Reverberation chambers have been used for measuring antenna efficiencies. In this letter, the measurement uncertainty of the efficiency measurement is analyzed and a simple uncertainty model is given. The model is verified by extensive measurements

Chen, Xiaoming

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Measurement Uncertainty of Antenna Efficiency in a Reverberation Chamber

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Measurement Uncertainty of Antenna Efficiency in a Reverberation Chamber
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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 55, NO. 6, DECEMBER 2013 1331
Letters
Measurement Uncertainty of Antenna Efficiency in a Reverberation Chamber
Xiaoming Chen
Abstract—Reverberation chambers have been used for measur-
ing antenna efficiencies. In this letter, the measurement uncertainty
of the efficiency measurement is analyzed and a simple uncertainty
model is given. The model is verified by extensive measurements.
Index Terms—Antenna efficiency, independent sample number,
measurement uncertainty, reverberation chamber (RC).
I. INTRODUCTION
THE reverberation chamber (RC) has traditionally beenused for electromagnetic compatibility tests [1]. Over the
past decade, it has found applications in various over-the-air
(OTA) tests such as measuring antenna efficiencies [2], diversity
gains and ergodic capacities of multiantenna systems [3], and
bit error rates of telecommunication systems [4], [5]. Almost
all of these applications necessitate a reference measurement
(to determine the average power transfer function Pref [6]) for
calibrating the chamber. Therefore, the measurement accuracy
of Pref affects the measurement accuracies of all the OTA tests.
The measurement uncertainty of Pref was studied in [6] and [7].
However, the actual uncertainty effects of Pref on OTA tests
have not been studied quantitatively yet.
In this letter, we will study the measurement uncertainty of
antenna efficiency in an RC. In the literature, there are quite
a few different methods for measuring antenna efficiency in
an RC, e.g., [8]–[11]. A time-reversal method was presented
in [8] for measuring the antenna efficiency in the time domain.
However, the time-reversal method is limited to ultrawideband
antennas. To eliminate the use of a reference antenna [2], [6],
different approaches were introduced in [9]–[11]. In [9], antenna
efficiencies were measured by reflection measurements, which
sometimes suffer poor accuracy in the return loss. By using
two identical antennas under test (AUT), the antenna efficiency
can be measured by estimating the quality factor of the cham-
ber [10]. Several methods for measuring antenna efficiencies
by estimating the chamber decay time were presented in [11].
Except for the empirical uncertainty study in [11], efficiency
measurement uncertainties were generally overlooked in these
studies. In this letter, measurement uncertainties of antenna ef-
ficiencies will be studied and modeled for the standard antenna
Manuscript received May 7, 2013; accepted May 10, 2013. Date of publica-
tion June 11, 2013; date of current version December 10, 2013.
The author is with the Department of Signals and Systems, Chalmers Uni-
versity of Technology, 41258 Gothenburg, Sweden (e-mail: xiaoming.chen@
chalmers.se).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TEMC.2013.2265042
efficiency method, e.g., [2], [12]. (Note that the uncertainty anal-
ysis in [12] is actually for Pref instead of antenna efficiency.)
The uncertainty model is verified by extensive measurements in
an RC.
II. MEASUREMENT UNCERTAINTY OF ANTENNA EFFICIENCY
The standard method for measuring the antenna efficiency is
to first measure the average power transfer function Pref of a
reference antenna (with known antenna efficiency eref ) and then
measure that of the AUT, PAUT . During both measurements,
both the reference antenna and the AUT must be placed in the
chamber in order to keep the same RC loading [7]. The total
radiation efficiency of the AUT can be estimated as
eAUT =
PAUT
Pref /eref
=
X
Y
(1)
where Y = Pref/eref (normalized average power transfer func-
tion of the reference antenna) and X = PAUT (average power
transfer function of the AUT).
Denote μx = E [X] , μy = E [Y ], where E denotes expecta-
tion, and f(x, y)= eAUT(X = x, Y = y). Taking the first-order
Taylor expansion of (1), one obtains
f(x, y) ≈ f(μx, μy)+ ∂f
∂x
∣
∣
∣
∣
x = μ x
y = μ y
(x−μx)+ ∂f
∂y
∣
∣
∣
∣
x = μ x
y = μ y
(x−μy).
(2)
The variance of eAUT is
var[eAUT] ≈ E[(f(x, y)− f(μx, μy ))2 ]
≈
(
∂f
∂x
∣
∣
∣
∣
x = μ x
y = μ y
)2
E[(x− μx)2 ]
+
(
∂f
∂y
∣
∣
∣
∣
x = μ x
y = μ y
)2
E[(y − μy )2 ]
+ 2
∂f
∂x
∣
∣
∣
∣
x = μ x
y = μ y
∂f
∂y
∣
∣
∣
∣
x = μ x
y = μ y
E [(x− μx)(y − μy )]
=
(
1
μy
)2
var[X] +
(−μx
μ2y
)2
var[Y ]
+ 2
∂f
∂x
∣
∣
∣
∣
x = μ x
y = μ y
∂f
∂y
∣
∣
∣
∣
x = μ x
y = μ y
cov[X,Y ]
=
(
μx
μy
)2 (
var[X]
μ2x
+
var[Y ]
μ2y
)2
(3)
0018-9375 © 2013 IEEE
1332 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 55, NO. 6, DECEMBER 2013
where var represents the variance and cov denotes the cross-
covariance. The last equality holds under the assumption that X
and Y are independent. Note that this is not a strong assump-
tion in that the average power transfer functions of the reference
antenna and the AUT are obtained via two independent mea-
surements.
The average power transfer functions Y is the average of Nind
independent and identically distributed (i.i.d.) power transfer
functions of the reference antenna (normalized by eref ). In a
well-stirred RC, these i.i.d. power transfer functions are expo-
nentially distributed [1] with a mean P0 . As a result
μy = P0 ,
var[Y ] = P0/Nind . (4)
Denote E[eAUT ] = e0 . Note that i.i.d. exponentially distributed
power transfer function seen by the AUT is attenuated by e0 .
Thus, the mean of the power transfer faction is e0P0 . Similarly,
one obtains
μx = e0P0 ,
var[X] =
e0P0
Nind
. (5)
Substituting (4) and (5) into (3), one obtains
var[eAUT] ≈ 2e
2
0
Nind
. (6)
The standard deviation (STD) of eAUT is
std[eAUT] ≈ e0
√
2
Nind
(7)
where std denoting the STD. The independent sample number
Nind can be estimated (based on one reference measurement)
using, e.g., the degree-of-freedom (DoF) method [7].
As can be seen in the next section, the estimated Nind using
the DoF method takes the RC loading into account. It is known
that RC loading (without blocking the line-of-sight path between
transmitting and receiving antennas) increases the ratio of the
unstirred power to the stirred power. Thus, the uncertainty due to
the unstirred component is included in the uncertainty model (7).
Note that this letter does not consider the systematic uncertainty,
which should be investigated in the future work.
III. MEASUREMENTS
In order to verify the efficiency uncertainty model (7), ex-
tensive measurements were performed from 700 to 3000 MHz
(covering the most interesting telecommunication bands) in the
Bluetest HP RC [6]. The RC used has a size of 1.80 m ×
1.75 m × 1.25 m (a drawing of which is shown in Fig. 1). It
has two mode-stirring plates, a turn-table platform (on which
a wideband discone antenna, used as the reference antenna, is
mounted), and three half-bow-tie antennas mounted on three or-
thogonal walls (referred to as wall antennas hereafter). During
the measurement, the turn-table platform was moved stepwise
to 20 platform-stirring positions (evenly distributed over one
complete platform rotation); at each platform-stirring position,
the two plates were moved simultaneously and stepwise to 50
Fig. 1. Drawing of the RC with two mechanical plate stirrers, a platform, three
wall antennas, and a head phantom.
positions (equally spanned on the total distances that they can
travel along two walls inside the RC). At each stirrer position
and for each wall antenna, a full frequency sweep was per-
formed by a vector network analyzer with a frequency step of
1 MHz, during which the scattering parameters (S-parameters)
are sampled (as a function of frequency and stirring position).
Hence, for each measurement, we have three wall antennas, 50
plate-stirring positions, and 20 platform-stirring positions.
To facilitate the estimation of the antenna efficiency un-
certainty and without loss of generality, the same measure-
ment sequence is repeated 12 times, each time with a different
height/orientation of the reference antenna on the platform, i.e.,
the reference antenna was placed at four different heights and
at each height it is placed with one vertical and two horizontal
orientations (in radial and tangential directions of the platform),
respectively. The heights and orientations are chosen to ensure
independent measurements. In postprocessing, arbitrary pairs
of antenna heights/orientations are chosen as the AUTs and the
reference antennas, respectively, for estimating eAUT , and we
introduce 0-, 1-, 2-, and 3-dB attenuator to the AUT (whose
negative value in decibels is the expectation of eAUT ).
Note that it is nontrivial to find the maximum number of
independent eAUT samples from the 12 measurements due to the
complicated distribution of eAUT , i.e., the ratio of two nonzero
Gaussian variables [13]. To be safe, we choose six pairs of
distinct measurements to obtain six independent eAUT , which
are used for estimating the measurement uncertainty of eAUT .
In order to see the RC loading effect on the measurement
uncertainty of eAUT , the measurement procedure was repeated
for two loading configurations: load0 (unloaded RC) and load1
(a head phantom that is equivalent to a human head in terms
of microwave absorption). Hereafter, measured data from these
different loading configurations are simply referred to as load0
and load1 data, whose quality factors are around 1000 and 550,
respectively. Hence, the STD of eAUT is estimated from these six
eAUT samples (for each e0 case and each loading configuration)
at each frequency point. Since Nind can be estimated based
on one reference measurement using the DoF method [7], the
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 55, NO. 6, DECEMBER 2013 1333
700 1000 2000 3000
0
200
400
600
800
1000
1200
1400
1600
Frequency [MHz]
Es
tim
at
ed
 In
de
pe
nd
en
t S
am
pl
e 
Nu
m
be
r
load0
load1
Fig. 2. Estimated independent sample number N using the DoF method.
700 1000 2000 3000
0.05
0.1
0.2
0.4
0.8
ST
D 
[dB
]
e0 = 0 dB
load1
load0
700 1000 2000 3000
0.05
0.1
0.2
0.4
0.8
ST
D 
[dB
]
e0 = −1 dB
load1
load0
700 1000 2000 3000
0.05
0.1
0.2
0.4
0.8
ST
D 
[dB
]
e0 = −2 dB
load1
load0
700 1000 2000 3000
0.05
0.1
0.2
0.4
0.8
ST
D 
[dB
]
Frequency [MHz]
e0 = −3 dB
load1
load0
Fig. 3. Estimated antenna efficiency uncertainties based on nine independent
measurements (solid) and the uncertainty model (7) (dotted). The four graphs
from top to bottom correspond to e0 = 0, −1, −2, and −3 dB, respectively,
where e0 = E [eAUT ].
eAUT STD can also be estimated using the uncertainty model
(6). Fig. 2 shows the estimated Nind as a function of frequency
for both load0 and load1.
Fig. 3 shows the estimated STDs based on the 12 indepen-
dent measurements and the uncertainty model (7) from a single
measurement (with estimated Nind using the DoF method [7]),
respectively. Note that the DoF method inherently takes the
RC loading (and therefore K-factor [6]) effect into account [7].
Therefore, there is no need in decomposing the STD into stirred
and unstirred components as in [6]. Also note that, for clear
exhibitions, the estimated STD is plotted using the following
dB-transformation [6]
σdB = 5 log10
(1 + σ)
(1− σ) (8)
and that a 50-MHz frequency smoothing is applied to the STD
curves before plotting. As can be seen, there is good agreement
between the uncertainty model (based on one measurement) and
the direct STD estimate (based on the 12 independent measure-
ments). It can also be seen that eAUT measurement uncertainty
decreases with decreasing total radiation efficiency of the AUT.
The later observation can be readily explained from eAUT un-
certainty model (7).
Note that the DoF method for estimating Nind proposed in [7]
is suitable to RCs with different (separable) stirring mecha-
nisms. In case the stirring sequences generated by different stir-
ring mechanisms are not separable, or there is only one stirring
mechanism, one can either use the frequency-domain samples
for estimating Nind (at the cost of sever degradation of the fre-
quency resolution) [14] or simply use suitable Nind estimators,
e.g., [15], [16].
IV. CONCLUSION
In this letter, the measurement uncertainty of antenna ef-
ficiency estimated using the standard efficiency measurement
approach [2] is analyzed. A simple efficiency uncertainty model
(approximation) is given. And the model is verified based on ex-
tensive RC measurements. The good agreement indicates that
the model can be used for predicting the measurement uncer-
tainty of the antenna efficiency based on a single reference
measurement.
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