L-UNIVERSITA` TA` MALTA
Msida − Malta
UNIVERSITY OF MALTA
Msida − Malta
DIPARTIMENT TA’ L-INGINERIJA
TAL-KOMUNIKAZZJONI U KOMPJUTER
DEPARTMENT OF COMMUNICATIONS
AND COMPUTER ENGINEERING
5th April 2010
CCE 5303 – Radio Propagation and QoS
Assignment
Discuss the contents of the paper “On the K-Factor Estimation for Rician Channel
Simulated in Reverberation Chamber” attached. Argue on the validity of the
technique used to obtain the k-factor.
Use any published material to sustain your arguments.
The submitted report should follow A4 IEEE double column format with singlespaced, twelve-point font in the text. The maximum report length is four (4) pages.
Reports in excess of four pages will not be read and a zero mark will be assigned. All
figures, tables, references, etc. are included in the page limit.
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Hard deadline for the submission of the assignment: 31st May 2011 at 12:00, please
submit the assignments at the Department’s secretary office.
No Assignment will be accepted after this date and time. Assignment can be
submitted in groups of not more than two students.
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 3, MARCH 2011
K
1003
On the -Factor Estimation for Rician Channel
Simulated in Reverberation Chamber
Christophe Lemoine, Emmanuel Amador, Student Member, IEEE, and Philippe Besnier, Senior Member, IEEE
Abstract—Reverberation chambers were recently proposed to
-factor.
simulate Rician radio environment with controllable
The -factor is also a parameter that may tell how ideal may be a
reverberation chamber when it is used for other more conventional
purposes. This paper is dedicated to the problem of the correct
in a reverberation chamber given a set of data
estimation of
measured along a stirring process.
Index Terms— -factor, reverberation chamber, Rician channel,
statistical estimation.
I. INTRODUCTION
I
N many radio propagation environments, the time varying
envelope of the received signal can be statistically described
by a Rician distribution [1]–[4]. When there is a line of sight
(LOS) between the transmitter and the receiver, the received
signal can be written as the sum of a complex exponential and
a narrowband Gaussian process, which are known as the LOS
component and the diffuse component respectively. The relative
strength of the direct and scattered components of the received
signal is expressed by the Rician -factor. Recently, reverberation chambers (RC) have been proposed to simulate a controllable Rician radio environment for testing wireless devices [5].
A reverberation chamber generally consists of a metallic
cavity and an electrically large metallic paddle called a stirrer
which enables to change the boundary conditions in the cavity
(Fig. 1). The rotation of the stirrer supplies the stirring process.
If this Faraday cage is overmoded enough, the field can be
described as a combination of numerous modes. Stochastic
field is the result of the stirring process [6]. Statistics provide
appropriate methods for the evaluation of the main characteristic parameters of an RC [7], [8]. In ideal conditions, any
rectangular component of the electric field follows a Rayleigh
distribution [9]. Direct coupling paths between transmitting and
receiving antennas must be minimized to favor this Rayleigh
distribution. This is a common approach when using RC for
electromagnetic compatibility (EMC) purposes.
Manuscript received July 29, 2009; revised July 20, 2010; accepted
November 15, 2010. Date of publication December 30, 2010; date of current
version March 02, 2011. This work was supported in part by the French
Ministry of Defence DGA (Délégation Générale de l’Armement), “REI” under
Grant 2008 34004. The work of E. Amador was supported by a Ph.D. Grant
delivered by the DGA.
The authors are with the Université Européenne de Bretagne, France,
INSA, IETR, UMR CNRS 6164, F-35708 Rennes, France (e-mail:
christophe.lemoine@insa-rennes.fr).
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/TAP.2010.2103003
Fig. 1. Reverberation chamber in IETR research laboratory (2.9 m
8.7 m). The lowest usable frequency is approximately 250 MHz.
2
2 3.7 m
In order to extend the use of RC some authors [5], [10]–[13]
have investigated the potential of RCs for the emulation of controlled Rician propagation channels. Rician environment may
be reproduced by adjusting levels of direct coupling paths and
scattered paths in the chamber [14]. -factor is one of the key
parameter of a Rician propagation channel since it represents
the ratio of the first paths to the second ones. The use of mechanical stirring is analogous to a static LOS component in mobile channel; whereas the use of mechanical stirring combined
with electronic stirring is analogous to a moving terminal, i.e.,
the phase of the LOS component is constantly changing with
time. However, evaluation of Rician -factor out of RC measurements must be analyzed very carefully. A rough estimation
was proposed in [5] from parameters measurements, but the
estimator is biased and the statistical uncertainties of this estimation are not provided. In a recent paper [15], authors prostarting from the goodness-of-fit test
posed an estimator of
of the normal distributions of both real and imaginary parts of
parameters between antennas. However, the
transmission
statistics of was not deeply investigated neither the accuracy
of estimation as a function of the number of individual measurements and the number of samples.
To estimate -factor, some methods use the measured power
signals—amplitude only, no phase—while others use complex
in-phase and quadrature
signals-amplitude and phase or,
components, or only the fading phase. Various approaches were
done to find -factor using a maximum likelihood method.
Greenwood and Hanzo [16] have proposed to compute the distributions of the envelope, then to compare the probability density function of the measured data with a set of hypothetical
0018-926X/$26.00 © 2010 IEEE
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 3, MARCH 2011
can be expressed as a sum
The complex transfer function
associated with the unstirred paths
of a direct component
associated with the stirred energy
and a stirred component
in the cavity
(1)
with the complex form
(2)
(3)
Fig. 2. Configuration with a dominant LOS path.
parameter is the sum of
Each real and imaginary part of the
a deterministic term and a stochastic term (1)–(3)
distributions using a suitable goodness-of-fit test. The disadvantage of this method is complexity of implementation, which
is a time-consuming and computationally extensive procedure
[2], [17]. In these conditions, not being suited for online implementation, this approach is more useful for testing whether
the measured envelope is Rician distributed, rather than estifrom independent and
mating [16]. A method to extract
identically distributed complex Rician channel samples was presented in [18]. Using samples of the phase and envelope of the
received signal to estimate , in [19], two estimators are proposed. The best of them will catch our attention in this paper for
-factor estimation in a reverberation chamber from
measurements. Other methods simpler than the alternatives menestimation.
tioned above are moment-based estimators for
These techniques are used to estimate -factor based on measurements of the received fading envelope [2], [16], [20]. Such
methods are not optimal in coherent wireless systems because
they do not take into account the additional phase information
provided by complex baseband realizations. Moreover, in [21]
a new Rician -factor estimator was derived using correlated
channel samples in a noiseless channel, based on samples of
the fading instantaneous frequency, representing the derivative
of the phase oscillation of fading with time. The main disadvantage of this estimator is represented by the cost of time and
estimation computational resources needed for estimation.
The purpose of this paper is to provide all necessary theoretin a reverberation chamber.
ical background for estimating
extraction from RC measurements is revisIn Section II,
ited in details. In particular, the first and second moments of
estimator are derived from theoretical statistics. The proposed
method is then validated through Monte-Carlo (MC) analyses
and experiments in RC (Section III).
II. REVISITING
(4)
(5)
Both stirred components follow independent zero mean normal
distributions, with the same standard deviation [5], [9]
(6)
(7)
In the simplified case where all wall reflections interact with the
stirrer, the only unstirred component is the direct coupling term
between antennas. Then the direct component identifies with the
LOS path. On the other hand, if there is no multipath scattering
only
involving the paddle, the stirred component is null and
. Moreover, if we assume absolutely
has a direct component
no reflection, we obtain the anechoic chamber (AC) situation,
i.e.,
(8)
Under the hypothesis of an ideal reverberation environment,
many authors [5], [22]–[24] have shown that the scattering cofollows the same statistics as a rectangular comefficient
ponent of the electric field. In the case of an overmoded cavity,
is Rayleigh distributed and the phase
the modulus
follows a uniform distribution. In addition, the real
and
parts follow independent zero mean normal disimaginary
tributions, with the same standard deviation . Therefore, from
(1) it appears clearly that
EXTRACTION FROM RC MEASUREMENTS
(9)
A. Overview of
-Factor Formulations in RC
In the paper, we consider a basic RC configuration, where
two antennas are located in the cavity. A strong direct antenna
coupling can be introduced for instance when decreasing the
separation distance between both opposite antennas (Fig. 2).
A 2-port vector network analyzer (VNA) gives access to
measurements.
denoting the mean operator of data where
is the
with
number of independent stirrer positions1. In the same way as
, we will denote
for the complex stirred component
and
respectively the real and imaginary parts of the direct
.
component
N is systematically implicit and not written in order to simplify the notation.
1
LEMOINE et al.: ON THE
-FACTOR ESTIMATION FOR RICIAN CHANNEL SIMULATED IN REVERBERATION CHAMBER
For a multipath environment, -factor is defined as the ratio
of the unstirred energy and the stirred energy [5], [15], [25]
Moreover, as far as the estimation of
is concerned, (11) leads to [15]
(11)
-factor denominator
(20)
(10)
We can also write -factor as a function of the standard deviation of each real and imaginary part of the
transfer function. With (see (1))
1005
is an a Priori Known Parameter: Now,
1) Assuming
let
be independent, normally distributed,
.
random variables, with mean zero and same variance
be constant values. It is shown in [27]
Let also
is a noncentral
that the distribution of
distribution with degrees of freedom and noncentrality pa. Moreover, from decomposition of
rameter
transmitting parameter (1), (4)–(15), we have the following
distribution functions:
where “Var” denotes the variance operator, we have
(21)
(22)
(12)
and therefore2 [5]
(13)
B.
Therefore, when becomes sufficiently large, in practice [28]
, the central limit theorem (CLT) provides the following
distribution functions [29], [30]:
(23)
Estimation in RC
As far as deterministic components are concerned, [15], [26]
and
shows, with
(14)
(15)
This is equivalent to write that (2)
(16)
(17)
(24)
follows a noncenConsequently using (19), the ratio
distribution with 2 degrees of freedom and noncentrality
tral
parameter
.
distribuFurthermore, both first moments of a noncentral
tion with degrees of freedom and noncentrality parameter
are well-known [27]
(25)
complex parameter.3
denoting the phase of the
with
From (13) and (16), we have
and
Now, using the trigonometric property
(14) and (15) is [15]
best estimator for
(18)
(26)
, the
and
the first and second moments respectively.
with
is the expected value4 of the
Hence using (25),
. As a result, for a large number
of measureratio
ments we find [28] the following expected value:
(19)
with
2Some
denoting the estimated value of
.
(27)
papers deal with the direct-to-scattered ratio (DSR) [15], [25], [26]
j
j
.
which is very similar to -factor:
3Assuming that all wall reflections interact with the paddle, j
j is equivalent to the free-space coupling term.
K
DSR = S
=
S
4And
not
v N= !
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 3, MARCH 2011
Fig. 3. Biased estimation of
dependent stirrer positions.
Now let
K -factor using K
be an estimator of
with different numbers of in-
-factor defined by5
(28)
Therefore, assuming
is a priori known, we show here that
estimation is not the -factor, but
the expected value of
since the distribution of is not a centered distribution
Fig. 4. Numerical estimation of the 95% confidence interval of
30.
K
with
N=
by MC simulations give a precious idea of the goodness of the
estimation of the -factor one should expect for a given and
a given number of stirrer positions .
is an a Priori Unknown Parameter: In the
2) Assuming
is a stochastic parameter, therefore we need to
real case
evaluate both and
in order to estimate . Consequently
is more appropriate
the following estimator
(31)
(29)
MC simulations are conducted to simulate the estimation of
the -factor for a given direct component and a given number
of independent stirrer positions . Our approach is to replicate
parameter according to
our measurements by generating a
the (21) and (22). The purpose of these simulations is to estimate the -factor and its accuracy by evaluating its confidence interval (CI). Every MC simulation is conducted using
scenarios. Let
be an estimator of a given
scenarios, we can
random variable . Using
assume that the sample mean
is approximately equal
to the expected value of
affects
Compared to Figs. 4 and 5 shows that evaluating
when
is relatively high. It means that even with
only
high values of the -factor, the accuracy of the estimation is
. MC simulations show that
diminished by the estimation of
, the CI of
is at least 3 dB.
with
Baddour [18] has proposed an analytical expression of
-factor estimation, based on the biased estimator of variance
. Since the RC community generally prefers using unbiased
estimators, we provide here the appropriate -factor estimator
following the same developments as Baddour in [19]. Thus,
using the following unbiased estimator of the variance in (20):
(30)
(32)
Fig. 3 shows how the accuracy of the estimation of the
-factor is affected by the number of independent stirrer
and
, the estimation is
positions. For both
. Moreover, the
significantly biased for values under
lower is the more the bias is significant. One should keep in
mind that the number of independent stirrer positions in a RC at
a given frequency is typically limited to several tens [30]. Fig. 4
for a given and
shows the quantiles of the estimation of
independent stirrer positions. The quantiles estimated
5We
would like to draw readers’ attention to the fact that although the ratio
follows a noncentral
distribution with 2 degrees of freedom and
noncentrality parameter
,
does not follow a noncentral
distribution with 2 degrees of freedom and noncentrality parameter .
v N=
v N= K
K
with
of mean
estimator of
, being observations of the random value
, one can show analytically that the unbiased
-factor is the following6
(33)
We denote
the correction factor and
the correction term in (33). It is the first time in
6In [18], the author has used the common biased estimator of the variance,
see [19]. Here we actually take into account the standard unbiased estimator of
the variance (32).
LEMOINE et al.: ON THE
-FACTOR ESTIMATION FOR RICIAN CHANNEL SIMULATED IN REVERBERATION CHAMBER
1007
S
Fig. 6. Variations of the phase of the
transmitting complex parameter as
a function of frequency. Here the distance between horn antennas is 1 m (Fig. 2).
Fig. 5. Numerical estimation of the 95% confidence interval of
30.
K
with
N=
transmitted to the chamber can
Moreover, the total power
using denoting the insertion
be related to the mean power
loss parameter of the cavity [32]–[34]
a paper dedicated to the emulation of propagation channels in
RC, that this appropriate estimator (33) is proposed.
C. Increasing Accuracy With Electronic Stirring
(37)
Thus we have,
Many practical situations can benefit from electronic stirring
in addition to mechanical stirring, in order to improve the estimation of -factor. We have shown previously that only using
mechanical stirring may not suffice to have an acceptable level
of uncertainty over the evaluation of . The idea here is to add
samples using electronic stirring, in order to reduce this level of
uncertainty. However, it is shown in this Section that we must
be very careful with the use of electronic stirring for estimating
-factor.
1) Preliminary Hypothesis: The first question is how the
is as a function of frequency in a RC. If the
variation of
-factor can be considered as a constant value in a
frequency bandwidth, then applying electronic stirring is appropriate. Using as the total power transmitted by the emitter to
the chamber, and assuming the transmitting antenna has a directivity , the Friis’ transmission formula provides [31], [32]
(34)
where is the distance between the transmitting and receiving
antennas and is the free-space impedance. On the other hand,
Hill [9] provides the following relationship between the stirred
and the mean power
received over a stirrer
component
revolution
(35)
leading to [5]
(36)
(38)
It is well established in RC literature that the insertion loss
parameter evolves in
[35], [36]. Moreover, assuming a
simple but realistic case where the antenna directivity is unbandwidth, we can deduce that the -factor is
changed in a
a function of the square root of the frequency
(39)
, then variations of
As shown in Appendix, if
. Translated in deciBel, this
the -factor are very low
, then extreme
means that in the bandwidth
values of -factor differ only in 0.2 dB. Therefore the assumpin a reasonable frequency
tion of insignificant variations of
bandwidth is consistent. This conclusion remains valid only if
(34) is strictly satisfied (see Section III).
measurements in
2) MC Simulation Analysis: Analyzing
the electronic stirring case requires particular cautions. It does
and
in the same way as for menot sum up to estimate
chanical stirring [26]. Indeed the phase (17) changes with frefrequency bandquency and is uniformly distributed over a
width (Fig. 6). Considering one sample instead of
samples
may lead to underestimate the -factor of more than 20 dB!
This will be illustrated in Section III-B. This is the reason why
we adopt the following steps to estimate a global -factor simulated in RC using electronic stirring in the frequency band
.
for each7 selected independent fre• First we estimate
, following the approach developed in (33).
quency in
7Therefore
K -factor.
we do not combine frequencies for estimating directly the
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 3, MARCH 2011
K
Fig. 7. Confidence interval associated with the estimation of h i , with
30 independent stirrer positions and
= 1, 30 and 100 independent
frequencies.
N=
N
• Second, we gather all
estimations to provide a global
estimation of factor in
. Using
independent fre, we estimate the -factor as follows:
quencies in
K
Fig. 8. 95% confidence interval associated with the estimation h i of
-factor using a MC simulation in comparison with the CI (denoted “th.” in
legend) calculated from the 2.5% and 97.5% quantiles which are analytically
= 100. It is
expressed respectively in (43)–(44), using
= 50 and
clearly shown that for estimating low values of -factor we need to increase
in order to satisfy the desired CI.
and/or
K
N
N
N
K
N
Therefore, the 95% CI associated with the estimation of
is defined by the following quantiles [28]
(40)
with
denoting the mean operator over
data where
is the number of independent frequencies used in the
narrowband
.
The advantage of electronic stirring is to provide many estimations of the underlying -factor emulated from mechanical
stirring. In comparison with the 95% confidence interval of a
-factor estimation based only on mechanical stirring (Fig. 5),
Fig. 7 shows the significant reduction of the uncertainty level
. As
when adding electronic stirring, over the estimation
shown in Fig. 7, in the limit case of a Rayleigh channel situation,
using only one frequency8 to estimate leads to more than 20
dB of uncertainty. But with
independent frequencies
(e.g.,
at 1 GHz [30]) we only have apin
proximately 4 dB of uncertainty which is a great improvement9.
As in [19], one can demonstrate that the variance of the esti(33) is
mator
(41)
Then, using the CLT [28] the distribution of
normal distribution with mean and variance
tends to a
(42)
8i.e.,
N
:
(43)
(44)
Fig. 8 compares the 95% CI obtained from MC simulation with
the one calculated from the 2.5% (43) and 97.5% (44) quantiles
and
. It shows that the analytical forwith
mulations (43) and (44) match perfectly the result of MC simulations. Fig. 8 shows that if we need a more satisfying confidence interval for very low -factor, we have to select more
independent stirrer positions
and more independent frein
. As indicated in (42), increasing
is
quencies
to reduce the CI associated
more efficient than increasing
with the estimation of . Nonetheless, one must keep in mind
that mechanical stirring (especially in mode-tuning) takes gennarrow
erally more time than electronic stirring. So, given a
bandwidth, a right method may be to first use the maximum
of independent frequencies which are available in
number
, and second to adjust the number of independent stirrer
positions in order to be consistent with the desired CI.
III. EXPERIMENTAL RESULTS
=1
9About more conventional RC purposes, when we are looking for optimizing
the mechanical stirring efficiency, we try to decrease -factor as most as possible in order to eliminate the entire direct component and favor as most as possible the stirred component. Without the knowledge of the associated confidence
interval, the evaluation of the -factor can be strongly inaccurate.
K
K
This Section gives four different experimental results, in
order to illustrate the consequence and advantages of the
previous statistical analysis provided in the case of both mechanical and electronic stirring.
LEMOINE et al.: ON THE
-FACTOR ESTIMATION FOR RICIAN CHANNEL SIMULATED IN REVERBERATION CHAMBER
K
Fig. 9. Experimental estimation of -factor in RC using horn antennas (cf.
Fig. 2) compared to measurements in an anechoic chamber, at 3.5 GHz, with
30 independent stirrer positions.
N=
A. Comparison With Anechoic Chamber Measurements for
the Direct Path
collected data for a
Here the measurement consists of
single frequency, where is the number of independent stirrer
of the direct component is
positions. The phase
invariant as long as the line of sight is cleared during the rotation of the stirrer. Compared to electronic stirring, mechanical
stirring is more time-consuming and the number of stirrer positions over a complete rotation should be chosen meticulously
[30].
In order to confirm the results obtained by our MC simulations, we conducted measurements both in AC and in RC. Measurements in AC give us what can be interpreted as true values
of the direct component for a given distance . We use two
wide-band horn antennas at 3.5 GHz separated by a distance .
By modifying the distance we change the direct component of
parameter and thus we change the -factor
the measured
for each distance .
in RC. In RC we estimate both and
parameter consists only in an unstirred
In AC the measured
component. In order to compare the results of RC measurements
with those issued from AC we build a -factor using the mean
obtained in RC. Fig. 9 shows that for relaof the values of
tively high values of mechanical stirring allows a rough estimation of the -factor whereas for low values the estimation
is inaccurate. These results corroborate our MC simulations.
More accuracy means more independent samples, but the
number of independent stirrer positions at a given frequency
is limited [30]. By adding electronic stirring, we can increase
substantially the number of samples and expect a more accurate
estimation of the -factor.
B. Electronic Stirring With Horn Antennas
The same experiment is here performed using electronic stirring in addition to mechanical stirring. We show in Fig. 10 that
1009
K
:
Fig. 10. Experimental estimation of -factor in RC with horn antennas (cf.
= 30 indeFig. 2), using electronic stirring in [3 45 GHz; 3 55 GHz] (
= 100 independent frequencies). The green
pendent stirrer positions and
asterisks correspond with a wrong estimation, highlighting that we cannot use
for electronic stirring combined with mechanical stirring, the same method as
for mechanical stirring only.
N
:
N
frequency stirring leads to reduce statistical fluctuations. However the improvement which is brought by both electronic stirdoes not seem greatly signifiring and the correction term
cant since we can emulate only high -factors with the configuration in Fig. 2. We choose this configuration in order to have
using an
a reference measurement of the direct component
anechoic chamber. With high -factor values, the associated CI
of the estimation remains relatively small either with mechanical stirring only or with combined mechanical and electronic
stirring (Fig. 7). The green asterisks correspond with a wrong
estimation, highlighting that we cannot use for electronic stirring combined with mechanical stirring, the same method as for
mechanical stirring only. The reason is that the phase (17)
changes with frequency and is uniformly distributed over a
frequency bandwidth (Fig. 6). The next experiment aims to generate lower controllable -factors in order to highlight a significant improvement in the estimation of -factor using electronic
stirring combined with mechanical stirring.
C. Electronic Stirring With Discone Antennas
A new experiment similar to the one with horn antennas, but
with discones, may simulate lower -factor values since discones are less directive than horn antennas. The result is drawn
in Fig. 11. The curve related to AC measurements gives the reference -factor, and is consistent with the direct component
calculated from Friis’ transmission formula [31]. However, both
results coming from RC measurements do not fit the reference.
The reason is clear: in RC our direct component originates in the
free-space propagation but also in the numerous reflected paths
which are not affected by the stirrer. Our discone antennas are
indeed characterized by a very low directivity, and this explains
why RC measurements cannot fit AC measurements. The evaluation of RC -factor is not wrong, as the evaluation of the direct
component in AC is not biased too, but each chamber does not
1010
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 3, MARCH 2011
K
:
Fig. 11. Experimental estimation of -factor in RC with discone antennas (cf.
Fig. 2), using electronic stirring in [3 5 GHz; 3 6 GHz] ( = 50 independent
stirrer positions and
= 100 independent frequencies). Differences between
the reference curves (Friis or AC measurements) and RC measurements are due
to the fact that in RC there are multiple unstirred paths and not only the direct
path observed in AC.
N
:
Fig. 12. Experimental configuration for controlling
Wilkinson coupler.
N
Fig. 13. Experimental configuration leading to optimize mechanical stirring
(
=
18 dB).
K 0
K -factor in RC using a
Fig. 14. Experimental characterization of the influence of the correction term
= 1
from RC measurements, for estimating -factor in the case of electronic stirring. One thousand frequencies have been used in [3 GHz 3 1 GHz].
C
simulate the same direct component so we cannot have comparable results. However, this result highlights the limited performances of mechanical stirring, through the -factor value
which cannot be easily reduced, even if the distance between
transmitting and receiving antennas is quite long.
D. Experimental Validation of the Correction Term
We introduced two corrections in the estimation of -factor
has the main impact on the quality
(33). The correction term
of estimation, particularly for low values. For RC purposes,
this correction term has never been used so far [5]. In order
to show empirically the impact of the correction term
, on the estimation of -factor we perform the following
experiment. For controlling -factor, we carry out two series of
measurements as shown in Fig. 12. First we measure the stirred
, and second we artificially add a
component
direct component
. The underlying goal is to have
therefore a -factor reference value which we tend to recover
parameter measurements using the best estimator (40).
from
On the one hand, we try to find the best configuration of mechanical stirring in RC with two horn antennas, decreasing as
much as possible -factor, in order to have an insignificant di. We obtain
rect component and therefore assume
=N
K
; :
using the configuration exposed in Fig. 13 with
antennas in cross-polarization. In particular, to improve the stirring efficiency we place an additional panel in order to reflect
waves toward the mechanical stirrer. By this way, we manage to
reduce -factor of 5 dB with regards to the same configuration
but without the additional panel.
On the other hand, we control the direct component
linking both transmitting and receiving cables and changing
the attenuation level of the transmission (Fig. 12). Therefore,
transmitting parameter issued from RC
adding the complex
transmitting parameter meameasurements, to the complex
sured with linked cables (Fig. 12), we can control the resulting
-factor [37]. Using electronic stirring with a high number
of independent frequencies in a small bandwidth
, we obtain an accurate evaluation of the expected value of
the -factor. The effect of the correction term
is
clearly shown experimentally in Fig. 14.
IV. CONCLUSION
Following a meticulous development based on a theoretical
analysis combined with simulations results, we endeavour to
LEMOINE et al.: ON THE
-FACTOR ESTIMATION FOR RICIAN CHANNEL SIMULATED IN REVERBERATION CHAMBER
Moreover, one can demonstrate that
TABLE I
VARIATIONS OF
1011
K -FACTOR AS A FUNCTION OF 1f=f
(48)
Therefore, (47) provides
(49)
provide all necessary elements to estimate correctly the Rician
-factor using mechanical and frequency stirring in RC.
Many experimental results are presented and show different
important points. First, trying to relate RC measurements to AC
measurements is quite easy when the direct coupling is very
strong compared to all stirred paths. When the direct component
is not only due to the direct coupling path, i.e., when unstirred
reflections become significant, then we cannot have a reference
situation in AC. Moreover, we show that in RC very different
scenarios of propagation can be emulated, the lowest -factor
being limited by the stirring efficiency. In our RC, we managed
. On the other hand,
to decrease the -factor down to
it is necessary to take into account the corwhen
rection term
due to the non-central distribution of the direct
component, in order to have the best estimation of -factor.
The use of frequency stirring is highly recommended by the
authors, particularly for reducing the confidence interval associ(40),
ated with a -factor estimation. With the estimator
using only a few number of independent stirrer positions and
of independent frequencies in a narrow
a few number
bandwidth, we can significantly improve the accuracy of the
estimation.
APPENDIX
Here we estimate the impact of a bandwidth
centered in a frequency
over the variations
,
and
be respectively the
of -factor. Let
minimum, the average and the maximum of the -factor that
. Starting from (39)
can be emulated in
(45)
, we can
and supposing that the directivity is invariant in
find in a good approximation the interval
of variations
in
, defined by
of
(46)
With
we find
(47)
Some typical variations of
in
are indicated in Table I.
ACKNOWLEDGMENT
The authors would like to thank J. Sol for assistance with the
measurements.
REFERENCES
[1] W. C. Jakes, Microwave Mobile Communications. New York: IEEE
Press, 1974.
[2] L. Greenstein, D. Michelson, and V. Erceg, “Moment-method estimation of the Ricean -factor,” IEEE Commun. Lett., vol. 3, pp. 175–176,
Jun. 1999.
[3] J. D. Parsons, The Mobile Radio Propagation Channel, 2nd
ed. Chichester, U.K.: Wiley, 2000.
[4] R. Vaughan and J. B. Andersen, Channels, Propagation and Antennas
For Mobile Communications. London, U.K.: Inst. Elect. Eng. Electromagnetic Wave Series, No. 50, 2003.
[5] C. L. Holloway, D. A. Hill, J. M. Ladbury, P. F. Wilson, G. Koepke, and
J. Coder, “On the use of reverberation chambers to simulate a Rician
radio environment for the testing of wireless devices,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3167–3177, Nov. 2006.
[6] P. Corona, G. Ferrara, and M. Migliaccio, “Reverberating chambers
as sources of stochastic electromagnetic fields,” IEEE Trans. Electromagn. Compat., vol. 38, no. 3, pp. 348–356, Aug. 1996.
[7] L. R. Arnaut, “Statistics of the quality factor of a rectangular reverberation chamber,” IEEE Trans. Electromagn. Compat., vol. 45, pp. 61–76,
Feb. 2003.
[8] P. Corona, G. Ferrara, and M. Migliaccio, “Generalized stochastic
field model for reverberating chambers,” IEEE Trans. Electromagn.
Compat., vol. 46, no. 4, pp. 655–660, Nov. 2004.
[9] D. A. Hill, “Plane wave integral representation for fields in reverberation chambers,” IEEE Trans. Electromagn. Compat., vol. 40, no. 3, pp.
209–217, Aug. 1998.
[10] J. Valenzuela-Valdés, M. García-Fernández, A. Martínez-González,
and D. Sánchez-Hernández, “Non-isotropic scattering environments
with reverberation chambers,” in Eur. Conf. Antennas Propagation,
Edinburgh, U.K., Nov. 2007, pp. 1–4.
[11] J. Valenzuela-Valdés, A. Martínez-González, and D. SánchezHernández, “Emulation of MIMO nonisotropic fading environments
with reverberation chambers,” IEEE Antennas Wireless Propag. Lett.,
vol. 7, pp. 325–328, 2008.
[12] E. Genender, C. Holloway, K. Remley, J. Ladbury, G. Koepke, and H.
Garbe, “Use of reverberation chamber to simulate the power delay profile of a wireless environment,” in Proc. EMC Eur., Hamburg, Germany, Sep. 2008, pp. 1–6.
[13] J. Valenzuela-Valdés, A. Martínez-González, and D. SánchezHernández, “Diversity gain and MIMO capacity for nonisotropic
environments using reverberation chamber,” IEEE Antennas Wireless
Propag. Lett., vol. 8, pp. 112–115, 2009.
[14] N. K. Kouveliotis, P. T. Trakadas, I. Hairetakis, and C. N. Capsalis,
“Experimental investigation of the field conditions in a vibrating intrinsic reverberation chamber,” Microwave Opt. Technol. Lett., vol. 40,
pp. 35–38, Jan. 2004.
[15] C. Lemoine, P. Besnier, and M. Drissi, “Advanced method for estimating direct-to-scattered ratio of Rician channel in reverberation
chamber,” Electron. Lett., vol. 45, no. 4, pp. 194–196, Feb. 2009.
[16] D. Greewood and L. Hanzo, Characterization of Mobile Radio Channels, R. Steele, Ed. London, U.K.: Mobile Radio Communications,
1992.
k
1012
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 3, MARCH 2011
[17] K. Talukdar and W. Lawing, “Estimation of the parameters of the rice
distribution,” J. Acoust. Soc. Amer., vol. 89, no. 3, pp. 1193–1197, Mar.
1991.
[18] K. E. Baddour and T. J. Willink, “Improved estimation of the Ricean
k -factor from I/Q fading channel samples,” IEEE Trans. Wireless
Commun., vol. 7, pp. 5051–5057, Dec. 2008.
[19] K. Baddour and T. Willink, “Improved estimation of the Ricean factor
from I/Q samples,” in IEEE 66th Vehicular Technology Conf., 2007,
pp. 1228–1232.
[20] A. Abdi, C. Tepedelenlioglu, M. Kaveh, and G. Giannakis, “On the
estimation of the k parameter for the Rice fading distribution,” IEEE
Commun. Lett., vol. 5, pp. 92–94, Mar. 2001.
[21] G. Azemi, B. Senadji, and B. Boashash, “Ricean k -factor estimation
in mobile communication systems,” IEEE Commun. Lett., vol. 8, pp.
617–619, 2004.
[22] D. A. Hill, Electromagnetic theory of reverberation chambers National
Institute of Standards and Technology (NIST), 1998, Tech. Note 1506.
[23] P. Corona, “Reverberating chamber electromagnetic field in presence
of an unstirred component,” IEEE Trans. Electromagn. Compat., vol.
42, no. 2, pp. 111–115, May 2000.
[24] C. Lemoine, P. Besnier, and M. Drissi, “Investigation of reverberation
chamber measurements through high power goodness of fit tests,” IEEE
Trans. Electromagn. Compat., vol. 49, pp. 745–755, Nov. 2007.
[25] P. Hallbjörner, “Reverberation chamber with variable received signal
amplitude distribution,” Microwave Opt. Technol. Lett., vol. 35, no. 5,
pp. 376–377, Dec. 5, 2002.
[26] G. Lerideau, E. Amador, C. Lemoine, and P. Besnier, “Quantifying
stirred and unstirred components in reverberation chamber with appropriate statistics,” presented at the IEEE Int. Symp. EMC, Austin, Aug.
2009.
[27] N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate
Distributions. New York: Wiley, 1995.
[28] A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill, 1965.
[29] J. D. Hamilton, Time Series Analysis. Princeton, NJ: Princeton Univ.
Press, 1994.
[30] C. Lemoine, P. Besnier, and M. Drissi, “Estimating the effective sample
size to select independent measurements in a reverberation chamber,”
IEEE Trans. Electromagn. Compat., vol. 50, no. 2, pp. 227–236, May
2008.
[31] C. Balanis, Antenna Theory—Analysis and Design, 2nd ed. New
York: Wiley, 1997.
[32] V. Fiumara, A. Fusco, V. Matta, and I. M. Pinto, “Free-space antenna
field/pattern retrieval in reverberation environments,” IEEE Antennas
Wireless Propag. Lett., vol. 4, pp. 329–332, 2005.
[33] Reverberation chamber test methods Int. Electrotech. Commission
Std., 2003, IEC 61000-4-21.
[34] V. Rajamani, C. Bunting, and J. West, “Sensitivity analysis of a reverberation chamber with respect to tuner speeds,” in Proc. IEEE Int.
Symp. EMC, Honolulu, HI, Jul. 2007, vol. 1, pp. 1–6.
[35] L. Kone, B. Démoulin, and S. Baranowski, “Application des chambres
réverbérantes à brassage de modes à la caractérisation des émissions
rayonnées par un équipement,” presented at the CEM 08, Paris, France,
May 2008.
[36] J. M. Ladbury, G. H. Koepke, and D. Camell, “Evaluation of the NASA
Langley Research Center mode-stirred chamber facility,” U.S. Dept.
Commerce, 1999, NIST Tech. Note 1508.
[37] E. Amador, G. Delisle, and D. Grenier, “Reverberation chamber as
a synthesis instrument for Rayleigh and Rice channels,” in Proc. 5th
IASTED Int. Conf. Antenna, RADAR, Wave Propagation, Baltimore,
MD, Apr. 2008, pp. 65–68.
Christophe Lemoine received the Diplôme d’Ingénieur degree from Ecole Nationale Supérieure
de l’Aéronautique et de l’Espace (SUPAERO),
Toulouse, France, in 2004, the Master degree in
financial risk management in 2005, and the Ph.D.
degree in electronics from the Institut National des
Sciences Appliquées (INSA), Rennes, France, in
2008.
He is now an Assistant Professor at INSA of
Rennes, France. His current research interest at the
Institute of Electronics and Telecommunications of
Rennes (IETR), Rennes, France, includes new theoretical and experimental
approaches of mode-stirred reverberation chambers for EMC, propagation
channels and antenna measurement applications.
Emmanuel Amador (S’10) received the Diplôme
d’Ingénieur degree from the Institut National des
Télécommunications (INT), Evry, France, in 2006
and the M.Sc. degree in electrical engineering from
Laval University, Quebec, QC, Canada, in 2008. He
is currently working toward the Ph.D. degree at the
Institute of Electronics and Telecommunications of
Rennes (IETR), INSA, Rennes, France.
Philippe Besnier (SM’10) received the diplôme
d’ingénieur degree from Ecole Universitaire d’Ingénieurs de Lille (EUDIL), Lille, France, in 1990 and
the Ph.D. degree in electronics from the University
of Lille, in 1993.
Following a one year period at ONERA, Meudon,
as an Assistant Scientist in the EMC Division, he
was with the Laboratory of Radio Propagation and
Electronics, University of Lille, as a researcher at
the Centre National de la Recherche Scientifique
(CNRS) from 1994 to 1997. From 1997 to 2002, he
was the Director of Centre d’Etudes et de Recherches en Protection Electromagnétique (CERPEM), a non-profit organization for research, expertise and
training in EMC, and related activities, based in Laval, France. He co-founded
TEKCEM in 1998, a private company specialized in turn key systems for
EMC measurements. Since 2002, he has been with the Institute of Electronics
and Telecommunications of Rennes, Rennes, France, where he is currently a
Researcher at CNRS heading EMC-related activities such as EMC modeling,
electromagnetic topology, reverberation chambers, and near-field probing.