Multivariate Statistics Applied to Assess Measurement Uncertainty of Complex Reﬂection Coefﬁcient
M. Benjam´ın, H. Silva, G. Monasterios, N. Tempone, and A. Henze Instituto Nacional de Tecnolog´ıa industrial (INTI), Electro´nica e Informa´tica, Lab. Metrolog´ıa RF & Microondas
metrologiarf@inti.gob.ar
Abstract—In this paper we show an alternative mathematical interpretation of the error propagation law for complex quantities established in supplement 2 of the GUM [1]. We use this interpretation to study VNA’s one port reﬂection measurement which includes several terms that represent complex quantities. We show two different approaches to solve the problem that arises when trying to establish the variance matrix of the sum of two complex quantities. We also give explicit formulae to estimate uncertainty with both approaches.
Keywords—GUM, multivariate statistics, RF metrology, VNA measurement, complex quantities, measurement uncertainty.
I. THEORY
Supplement 2 of the GUM [1] establishes a method for
the expression of uncertainty in multiple output measurements.
This means determining the variance matrix of the output
vector. In the special case where measurand and input quan-
tities are complex, an alternative formulation of the error
propagation law is presented in [2]. This formulation is a
matrix analogy of the well known law in the univariate case. Let f : Cm → C be an analytic function and the complex
measurand Y = f (z1, . . . , zm), [2] shows that the variance matrix of Y is
mm
var [Y ] =
J (ci)cov [zi, zj] J (cj)t
(1)
i=1 j=1
Wrephreerseenctia=tion∂∂zfai nadreJs(ecnj)stititvhietytrcaonesfpﬁocsieenotfs,JJ((ccji)) their matrix
J (ci) =
Re(ci) Im(ci)
−Im(ci) Re(ci)
(2)
The covariance matrix of two complex variables is
cov [zi, zj] =
u[Re(zi),Re(zj )] u[Re(zi),Im(zj )] u[Im(zi),Re(zj )] u[Im(zi),Im(zj )]
(3)
Where u[Re(zi), Re(zj)] is the covariance between Re(zi) and Re(zj). u[Re(zi), Re(zi)] = u2[Re(zi)] is the variance of Re(zi). If z and w are complex variables and c and d are constant
complex values, covariance matrices have the following prop-
erties
var [z] = cov [z, z]
(4)
cov [z, w] = cov [w, z]t
(5)
cov [cz, dw] = J(c)cov [z, w] J(d)t
(6)
cov [z1 + z2, w] = cov [z1, w] + cov [z2, w]
(7)
var [z + w] = var [z] + var [w] + cov [z, w] + cov [w, z] (8)
var [−z] = var [z]
(9)
Using (4), (6) and (8) in equation (1) the variance matrix of the measurand can be interpreted as
mm
m
var [Y ] =
cov [cizi, cjzj] = var
cizi (10)
i=1 j=1
i=1
The complex variables z and w are uncorrelated if
cov [z, w] =
0 0
0 0
(11)
From (8) we get that
var [z + w] = var [z] + var [w]
(12)
We say z is “circular” and has a circular variance matrix if
var [z] = u2z
1 0
0 1
(13)
Where u2z = u2[Re(z)] = u2[Im(z)]. If c is a constant complex
value,
var [cz] = |c|2uz2
1 0
0 1
(14)
II. MEASURMENT MODEL
The following equation expresses the true reﬂection coefﬁcient Γ of a one port device measured with a VNA.
Γ=
Γm − D M (Γm − D) + T
−R
(15)
Here Γm is the measured value of Γ. D, M and T are the residual directivity, source-match and reﬂection tracking respectively. R accounts for external error terms, such as
cable ﬂexibility, stability, etc. All terms are complex quantities. The following approximations can be considered: T ≈ 1, M, D, R ≈ 0 and Γm ≈ Γ. After complex differentiation and using the approximate values,
∂Γ ∂D
=
−1
∂Γ ∂T
=
−Γ
∂Γ ∂M
= −Γ2
∂Γ ∂R
=
−1
(16)
Replacing in (10) and using (9), we have that
var [Γ] = var D + Γ2M + ΓT + Γm + R
(17)
III. UNCERTAINTY ANALYSIS
All input variables are assumed uncorrelated , and except for Γm they all satisfy (13). The Ripple technique described in [3] is a standard approach to assess var [D] and var [Mef ]. Where Mef , shown in Fig.1, is the “Effective Test Port Match ”. This quantity is similar to M and is assumed to satisfy (13). Guidelines in [3] establish
978-1-4799-2479-0/14/$31.00 ©2014 IEEE
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Fig. 1. Ripple signal ﬂow graph for Mef
to use Mef as M , and warns that Mef could be correlated with D. Using (12) and (14) in (17) we get
var [Γ] = var D + Γ2Mef +|Γ|2var [T ]+var [Γm]+var [R] (18)
The last three terms of the sum do not present any analytical difﬁculties. For the ﬁrst one, [3] neither explains how to calculate the covariance needed nor gives a bivariate treatment to the problem. In order to overcome this situation we present two different approaches we developed. One that overestimates var D + Γ2Mef and another where the covariance between Mef and D is calculated. Although the second approach is more rigorous, it makes more assumptions about the model. The ﬁrst approach is also of general interest beyond the case of the reﬂection coefﬁcient measurement. It serves as an example of what can be done to establish the variance matrix of the sum of two complex quantities when their covariance matrix is unknown.
A. Overestimation of the Variance of the Sum
Using (8), the fact that D and Mef satisfy (13), and assuming that u[Re(D), Im(Mef )] = u[Im(D), Re(Mef )] = 0
var [D+Γ2Mef ] =
u2[Re(D)+Re(Γ2Mef )]
0
0
u2[Im(D)+Im(Γ2Mef )]
(19)
Due to missing information about covariance, diagonal ele-
ments can not be calculated. Cauchy-Schwarz inequality for
random scalars is used to bound them in order to get a worst
case variance matrix as follows
var [D+Γ2Mef ] = (uD + |Γ|2uMef )2
1 0
0 1
(20)
B. Covariance Matrix of D and Mef
Solving the ﬂow graph in Fig. 1 yields
Mef ≈ M + D + L
(21)
Where L is the reﬂection coefﬁcient of the air-line used for the Ripple technique. L is assumed to satisfy (13) and is
uncorrelated with all input variables. Using (4), (7) and (21)
cov [Mef , D] = cov [M, D] + cov [D, D] + cov [L, D] = var [D]
(22)
Using (5), (6), (8), (14), (22) and assuming that D is circular, it is possible to obtain the following equality
var D + Γ2Mef
=var [D] + |Γ|4var [Mef ] + . . . . . . + (J (Γ2) + J (Γ2)t)var [D]
(23)
Finally, this can be expressed as
var D + Γ2Mef = (1 + 2Re(Γ2))u2D + |Γ|4uM 2 ef
10 01
(24)
IV. MEASURAND’S VARIANCE MATRIX
In this section we establish var [Γ] for both approaches. In order to get a circular variance matrix for the measurand we overestimate var [Γm] with a circular variance matrix. A common practice is to do this with
uΓ2 m = u2(Re(Γm)) + u2(Im(Γm))
(25)
For the ﬁrst approach, using (20) in equation (18) we get an overestimated variance matrix
var [Γ] = uc21
1 0
0 1
(26)
where
uc21 = (uD + |Γ|2uMef )2 + |Γ|2uT2 + u2Γm + u2R
(27)
For the second approach, using (24) in equation (18) we get
var [Γ] = uc22
1 0
0 1
(28)
where
uc22 = (1+2Re(Γ2))u2D +|Γ|4uM 2 ef +|Γ|2uT2 +uΓ2 m +uR2 (29)
In order to compare both matrices it is enough to compare uc1 and uc2.
V. COMPARISION OF BOTH APPROACHES
We show uc1 and uc2 deﬁned in (27) and (29) for three real measurement of low, medium and high reﬂection coefﬁcients at 18 GHz with a type N connector
Γlow
Γmed
Γhigh
|Γ| 0.058 0.559 0.980
The values obtained for uc1 and uc2 were
×10−3 Γlow Γmed Γhigh uc1 7.8 11.9 20.5 uc2 7.8 7.8 12.2
VI. CONCLUSION
Formulating the variance matrix of a measurand as in (10) allows a better understanding of the error propagation law. The approach shown in (III-A) is a useful resource when the covariance between two complex variables is unknown. In (V) we showed that overestimation was too big for measurements with medium or high reﬂection values. Therefore, overestimation should be avoided whenever possible, for example using an approach similar to (III-B). Equations (28) and (29) are of special interest for one port reﬂection measurements as they establish an explicit expression for the measurand’s variance matrix.
REFERENCES
[1] BIPM, IEC, IFCC, ISO, IUPAC, UPAP and OIML “Evaluation of Measurement Data - Supplement 2 to the ‘Guide to the Expression of Uncertainty in Measurement’. Extension to any number of output quantities.” JCGM 102:2011.
[2] B. D. Hall “On the propagation of uncertainty in complex-valued quantities,” Metrologia, vol 41 pp 173 - 177. 2004
[3] EURAMET cg-12 v2.0. “ Guidelines on the Evaluation of Vector Network Analysers (VNA),” European co-operation for Accreditation, 2011.
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