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ADAPTIVE ANTENNAS AND
MIMO SYSTEMS FOR WIRELESS SYSTEMS
Correlation and Capacity of
MIMO Systems and Mutual Coupling,
Radiation Efficiency, and
Diversity Gain of Their Antennas:
Simulations and Measurements in a
Reverberation Chamber
Per-Simon Kildal, Chalmers University of Technology
Kent Rosengren, Flextronics Design
ABSTRACT
MIMO systems are characterized by their
maximum available capacity, which is reduced if
there is correlation between the signals on different channels. The correlation is primarily
caused by mutual coupling between the elements
of the antenna arrays on both the receiving and
transmitting sides. Similarly, diversity antennas
can be characterized by a diversity gain that also
is affected by mutual coupling between the
antennas. We explain how such MIMO and
diversity antennas with mutual coupling can be
analyzed by classical embedded element patterns
that can be computed by standard computer
codes. In the MIMO example under investigation, the mutual coupling causes both reduced
correlation, which increases the capacity, and
reduced radiation efficiency, which decreases it,
and the combined effect is a net capacity reduction. We also explain how radiation efficiency,
diversity gain, correlation, and channel capacity
can be measured in a reverberation chamber.
The measurements show good agreement with
simulations.
INTRODUCTION
Mobile and wireless terminals are subject to
strong fading due to multipath propagation, in
particular when used in urban and indoor environments. The performance of the terminals in
such environments can be significantly improved
by making use of spatial, polarization, or pattern
diversity. This means that the signals on, say, two
antennas (with different position, polarization,
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0163-6804/04/$20.00 © 2004 IEEE
or radiation pattern) are combined in such a way
that there are shallower fading minima on the
combined signal. This corresponds to an increase
in the signal-to-noise ratio (SNR) in the fading
dips, so the fading margins in the system link
budget can be reduced. The increased SNR can
alternately be used to increase the capacity of
the communication channel if the system allows
this. Then, particularly if we have several antennas on both the transmitter and receiver sides,
we obtain multiple-input multiple-output
(MIMO) systems. The present article describes
how to characterize antennas for both diversity
and MIMO systems.
The so-called reverberation chamber [1] has
for a couple of decades been used for some
types of electromagnetic compatibility (EMC)
measurements. It is a metal cavity that is sufficiently large to support many resonant modes,
which are perturbed with movable stirrers inside
the chamber, creating a fading environment. We
have previously shown that the reverberation
chamber represents an isotropic multipath environment of a similar type as that we find in
urban and indoor environments, but with a uniform elevation distribution of the incoming
waves [2]. The classical radiation efficiency characterizes the antenna performance in such a uniform and isotropic multipath environment, and
we showed in [3] that this can be measured fast
and accurately in a small reverberation chamber
(Fig. 1). Furthermore, in [4] we showed that the
diversity gain of a two-port antenna can be measured straightforwardly in a reverberation chamber. This has many advantages over alternative
measurement methods, which involves driving
IEEE Communications Magazine • December 2004
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The measurements
in the reverberation
chamber are in
comparison fast,
and repeatable even
in other reverberation chambers of
equal or larger size,
provided the
chambers have
efficient stirring
methods.
A
B
Three fixed
wall antennas
C
Switch
D
1
E
2
F
Network
analyzer
n Figure 1. Photo (upper) and illustration (lower) of the reverberation chamber. The photo shows a setup
for measuring radiated power of a mobile phone in talk position relative to a mobile phone. The illustration shows a setup for measuring a six-element monopole circular MIMO array. The chamber is equipped
with two mechanical plate-shaped stirrers. The six-element monopole array and reference dipole are located on a rotatable platform and rotated inside the chamber (platform stirring). The drawing also shows a
head phantom inside the chamber, which is used to load the chamber for more excited mode. The chamber is available from Bluetest AB (http://www.bluetest.se, patent pending).
IEEE Communications Magazine • December 2004
105
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The formula for
MED is the same as
for MEG, but the
“realized gain”
radiation field
functions in the
formula for MEG
must be replaced
by “directive gain”
radiation field
functions in the
formula for MED,
where the realized
and directive gains
are standard IEEE
definitions.
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around with the measurement equipment in an
urban environment, or moving it around in an
indoor environment. The measurements in the
reverberation chamber are, in comparison, fast
and repeatable even in other reverberation
chambers of equal or larger size, provided the
chambers have efficient stirring methods. Also,
the measurements in many such reverberation
chambers can be done by using only one single
channel receiver, because we can repeat exactly
the same environment in the chamber as many
times as we like, and therefore we do not need
to simultaneously measure the two channels and
the single isolated antenna reference. In a real
environment we need, in comparison, three measurement receivers to do diversity measurements, one for each branch and one for the
reference, or a fast switch to select the three signals one after the other. These advantages are
even greater when measuring MIMO systems,
because the MIMO antenna has many ports at
which the signals have to be measured simultaneously or otherwise under the same conditions.
In the present article we describe how such
reverberation chamber measurements can be
performed, and compare with simulated results.
First, however, in the next two sections we
explain which performance parameters characterize single antennas and array antennas,
respectively, in a multipath environment.
The reverberation chamber used in the present measurements is the same as in [4] (Fig. 1).
It has dimensions 0.8 m × 1.05 m × 1.6 m. The
chamber makes use of frequency stirring, platform stirring [3], and polarization stirring [5] to
improve accuracy. In all measurements we used
25 platform positions and two mechanical stirrer
positions for each platform position. With 50 different stirrer positions in total, we also used 25
MHz frequency stirring and polarization stirring
using three perpendicular monopoles on the
inside of the chamber walls. The measurements
were done at 900 MHz.
CHARACTERIZATION OF
SINGLE ANTENNAS IN A
MULTIPATH ENVIRONMENT
DIFFERENT FADING ENVIRONMENTS
The fading environment at the receive side can
be characterized by several independent incoming plane waves. This independence means that
their amplitudes, phases, and polarizations are
arbitrary relative to each other. If the number of
waves is large enough (typically a few hundred)
or we move the antenna large distances in a less
rich environment, the in-phase and quadrature
components of the received signal become normally (Gaussian) distributed, and from this their
associated magnitudes get a Rayleigh distribution, and the phases gets a uniform distribution
over 2π. In addition, the arriving waves may
have a certain distribution in the elevation and
azimuth planes. It is natural to assume that the
mobile terminal can be oriented arbitrarily relative to directions in the horizontal plane, which
means that the azimuth angle is uniformly distributed. The terminals may have a certain given
106
orientation relative to the vertical axis, and common environments (especially outdoor) have
larger probability of waves coming in from close
to horizontal directions than from close to vertical ones. Therefore, we may need an elevation
distribution factor to describe real multipath
environments. However, it is always desirable to
have an isotropic reference environment with a
uniform elevation distribution, in which all directions of incidence over the whole unit sphere are
equally probable. This is convenient because it
simplifies characterization of the antenna in the
sense that performance becomes independent of
orientation of the antenna in the environment.
The reverberation chamber simulates such an
isotropic environment.
MEG, MED, AND RADIATION EFFICIENCY
Antennas in fading environments are sometimes
characterized by the so-called mean effective gain
(MEG). This can be calculated from the radiation field function of the antenna, and it is a
function of the orientation of the antenna, its
polarization, and the azimuth, elevation, and
polarization distributions of the incoming waves
in the environment. For the isotropic environment the MEG becomes equal to half the classical radiation efficiency. Actually, the MEG can,
for an arbitrary environment, be separated into
two factors, classical radiation efficiency and
mean effective directivity (MED). In this way
the MED would solely contain the effect of the
environment and the shape of the radiation pattern, whereas the radiation efficiency contains all
losses due to absorption and the impedance mismatch. The formula for MED is the same as for
MEG, but the “realized gain” radiation field
functions in the formula for MEG must be
replaced by “directive gain” radiation field functions in the formula for MED, where the realized and directive gains are standard IEEE
definitions [6]. Thus, the relation between MEG
and MED becomes the same as between the
classical realized and directive gains for line-ofsight systems. The radiation efficiency can be
measured in a reverberation chamber.
CHARACTERIZATION OF ARRAYS IN A
MULTIPATH ENVIRONMENT
ISOLATED ELEMENT
PATTERN AND CORRELATION
Classical array antennas for line-of-sight systems
are characterized by their realized gain and radiation patterns. The radiation patterns can also
be referred to as realized gain function. The realized gain function is the directive gain multiplied
by the radiation efficiency. The latter has contributions due to absorption in lossy material in the
antenna and its environment and mismatch at
the antenna port. The radiation pattern of a
classical array antenna is the product of two factors, the isolated element pattern (which is equal
for equal elements at different locations and the
array factor. The isolated element pattern as
defined in [7] is the radiation from one element
of the array when all the others are removed,
but with the ground plane on which they are
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located present (if any). In theoretical work on
diversity and MIMO systems the elemental
antennas are often treated as isolated, but in the
next paragraph we explain that this is not a valid
description [8, 9]. The array factor cannot be
used for MIMO antennas because the ports have
uncorrelated signals.
EMBEDDED ELEMENT PATTERN AND
RADIATION EFFICIENCY
In classical array analysis the so-called active element pattern is introduced [7]. This is the radiation pattern of a single element when all the
other elements are present, but are not excited
and instead terminated with loads representing
the source impedance on their ports. This is now
more commonly and descriptively referred to as
an embedded element pattern, a term already
used in [10]. The excited antenna will induce
radiating currents on the terminated non-excited
antennas. Therefore, the embedded element pattern may be very different from the isolated element pattern. The embedded element pattern is
used to describe blindness in classical arrays,
whereas in MIMO antennas it plays an even
more significant role. In a multipath environment the received signal on each port is transmitted or detected independent of the other
ports; therefore, each port transmits or receives
signals through their embedded element patterns. In such a way, the radiation efficiency at
each port, as well as the correlation between the
signals at all ports, are determined by the embedded element pattern. Both these quantities (radiation efficiency and correlation) are needed in
order to theoretically predict diversity gain and
maximum capacity. The embedded element patterns were used in the analysis of the diversity
antenna in [8]. The embedded element pattern
of an excited antenna port can be computed by
most commercial computer programs, by terminating all non-excited ports with 50 Ω. The
embedded element pattern can also be measured in a normal anechoic chamber. However,
it is possible to characterize diversity and MIMO
antennas without knowing the embedded element patterns explicitly. For instance, the reverberation chamber provides a way of measuring
radiation efficiency and correlation, and also
estimates of a complete communication channel
without explicitly using the embedded element
patterns, and from these channel estimates the
diversity gain and capacity can be obtained.
EQUIVALENT CIRCUITS OF RECEIVE ANTENNAS
The embedded element patterns are, like all
radiation patterns, easiest to describe on transmit, when there is a current or voltage source
connected to the antenna port, and the radiation
efficiency is well defined. However, it is well
known that reciprocity applies, so the performance of the same antenna for the receive case
is described in terms of the same embedded element pattern and radiation efficiency as on
transmit. Actually, if we mathematically formulate the embedded radiation field function
(caused by a given source current at the port of
an element) as a complex vector field function,
we can include this in the equivalent circuit of
IEEE Communications Magazine • December 2004
the same antenna on reception, as given for an
arbitrary antenna in [6, p. 78]. The complete
complex radiation field function (also containing
phase information) is needed in theoretical characterization in order to add the received voltages
from several incident waves from different directions with correct amplitudes and phases.
MEASUREMENT PROCEDURE IN
REVERBERATION CHAMBER
The uniform multipath environment can be generated artificially in a reverberation chamber,
which thereby provides a statistically repeatable
laboratory-produced environment for characterizing mobile terminals and their antennas. The
reverberation chamber can be used to measure
radiation efficiency, which characterizes the performance of single antennas in an isotropic multipath environment according to the above. It
can also be used to measure the diversity gain as
described in [4], and channel capacities of
MIMO antennas, described later.
The procedure for measuring a diversity or
MIMO antenna in a reverberation chamber is
briefly described as follows. The diversity or
MIMO array is located inside the reverberation
chamber in such a way that it is more than 0.5
wavelengths from the walls and mechanical stirrers. We also locate a single antenna with known
radiation efficiency far enough away from the
array to avoid significant direct coupling, to produce a reference level (for nondirective antennas
a spacing of a half to one wavelength is sufficient). We connect one of the array ports to a
source (i.e., a network analyzer), and terminate
all the other ports and the reference antenna in
50 Ω. We gather S-parameters between the port
and the three wall-mounted antennas (used for
polarization stirring) for all positions of the platform and mechanical stirrers and for all frequency points. The measurement procedure is then
repeated for every antenna port, also with the
uncorrected ports terminated in 50 Ω, for exactly
the same stirrer positions and position of the
array inside the chamber. Thus, the field environment is exactly the same when measuring
every port. The complex transmission coefficients S21 between the connected port and each
of the three fixed wall antennas, as well as the
reflection coefficients S11 of each wall antenna
and S 22 of the array port, are stored for every
stirrer position and frequency point. Finally, we
connect the reference antenna to the network
analyzer and perform the same measurements as
for the array. During the reference measurements, the array with all its ports terminated in
50 Ω must be present in the chamber. This is
necessary because the loading of the chamber
(and thereby the Q-factor) needs to be the same
during measurements of both the reference
antenna and the array, and because the array
itself loads the reverberation chamber noticeably
even when there is a lossy object such as a head
phantom inside the chamber. In a small chamber
it is advantageous to use frequency stirring
(averaging) to improve accuracy. In such cases
we correct the complex samples of S21 with mismatch factors due to both S11 and S22 before the
the reverberation
chamber provides a
way of measuring
radiation efficiency
and correlation,
and also estimates
of a complete
communication
channel without
explicitly using the
embedded element
patterns, and from
these channel
estimates the
diversity gain and
capacity can
be obtained.
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Two dipoles separated 15 mm
100
Ideal reference
Cumulative probability
Branch 1 and 2
separate
10–1
Theoretical
rayleigh
Radiation efficiency
branch 1 and 2
Selection combining
10–2
Effective diversity gain at 1%
Diversity gain at 0.5%
10–3
–30
–25
–20
–15
–10
–5
0
Relative power lever (dB)
n Figure 2. The cumulative probability distribution function of measured val-
ues of S12 in a reverberation chamber for an ideal reference antenna (corrected
for its finite radiation efficiency) and a diversity antenna consisting of two parallel dipoles.
frequency stirring, as explained in [5, Eq. 1]. We
also normalize the corrected S21 samples to the
reference level corresponding to 100 percent
radiation efficiency. This is obtained from the
corrected S 21 samples measured for the reference antenna and its known radiation efficiency.
We refer briefly to these corrected and normalized samples of S21 as the normalized S21 values.
The normalized S21 values represent estimates of
the channel matrix H of multipath communication channels set up between the wall antennas
and the MIMO array inside the chamber. Therefore, from the measured S-parameters the diversity gain and capacity can be obtained. This is
explained below.
APPARENT, ACTUAL, AND
EFFECTIVE DIVERSITY GAINS
Diversity means that we use two antennas
located sufficiently far from each other (space
diversity) or otherwise with low coupling
between them (polarization or pattern diversity). The received signal will then be uncorrelated on the two antennas, and it is very unlikely
that there will be a fading dip simultaneously
on both antennas. Therefore, by an appropriate combination of the two signals, the probability of a fading dip will be strongly reduced.
There are several different possible combination schemes, such as switch diversity, selection
combining, and maximum ratio combining, and
the improvement in the fading margin can be
as large as 10–12 dB. Diversity in the mobile
terminal exists in Korean and Japanese mobile
communication systems, and Universal Mobile
Telecommunications System (UMTS) is prepared for it.
108
As an example we treat two parallel dipoles,
with a given separation, at 900 MHz. Figure 2
shows the cumulative distribution function
(CDF) of the measured normalized S21 samples
in the reverberation chamber. We see that the
curve of the ideal reference antenna (a dipole
with known radiation efficiency) follows the theoretical Rayleigh distribution very closely
(because the curve is corrected for the dipoles’
known radiation efficiency), which is the guarantee for a rich scattering environment. The CDFs
of each branch of the diversity antenna also have
the same shape as a theoretical Rayleigh distribution, but they are shifted to the left because
their radiation efficiency is lower. Actually, the
horizontal spacing between the CDFs of the two
elements of the branches (i.e., dipoles) of the
diversity antenna and the CDF of the ideal reference is equal to the radiation efficiency. If we
apply selection combining of the normalized S21
samples of the branches, we get the improved
CDF marked “selection combining.” The diversity gain is the difference between the selectioncombined CDF and one of the other CDFs at a
certain CDF level, commonly chosen to be 1
percent. We can distinguish between apparent,
effective, and actual diversity gains [4], depending on whether we use as a reference one of the
branches of the diversity antenna, an ideal single
antenna (corresponding to radiation efficiency of
100 percent), or an existing practical antenna to
be replaced, respectively. In the latter case the
practical reference antenna shall be located in
the position relative to an object (e.g., a head
phantom) that corresponds to the desired position of operation of the existing antenna as well
as the replacing diversity antenna. Then the
actual diversity gain is the apparent diversity
gain multiplied by the radiation efficiency of the
single existing antenna the diversity antenna
shall replace. The effective diversity gain represents the gain over a single ideal reference
antenna with no additional antenna close to it.
By ideal we mean, as before, that it is
impedance-matched to the transmission line
feeding it and has no dissipative losses.
We see that in our example in Fig. 2 the
apparent diversity gain at 1 percent CDF level is
8 dB, whereas the effective diversity gain compared to the ideal single antenna reference is
only about 3 dB. This means that if in a system
we allow for a fading margin of 20 dB in order
to receive with sufficient quality 99 percent of
the time (i.e., 1 percent CDF level), we can
reduce the fading margin by 3 dB if we use this
specific diversity antenna (two parallel dipoles
with 15 mm spacing) instead of a very good single antenna. The diversity gain could easily be
made larger by using larger dipole spacing, or
choosing two orthogonal antennas. The theoretical maximum is 10 dB by selection combining.
The discrepancy between 3 and 10 dB is mainly
due to low radiation efficiency. In the example
this is caused by mutual coupling, giving large
absorption in the 50 Ω load of the opposite
dipole, as studied in [8]. The reduced diversity
gain due to correlation has a comparatively
small effect, as explained in the next paragraph.
Figure 3 shows measured and computed
apparent and effective diversity gains as a func-
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tion of the spacing between the dipoles. The difference between the two sets of curves is the
radiation efficiency. The computed values are
obtained using a classical embedded element
model with sinusoidal current distribution on
both dipoles [8]. The radiation efficiency is automatically accounted for by this model, as
explained earlier. The diversity gain is calculated
by calculating the correlation between the two
branches’ embedded radiation patterns and
transforming this to an apparent diversity gain
using standard formulas from [11]. The correlation causes the reduction of the apparent diversity gain for small dipole spacing.
It is also of interest to know the diversity gain
for antennas used near the human body. Therefore, we measured in the reverberation chamber
two parallel dipoles located 2 cm from a PVC
cylinder filled with tissue-equivalent liquid of the
same type used in head phantoms. The results
are shown in Fig. 4. We have now also plotted
the actual diversity gain, obtained by using as a
reference a single dipole located at the same 2
cm distance from the PVC cylinder. We see that
now we have actual diversity gains of more than
6 dB, even when the dipole separation is only 2
cm at 900 MHz. This is very promising for the
use of diversity in mobile phones. Diversity
antennas may give a large improvement in SNR
when used in mobile phones, even if the two
antenna branches are not orthogonally polarized, and even at 900 MHz.
12
Computed apparent diversity gain
11
Measured apparent diversity gain
10
Diversity gain (dB)
ROSENGREN LAYOUT
9
8
7
Computed effective diversity gain
6
Measured effective diversity gain
5
4
3
0
0.1
0.2
0.4
0.3
0.5
Dipole separation (d/λ)
n Figure 3. Computed and measured apparent and effective diversity gain of
two parallel dipoles as a function of their separation.
THE CHANNEL CAPACITY OF
MIMO ANTENNAS
IEEE Communications Magazine • December 2004
10
Diversity gain
9
8
Diversity gain (dB)
In future mobile communication systems (after
UMTS) it is proposed to use an array of antennas on both the base station and terminal sides
to form several communication channels (i.e., a
MIMO system). For instance, three and six
antennas on transmit and receive sides, respectively, form 3 × 6 = 18 possible communication
channels. The data is then on transmit distributed among the channels and combined
again after reception in such a way that the
overall channel capacity is maximized. This
means that those of the 18 receive-transmit
antenna combinations that provide fading maxima transfer many more bits per second than
those transmit-receive antenna combinations
that give minima.
The maximum possible average channel
capacity in a MIMO system can be calculated by
using the formula below. As an example we use
a 3 × 6 MIMO system setup in the reverberation
chamber, consisting of three theoretical uncoupled antennas on one side (the three wall mounted antennas) and on the other side six close
monopoles with a fixed radius and equal distance between each other located on a circular
ground plane (the MIMO array under test).
The three fixed wall antennas are mounted
on three different perpendicular walls in the
reverberation chamber far from each other, so
the direct coupling is small. The MIMO array,
on the other hand, is six closely located
monopoles that interact strongly, and the mutual
coupling depends on the spacing between them.
7
Actual diversity gain
6
5
Effective diversity gain
4
3
2
3
4
5
6
Distance between dipoles (cm)
n Figure 4. Measured apparent, effective, and actual diversity gains of two par-
allel dipoles located 2 cm from a PVC cylinder filled with tissue-equivalent liquid at 900 MHz.
109
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Vi = −
25
2 jλ
∑ G i ( θ k , ϕ k ) ⋅ Ek ,
ηI k = Sufficient
(2)
many plane
waves
se
20
d
te
ca
where η is the free space wave impedance, λ is
the wavelength, and G i (θ,ϕ) is the embedded
radiation field function of monopole i calculated
on transmit when the impressed current at the
input port is I. Thus, the unknown I is removed
when G i (θ,ϕ) is divided by I. The embedded
radiation field function is, as explained earlier,
the radiation field function of the monopole in
the presence of the other monopoles when they
are terminated by 50 Ω. In this way we include
that the monopoles are coupled, which comes
into the radiation pattern. We also need to
account for the mismatch at the antenna ports.
This is characterized by a reflection coefficient,
0.24 λ
la
rre
co
Capacity (b/s/Hz)
Un
0.14 λ
15
0.06 λ
3 x 6 MIMO
10
5
ρi = (Z0 – Zi)/(Z0 + Zi),
0
0
5
10
15
Signal-to-noise ratio
20
25
n Figure 5. Mean capacity for a 3 × 6 MIMO system calculated from measured
channel estimates at 900 MHz in the reverberation chamber. The MIMO array
consists of six monopoles on a circular ground plane, and the spacing between
neighboring monopoles are 0.24 λ, 0.14 λ and 0.06 λ.
(3)
where Z0 = 50 Ω in the termination impedance,
and Z i is the impedance of the embedded element. To account for the mismatch we correct
the received voltage Vi with the factor
2
1 − ρi ;
that is, we transform Vi to a wave amplitude
When we measure the MIMO array in the reverberation chamber with three fixed wall antennas
(as in our case), we can define 3 × 6 number of
channels, and we can find the combined capacity
of the 3 × 6 = 18 channels from the channel
matrix H3×6-MIMO formed by the normalized S21
values. The instantaneous maximum capacity of
the MIMO system takes the known form
 

SNR
C3 × 6 − MIMO = log2  det  I M +
HH *  ,
3

 
(1)
where H≡ H3×6-MIMO. We have measured channel estimates on the ports of the real antenna, so
the coupling is included in the channel estimate.
Therefore, we can use Eq. 1 directly and do not
need to account for the coupling as a separate
effect. We use all the measured samples of the 3
× 6 channel matrix H 3×6-MIMO to calculate the
mean capacity C 3×6-MIMO as a function of the
SNR by averaging all values of the instantaneous
capacity. We repeat this for three different distances between neighboring monopoles (20 mm,
46 mm, and 80 mm). The resulting mean capacities for these three cases are plotted in Fig. 5 as
a function of the SNR. We see that the capacity
for small monopole spacings is reduced by a significant amount from that of the uncorrelated
case (very large spacing).
In order to calculate the theoretical capacity,
we distribute a set {Ek} of k plane wave sources
randomly and uniformly distributed over a
sphere surrounding the MIMO array. The direction of incidence of source k is denoted θ k, ϕ k.
The sources Ek in the set have independent and
normal-distributed complex amplitudes of their
θ- and ϕ-polarized components. We calculate
the received voltage Vi on each monopole i due
to the set of such incident plane waves by using
the equivalent circuit on reception [6, p. 78],
110
Ci = Vi 1 − ρi
2
that corresponds to S 21 in an earlier section.
(Computer codes may give as output a radiation
pattern normalized to 0 dB in the center of the
main beam, a separate phase pattern Φ(θ,ϕ),
and a realized gain in dB in their center direction. In such cases the Gi(θ,ϕ) can be obtained
from the values, but the mismatch factor will
already be included in G i(θ,ϕ), so we shall not
correct for it afterward.)
For the present monopole example, we first
calculate G i (θ,ϕ) and Z i of each element by a
commercial moment method code IE3D
(www.zeland.se). Thereafter we generate 3 ×
1000 sets of sources, with k = 1, 2, …, 20 sources
in each set, and calculate 3 × 1000 values of
received amplitudes C i . The received wave
amplitudes are then put row-wise into channel
matrix H after being normalized to a reference
level. This is the square root of the average
received power level at a single dipole, which is
exposed by the same sets of sources as the simulated radiation patterns of the MIMO element.
In this way, the radiation efficiency and mutual
coupling are included in channel matrix H. We
get in total 1000 H3×6-MIMO matrices for the 3 ×
6 MIMO system.
In Fig. 6 we show the resulting channel capacities for SNR = 15 dB for the 3 × 6 MIMO system. We see that the capacity based on
measurements and simulations agree very well.
Finally, we show in Fig. 7 some theoretical
maximum channel capacities for the example
MIMO antenna obtained by classical calculation
of the embedded element patterns, assuming a
sinusoidal current distribution on each quarterwavelength monopole and an infinite ground
IEEE Communications Magazine • December 2004
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plane. The electromagnetic analysis is similar to
that in [8] for a diversity antenna, but the number of elements is now 6. If we use the isolated
element pattern and correct them with the radiation efficiency of the embedded element, we see
a curve that naturally is lower than the isolated
element curve for all element spacings. If we
account for both the correlation and the radiation efficiency by using the original embedded
radiation field functions (including losses and
mismatch), the curve changes somewhat from
the latter, but not significantly. Therefore, it is
the radiation efficiencies of the embedded elements that affect the channel capacity the most,
not the correlation between the received signals
on their ports (at least for this six-monopole
case). In order to save space we did not include
any specific results for the radiation efficiency
and its two contributions, dissipation and mismatch, but the larger contribution is, as
explained, the dissipation in the neighboring elements.
CONCLUSIONS
We have discussed how MIMO and diversity
antennas can be analyzed in terms of classical
embedded element patterns (when all other elements are terminated with dummy loads). These
determine the correlation between the signals on
the different ports. The radiation efficiency can
be obtained as the ratio between the radiated
power of the embedded element and the maximum available power at the input port. We have
explained how the diversity gain and channel
capacity can be found from the embedded element patterns in theoretical predictions.
The characterizing quantities are radiation
efficiency, correlation, diversity gain, and channel capacity. These can also be determined
directly by measurements in a reverberation
chamber, without having to determine the
embedded element pattern.
The radiation efficiency has much greater
influence on the channel capacity than the correlation for the example array. The theoretical
measured results for diversity gain and channel
capacity agree well.
3 x 6 MIMO: capacity at SNR 15 dB
16
Capacity at SNR 15 dB (b/s/Hz)
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14
Modeled with IE3D
12
10
8
Measured
0
0.3
0.2
0.1
Distance between neighboring monopole (d/λ)
n Figure 6. Simulated and measured maximum capacity of the six-element
monopole MIMO antenna used in a 3 × 6 MIMO system for SNR = 15 dB.
The measurements were done in the reverberation chamber.
3 x 6 MIMO
Uncorrelated symptote
18
Capacity at SNR 15dB (b/s/Hz)
ROSENGREN LAYOUT
16
nts
me
le
de
ate
l
Iso
14
Embedded elements
12
Embedded only
radiation efficiency
10
8
REFERENCES
[1] M. Bäckström, O. Lundén, and P.-S. Kildal, “Reverberation Chambers for EMC Susceptibility and Emission
Analyses,” Rev. Radio Sci.. 1999–2002, pp. 429–52.
[2] K. Rosengren and P.-S. Kildal, “ Study of Distributions
of Modes and Plane Waves in Reverberation Chambers
for Characterization of Antennas in Multipath Environment,” Microwave and Opt. Tech. Lett., vol. 30, no. 20,
Sept. 2001, pp. 386–91.
[3] K. Rosengren et al., “Characterization of Antennas for
Mobile and Wireless Terminals in Reverberation Chambers: Improved Accuracy by Platform Stirring,”
Microwave and Opt. Tech. Lett., vol. 30, no. 20, Sept.
2001, pp. 391–97.
[4] P.-S. Kildal et al., “Definition of Effective Diversity Gain
and How to Measure it in a Reverberation Chamber,”
Microwave and Opt. Tech. Lett.., vol. 34, no. 1, July 5,
2002, pp. 56–59.
[5] P.-S. Kildal and C. Carlsson, “Detection of a Polarization
Imbalance in Reverberation Chambers and How to
Remove it by Polarization Stirring when Measuring
Antenna Efficiencies,” Microwave and Opt. Tech. Lett.,
vol. 34, no. 2, July 20, 2002, pp. 145–49.
[6] P.-S. Kildal, Foundations of Antennas — A Unified
Approach, Studentlitteratur, 2000.
IEEE Communications Magazine • December 2004
0
0.1
0.2
0.3
0.4
0.5
Distance between neighboring monopole (d/λ)
n Figure 7. Theoretical capacities of the six-element monopole MIMO antenna
modeled by using sinusoidal current distribution on all monopoles, infinite
ground plane, and models based on different approximations. The embedded
elements curve represents the most complete model.
[7] R. C. Hansen, Microwave Scanning Antennas: Vol II,
Array Theory and Practice, Academic Press, 1964.
[8] P-S. Kildal and K. Rosengren, “Electromagnetic Analysis
of Effective and Apparent Diversity Gain of Two Parallel
Dipoles,” IEEE Antennas and Wireless Prop. Lett., vol. 2,
no. 1, 2003, pp. 9–13.
[9] J.W. Wallace and M. A. Jensen, “Termination-Dependent Diversity Performance of Coupled Antennas: Network Theory Analysis,” IEEE Trans. Antennas Prop.,
vol.52, no.1, Jan. 2004, pp. 98–105.
[10] A. C. Ludwig, “Mutual Coupling, Gain and Directivity
of an Array of Two Identical Antennas,” IEEE Trans.
Antennas Prop., Nov. 197, pp. 837–41.
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ROSENGREN LAYOUT
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[11] M. Schwartz, W. R. Bennett, and S. Stein, Communication System and Techniques, McGraw-Hill, 1965.
BIOGRAPHIES
PER-SIMON KILDAL [M’82, SM’84, F’95] ([email protected]) He
received M.S.E.E., Ph.D., and Doctor Technicae degrees
from the Norwegian Institute of Technology (NTH), Trondheim,in 1976, 1982, and 1990, respectively. From 1979 to
1989 he was with the ELAB and SINTEF, research institutes in Trondheim, Norway. Since 1989 he has been a
professor at Chalmers University of Technology, Gothenburg, Sweden, where he has educated 11 antenna doctors. He has held several positions in the IEEE Antennas
and Propagation Society: elected member of the administration committee 1995–1997, distinguished lecturer
1991–1994, associate editor of Transactions 1995–1998,
and associate editor of a special issue in Transactions in
2004. He has authored or co-authored more than 167
papers in IEEE or IEE journals and conferences, concerning
antenna theory, analysis, design and measurement. He
gives short courses and organizes special sessions at conferences, and has given invited lectures in plenary sessions
at several conferences. His textbook Foundations of
Antennas — A Unified Approach (Studentlitteratur, 2000)
(www.studentlitteratur.se/antennas) got an excellent
review in IEEE Antennas and Propagation. ELAB awarded
112
his work in 1984. He has received two best paper awards
in IEEE Transactions on Antennas and Propagation. He is
the originator of the concept of soft and hard surfaces in
electromagnetics. He has done the electrical design and
analysis of two very large antennas. The first was the 120
m long by 40 m wide cylindrical parabolic reflector antenna of the European Incoherent Scatter Scientific Association (EISCAT), located in North Norway. The second was
the Gregorian dual-reflector feed of the 300 m diameter
radio telescope in Arecibo, on a contract for Cornell University. He is presently involved in the design of feeds for
the U.S. proposal of the square kilometer array (SKA). He
holds many granted patents and patents pending, and
based on these he has founded three companies, including COMHAT AB, which since 2002 is COMHAT-Provexa
AB (www.comhat-provexa.com), and Bluetest AB
(www.bluetest.se).
K ENT R OSENGREN ([email protected])
received a Master of Science degree in physics from the
university of Gothenburg, Sweden, in 1996. Since January
1999 he is with Flextronics Design (former Intenna Technology), RF and Antenna Center, Kalmar, Sweden, as a
senior antenna designer. Included in his employment, he is
part-time at Chalmers University of Technology, Gothenburg, Sweden, where he is pursuing a Ph.D. on measurements of terminal antennas in a reverberation chamber.
IEEE Communications Magazine • December 2004