sample size simulation mode.
  conf_type Optional parameter. Enter 1 to
select symmetrical coverage interval type. Enter
2 to select shortest coverage interval type. This
is for compatibility with Supplement 1 to the
GUM. model_index Optional parameter for future
expansion.
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User Defined Function
Computed by MCM Computed by MCM Computed by GUF Computed by GUF
index Return parameter index Return parameter
1 expectation of measurand 1 101 expectation of measurand 1
2 standard uncertainty of measurand 1 102 standard uncertainty of measurand 1
3 low boundary of 95 confidence interval of measurand 1 103 effective degree of freedom of measurand 1
4 high boundary of 95 confidence interval of measurand 1 104 coverage factor of measurand 1
11 expectation of measurand 2 105 expanded uncertainty of measurand 1
12 standard uncertainty of measurand 2 111 expectation of measurand 2
13 low boundary of 95 confidence interval of measurand 2 112 standard uncertainty of measurand 2
14 high boundary of 95 confidence interval of measurand 2 113 effective degree of freedom of measurand 2
21 kp 114 coverage factor of measurand 2
22 kq 115 expanded uncertainty of measurand 2
23 correlation coefficient 121 correlation coefficient