18978-1-4799-2479-0/14/$31.00 ©2014 IEEE
Multivariate Statistics Applied to Assess Measurement Uncertainty of
Complex Reflection Coefficient
M. Benjamı´n, H. Silva, G. Monasterios, N. Tempone, and A. Henze
Instituto Nacional de Tecnologı´a industrial (INTI), Electro´nica e Informa´tica, Lab. Metrologı´a 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 mea-
surement 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
var [Y ] =
m∑
i=1
m∑
j=1
J(ci)cov [zi, zj ] J(cj)t (1)
Where ci = ∂f∂zi are sensitivity coefficients, J(ci) their matrix
representation and J(cj)t the transpose of J(cj)
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
var [Y ] =
m∑
i=1
m∑
j=1
cov [cizi, cjzj ] = var
[
m∑
i=1
cizi
]
(10)
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|2u2z
(
1 0
0 1
)
(14)
II. MEASURMENT MODEL
The following equation expresses the true reflection coef-
ficient 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 = −Γ
2 ∂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

19
?
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 dif-
ferent 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|4u2Mef
)(1 0
0 1
)
(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
u2Γm = u
2(Re(Γm)) + u2(Im(Γm)) (25)
For the first approach, using (20) in equation (18) we get an
overestimated variance matrix
var [Γ] = u2c1
(
1 0
0 1
)
(26)
where
u2c1 = (uD + |Γ|2uMef )2 + |Γ|2u2T + u2Γm + u2R (27)
For the second approach, using (24) in equation (18) we get
var [Γ] = u2c2
(
1 0
0 1
)
(28)
where
u2c2 = (1+2Re(Γ2))u2D+|Γ|4u2Mef+|Γ|2u2T+u2Γm+u2R (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, overestima-
tion 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 re-
flection 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.