MIMO capacity in free space and above perfect ground:
Theory and experimental results
Persefoni Kyritsi
Center for PersonKommunikation
Niels Jernes Vej 12
Aalborg Ø DK-9220
Denmark
Abstract-Multiple input- multiple output (MIMO)
systems are a promising technique for wireless
applications that require high data rates. In this paper,
we theoretically study their capacity potential for
propagation in free space propagation and over a ground
surface. We investigate how this depends on the array
geometry and the electric field polarization. Moreover we
validate our theoretical predictions with measurements
above a nearly perfectly flat surface. The experiment
shows good agreement with the theory for a large range of
distances and deviates from it when secondary effects
(roughness etc) become more significant.
I. INTRODUCTION
In recent years, a lot of attention has been drawn to systems
with multiple element transmitter and receiver arrays,
because they can achieve very high spectral efficiencies [1].
This is particularly significant for wireless applications that
are power, bandwidth and complexity limited. Several
techniques that require advanced signal processing at the
receiver (and possibly the transmitter) have been developed
and have been demonstrated to achieve a hefty portion of the
theoretically achievable capacity [2]. In this paper we
investigate the potential of multiple input- multiple output
(MIMO) systems under simple propagation conditions and
we present relevant experimental results.
Assume a system with M transmitters and N receivers.
Each transmitter i sends an independent data stream xi with
power Ex, so that the total transmitted power is Pt =M Ex. Let
x,y be the transmitted and the received signal vectors
respectively. In the case of a flat-fading channel (no variation
with frequency), the channel gain from transmitter j to
receiver i is a scalar quantity, denoted Tij. The transmitted
and received vectors are related by the equation y=Tx+n,
where n is the receiver noise vector. The matrix T (channel
transfer matrix) incorporates the channel transfer gains from
each transmitter to each receiver. It is assumed that the noise
at the receivers is Gaussian, of equal power σ2 and its
components are independent of each other, so that the noise
auto-correlation matrix is Rnn=σ2I (I: identity matrix).
The generalized Shannon capacity for this channel is given
by the formula (the proof is trivial information theory)
E
(1)
C = log 2 (det(I + 2x TT H )) ,
σ
0-7803-7589-0/02/$17.00 ©2002 IEEE
H
where T is the Hermitian (complex conjugate transpose) of
the matrix T.
All the signals used in our formulation are discrete-time
complex base-band, so the vectors x,y,n and the elements of
the channel transfer matrix T are complex. It is assumed that
perfect down-conversion, filtering and sampling have been
performed.
We define the average channel gain g as
(2).
g2=E[|Tij|2
2
The normalized channel transfer matrix H ( E H ij = 1 ) is
defined as T=gH
and the average signal to noise ratio ρ as
P
ρ = t2 g 2
σ
[ ]
(3),
(4).
Similarly to the Shannon formula for the single
transmitter/ single receiver case, the channel capacity can be
expressed in terms of the average signal to noise ratio ρ as
ρ
(5).
C = log 2 (det( I +
HH H ))
M
For a given ρ and system size, the channel capacity is
maximized when the matrix H is unitary. In the initial
theoretical studies certain statistical properties have been
assumed for the elements of H (a common assumption is that
of a rich scattering environment, where power arrives with a
uniform angle of arrival over [0, 2π], which leads to Gaussian
distributed entries of H). However we will demonstrate
simple propagation situations where the channel transfer
matrix is deterministic.
Moreover the capacity of line-of-sight systems is
commonly assumed to be lower than that of Gaussian
channels at the same signal to noise ratio. We will
demonstrate that this is not necessarily the case.
This paper is structured as follows. Section II presents the
theoretical predictions of MIMO systems capacity in free
space. Section III performs a similar analysis for propagation
above a ground plane and investigates the effect of
polarization, inter-element separation, and array orientation.
Section IV presents the experimental results for propagation
above a ground surface and section V summarizes our results.
For our purposes, all the distances are normalized to the
wavelength.
PIMRC 2002
M Transmitters
N Receivers
kth element:
(x , y , z )
y
k
k
k
z
x
Figure 1: Transmit and receive array geometry
Figure 2: Capacity of a 4x4 system in free space for ρ =20dB
(CMAX, 4x4 = 27bps/Hz, CGaussian 50% = 22bps/Hz)
CAPACITY OF MULTIPLE ELEMENT ANTENNA
SYSTEMS IN FREE SPACE
II.
Let us assume linear arrays of isotropic antennas, without
any mutual coupling, as shown in figure 1. We are assuming
that the arrays are in free space, and the transmitter array has
M elements and the receiver array has N elements. Each
element k is at a position (xk, yk, zk).
The signal received at the kth element in the receiving array
due to the excitation in the mth element of the transmitting
array depends only on the distance rkm between them (ground
plane not present). The channel transfer coefficient Tkm is
given by the equation
Tkm = G
e
− j2π r
km
rkm
,
(6)
rkm = (x k − x m ) 2 + (y k − y m ) 2 + (z k − z m ) 2
The constant G is independent of the number of elements in
the geometry and incorporates the effect of antenna gain,
wavelength etc. Also, the absolute value of the locations is
not significant since the parameter of interest is the distance
between the elements k and m.
Given the channel transfer gain from each transmitter to
each receiver, we can calculate the channel transfer matrix T.
From equation 5, we can compute the system capacity for any
reference signal to noise ratio. The parameters that affect the
system capacity are the geometrical details, i.e. the layout of
the arrays and the inter-element separation s.
Figure 2 shows the capacity of a system with 4 transmitters
and 4 receivers. The antennas are arranged in linear arrays
parallel to the y-axis at zk = 0, ∀ k. The inter-element
separation is s. The transmitting array is fixed at x = 0 and the
receiver array is moved along the x axis. The results are
presented for a constant reference signal to noise ratio of
20dB, and the distance along the x axis is in logarithmic scale
because we want to draw attention to the capacity behavior of
the system at small distances. As a measure of comparison
we have also plotted the maximum capacity of a 4x4 system
and the median capacity of Gaussian 4x4 channels.
At small distances, the system achieves capacities that are
higher than the median capacity of Gaussian channels. This is
because the channel gains have sufficiently different
amplitudes and phases. Also the spacing of the antennas
strongly affects the channel capacity. Indeed when the
antennas are spaced wide apart (s = 5λ) the channel capacity
remains high for a large range of transmitter-receiver
separations. However, independently of the spacing, at large
distances the system capacity saturates to the minimum
capacity, C min = log 2 (1 + 4ρ ) = 8.65bps/Hz , i.e. it benefits
from the multiple elements only in terms of power diversity.
III.
CAPACITY IN THE PRESENCE OF A SINGLE
REFLECTING SURFACE
In this section we use the geometrical configuration of
figure 1, but we assume that at the xz plane there is a
perfectly reflecting ground plane. We employ the method of
images for the prediction of the electric field.
A. Theoretical analysis
Let us concentrate on a single transmitting antenna as in
figure 3.
The method of images uses the solution of Maxwell’s
equations to predict the electric field in the half-space y≥0. It
utilizes the boundary condition on the ground plane,
according to which the electric field has to be perpendicular
to the perfectly conducting plane (E// = 0).
Without loss of generality, let us assume that the original
source is at the location (0,y0,0).
The field in the half-space y≤0 is identically zero (E=0).
According to the method of images, the field at any location
in the half-space y≥0 can be computed as the sum of the
fields from the original source and from an image source at a
location (0, -y0, 0). An alternate way to treat the signal from
the image is to consider it as signal from the original source
that was reflected off the ground plane.
y
y
h
h
x
x
h
 ε cosθ − ε − sin 2 θ
i
r
i
 r
, polarization ⊥ plane of incidence
 ε r cosθ i + ε r − sin 2 θ i
R=
2
 ε r − sin θ i − cosθ i
, polarization // plane of incidence

 ε r − sin 2 θ i + cosθ i
(10)
For polarization perpendicular to the plane of incidence,
there is a value of the incidence angle that makes the
reflection coefficient equal to 0. This is called the Brewster
angle and its value is given by
(11)
θ brewster = tan − 1
εr
At angles close to the Brewster angle, the reflected signal is
greatly attenuated. There is no such angle for polarization
parallel to the plane of incidence.
In the case of finite conductivity of the reflecting surface, the
relative dielectric constant in equations (10) is replaced by
the quantity
σ
(12)
ε 'r = ε r (1 − j
)
ωε r ε 0
B. Effect of geometrical layout
(
Figure 3: Reflection off perfect ground
Depending on the polarization of the transmitted signal, the
signal from the image is of either the same (polarization
perpendicular to the ground plane) or the opposite
(polarization parallel to the ground plane) sign relative to that
of the source.
So the field at any location (x,y,z) (y≥0 ) is given by
T=G
e
− j2π r
r
+ RG ′
′
e − j2πr
,
r′
(7)
r = x + (y 0 − y) + z , r ′ = x + (y 0 + y) + z
The factor G’ again incorporates effects such as antenna gain,
wavelength etc. If we assume isotropic antennas, G = G’.
The factor R corresponds to the reflection coefficient off the
ground plane.
 1, polarization // plane of incidence
(8)
R=
−
1,
polarizati
on
⊥
plane
of
incidence

If we assume arrays of isotropic antennas, without any
mutual coupling, and of known polarization, we can repeat
the above calculation for each transmitter-receiver pair and
compute the channel transfer gains from equation (7). From
these we can formulate the channel transfer matrix T and
compute the system capacity.
This analysis holds exactly for a perfectly conducting
ground plane. However we can use it as an approximation to
the exact solution for the case of a dielectric or imperfectly
conducting ground plane. In this case the signal from the
image has to be scaled by a reflection coefficient R that
depends on the dielectric/ conductive properties of the
boundary surface as well as the angle of incidence and the
type of polarization.
Let us assume that the reflecting surface has a relative
dielectric constant εr and that the angle of incidence is θi:
2
2
2
2
2
x 2 + z2
r′
Then the reflection coefficient R is given by, [3],
cosθ i =
2
(9)
)
We again consider a system with 4 transmitters and 4
receivers as in figure 1. The xz plane (y = 0) is a perfectly
conducting surface. The antennas are arranged in linear
arrays. The inter-element separation is s. The transmitting
array is fixed at x =0 and the receiver array is moved along
the x axis. The lowest element of each array is placed at y=h.
The added parameter that affects the channel capacity relative
to free space propagation is the height h at which the arrays
are placed.
Moreover, the orientation of the linear array with respect to
the ground plane becomes important. The antenna arrays are
oriented either along the y-axis (vertical array), or parallel to
the ground plane (horizontal array). The calculations are
performed for two values of the array height (h=10λ and
h=100λ), and for two values of the inter-element separation
(s=0.5λ and s=5λ).
Figure 4 shows how the system capacity varies with
distance between the transmitter and the receiver arrays if the
transmitted signals are polarized parallel to the ground plane
(R=-1), and figure 5 performs similar analysis for transmitted
signals polarized perpendicular to the ground plane (R = 1).
The results are presented for a reference signal to noise
ratio of 20dB, and the distance is in logarithmic scale because
we want to accentuate the effects at small distances. As a
reference we have added thick horizontal lines that
correspond to the maximum capacity of a 4x4 system at
ρ=20dB, and the median capacity of Gaussian channels at the
same signal to noise ratio. (CMAX, 4x4 = 27bps/Hz, CGaussian, 50%
= 22bps/Hz, ρ = 20dB)
Also, the vertical antenna array achieves higher capacities
for a wider range of distances than the horizontal antenna
array. This indicates that the spacing of the antennas should
be along the direction where the channel displays the most
variation. In our case this is the y axis.
Although the exact capacity values are different for the two
polarizations, the qualitative behavior is similar.
IV.
Figure 4: Capacity behavior of a 4x4 system above perfect
ground at ρ = 20dB (polarization // to the ground)
Figure 5: Capacity behavior of a 4x4 system above perfect
ground at ρ = 20dB (polarization ⊥to the ground)
For a range of distances the capacity of the system
outperforms the median capacity of Gaussian random
channels. This range depends on the particular antenna height
and element separation. At large distances the capacity for
any separation and height converges to the minimum capacity
of 8.65bps/Hz, i.e. it benefits only in terms of power receiver
diversity (Cmin =log2 (1+4ρ) = 8.65bps/Hz).
Larger inter-element separation and height tend to increase
the range of distances for which the system capacity is above
its minimum value.
EXPERIMENTAL COMPARISON
A. Description of the experiment
In order to verify the theoretical predictions, we performed
a set of measurements in the parking lot of the Lucent
Technologies- Bell Labs building in Crawford Hill, NJ.
The measurements were taken with a system of 12
transmitters and 15 receivers at a frequency of 1.95 GHz. The
antennas used are flat arrays of folded cavity backed slot
antenna elements mounted on 2’x2’ panels. They have a
hemispherical gain pattern, so they pick up energy from the
direction at which they are facing. The antenna elements were
either vertically or horizontally polarized and arranged in
alternate polarizations on 4x4 grids, separated by λ/2 (~8cm).
Figure 6 shows how the arrays look from the front (H/ V:
vertically/ horizontally polarized elements).
The system bandwidth was 30 kHz. The filters were raised
cosine filters with a bandwidth expansion factor of α = 0.23
(symbol rate = 24.3 ksymbols/sec). The constellation size
used was 4 QPSK.
The prototype used for the measurement campaign
processes data in bursts. Each burst consists of 100 symbols.
Out of these, the first 20 are training symbols and are used for
the measurement of the channel transfer matrix, using
orthogonal training sequences as suggested in [4].
The measurements were performed in the parking lot,
because it approximates the situation of an antenna array
above a single reflecting surface. The parking lot is a flat
asphalt-covered area in front of the building. In order to
minimize reflections off the building, the hill and vehicles,
the antennas were located at a distance of 20ft from the side
of the building, part of the parking lot was reserved for the
experiment, and no cars were allowed in the vicinity.
We repeated the measurements for two heights of the
antenna arrays: the lowest element of the array was placed at
a height of either 2m or 12cm off the ground. This
corresponds to a height of 13.16λ and 0.94λ respectively.
Measurements were taken for a limited set of transmitterreceiver separations, namely 1m, 1.5m, 2m, 5m, 10m and
15m.
TRANSMITTER SIDE
RECEIVER SIDE
H5
H5
V6
H1
V5
V5
H8
V2
H4
H4
V7
H7
V2
V4
V4
H3
V3
H2
V1
H2
H6
V3
V6
H3
Figure 6: Array layout
H6
V1
H1
B. Experimental results
Figures 7 and 8 show how the capacity of the measured
system (as calculated from the measured channel transfer
matrices) matches the predicted capacity from the method of
images. The results are shown for the reference signal to
noise ratio of 20dB.
For the theoretical predictions we assumed that the relative
dielectric constant of the parking lot material (asphalt) was εr
=3. Moreover we distinguished the horizontal from the
vertical polarization, so we calculated different reflection
coefficients for each and we assumed that there is no crosspolarization coupling. So at large distances we expect the
capacity to converge to the sum of the capacities of the two
independent polarization subsystems
C limit = C min, // + C min, ⊥ =
.
7
8
= log 2 (1 + 7 ρ) + log 2 (1 + 8 ρ) = 19.7bps/Hz
6
6
εr = 3
For some range of distances the theoretical predictions for
free-space propagation are very close to the predictions that
include the perfectly conducting ground plane. This is due to
the fact that the reflected waves only become comparable to
the direct waves, and therefore start to affect the capacity
results, after the antennas have been sufficiently separated
and the two paths have comparable strengths. Obviously the
distance at which this occurs is larger for large antenna height
as we observe from the comparison of figures 7 and 8.
The measurements agree fairly well with the predicted
values for the capacity in the presence of a reflecting surface,
especially at the small distances and for the large antenna
height. As the separation between the transmitter and the
receiver array increases, secondary effects become more
significant:
The surface of the parking lot is not smooth. This causes
back scattering, which would tend to increase the capacity of
the measured system. This effect is more pronounced in the
measurements at the low antenna height.
At larger distances, the reflections off the surrounding large
surfaces (building, hillside, cars etc) which we tried to
minimize become more significant.
However the general agreement of the measured and
theoretically predicted results encourages the use of the
image method for the prediction of channel capacity, even in
more complicated propagation environments.
V.
Figure 7: Experimental comparison with the method of
images for the 12x15 system at a height of 2m and ρ=20dB
εr=3
CONCLUSIONS
In this paper we have calculated the capacity of a MIMO
system under simple propagation conditions, namely for
propagation in free space and above a perfect ground. We
have demonstrated that the capacity depends on the array
layout (inter-element separation, height, and orientation) as
well as the polarization of the electric field. This simplified
approach showed the benefit of orienting a linear array in the
direction of the highest environment variability. Moreover we
have validated our theoretical predictions with measurements
above a nearly perfectly flat surface. The experiment shows
good agreement with the theory for a large range of distances
and deviates from it when secondary effects (roughness etc)
become more significant.
REFERENCES
Figure 8: Experimental comparison with the method of
images for the 12x15 system at a height of 12cm and ρ=20dB
[1] G.J. Foschini and M.J. Gans, “On limits of wireless
communications in a fading environment when using
multiple antennas”, Wireless Personal Communications,
Volume 6, No. 3, March 1998, p. 311-335.
[2] P.W. Wolniansky, G.J. Foschini, G.D. Golden and R.A.
Valenzuela, “V-BLAST: An architecture for realizing
very high data rates over the rich scattering wireless
channel”, Proceedings ISSSE ’98, September 1998.
[3] S. Ramo, J.R. Whinnery, T. Van Duzer, “Fields and
waves in communications electronics”, J. Wiley and
Sons, 3rd edition.
[4] T.L. Marzetta, “BLAST training: Estimating channel
characteristics for high capacity space-time wireless”,
Proceedings
37th Annual Allerton Conference on
Communication, Control and Computing, Monticello, IL,
Sept. 22-24, 1999.