ENRI Int. Workshop on ATM/CNS. Tokyo, Japan (EIWAC2010)[EN-028] On the use of MIMO in aeronauticalcommunications+

B. Holter J. E. Håkegård T. A. Myrvoll

Department of Communication Systems
SINTEF ICTTrondheim, Norway[bengt.holter|jan.e.hakegard|torandre.myrvoll]@sintef.noAbstract In this paper, it is shown that it is possible to use multiple-input multiple-output (MIMO) spatial multiplexing systems
in aeronautical communications over an extended range even in the presence of a strong line-of-sight (LOS) component. Results
originally derived to maximize the MIMO capacity in fixed range applications dominated by a LOS component is exploited
to show that a high rank channel matrix may not only be oﬀered for a fixed distance but can be maintained over an extended
range and area in a MIMO ground-to-air communication system with nT = 2 transmit antennas (ground terminal) and nR ≥ 2
receive antennas (aircraft). Numerical results are presented for an nR × nT MIMO system in a Ricean fading channel assuming
communication between a ground terminal and an aircraft in the en-route domain.
Keywords MIMO, Aeronautical communications, Chebyshev polynomials

1. INTRODUCTION

the spatial multiplexing gain of the system since several data streams can be separated by an equalizer
only if the fading processes of the spatial channels are(nearly) independent.
A possible solution to this challenge was indicatedin [2], stating that a linear increase in capacity could
be achieved by increasing the distance between theantennas at the transmitter to produce antenna patterns with nulls on all but one receiver antenna. By
this approach, independent channels on the same carrier frequency could be established. A similar technique was indicated in [3, Sec. III].
Motivated by the results in [2] and [3], the authors in [4] and [5] propose a design methodology toachieve a full-rank MIMO channel matrix in a LOS
environment. It is shown that orthogonality between
subchannels of the channel matrix can be related tothe product of the inter-element spacings of the antennas at the transmitter and receiver, assuming uniform
linear array (ULA) antennas at both ends. They both
introduce more general geometrical models than thatapplied in earlier works to gain additional insight, inparticular to quantify a reduction from optimal performance caused by misalignment between the transmitter and receiver antenna arrays. In [4], the performance is evaluated with respect to both ergodic andoutage capacity using a Ricean fading channel model.
The results show that even with some deviations froman optimal design, a LOS MIMO system may outperform a system operating on independent and identically distributed (i.i.d.) Rayleigh fading channels interms of Shannon capacity.

The introduction of multiple-input multiple-output(MIMO) systems has contributed to vast improvements
in capacity and reliability in wireless communications.
MIMO systems thrive in rich multipath environments,
since it leads to antenna decorrelation and a high rank
channel matrix. 1 In these conditions, a MIMO system can oﬀer a linear increase in capacity that is proportional to the minimum number of transmit and receive antennas. As a result, the majority of researchon MIMO communications have been focused on systems operating in a rich multipath environment.
MIMO signalling techniques are currently been
considered for future aeronautical communication systems within Europe and the United States. In particular, the future airport datalink system AeroMACS
will be based on IEEE802.16 technology involving
MIMO, and aeronautical satellite communications consider introducing MIMO techniques for use in highlatitudes and for low satellite elevation angles.
A more theoretical approach to improve the performance of future aeronautical communication systems by increasing the capacity and reliability in airto-ground links has been proposed in [1]. Aeronautical channels are typically Ricean fading channels,
characterized by the presence of a strong line-of-sight(LOS) component. A LOS component will usually reduce the performance of a MIMO system, since there
is a large probability that all the signals will be conveyed through the same channel. Eﬀectively, it meansthat the MIMO channel matrix in a LOS environment
becomes rank-deficient (not full rank). This reduces
1 In some cases, the channel matrix may still be of rank one
even though the antennas at both ends are uncorrelated. Such a
MIMO channel is commonly called a ”pinhole” channel.

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B. Holter, J. E. Håkegård, T. A. Myrvoll

average power. The path loss for all subchannels is
accounted for by the common parameter Ω. This normalization is convenient since it makes the average
signal-to-noise ratio (SNR) independent of H.
A general entry in H is denoted h m+1,n+1 , representing the complex channel gain between transmit
antenna n ∈ {0, 1, . . . , n T − 1} and receive antennam ∈ {0, 1, . . . , nR − 1}. The channel matrix may thenbe written as⎡⎤⎢⎢⎢ h1,1 h1,2 . . . h1,nT ⎥⎥⎥⎢⎢⎢ h⎥⎢⎢⎢ 2,1 h2,2 . . . h2,nT ⎥⎥⎥⎥⎥
H = ⎢⎢⎢ .
(2)..
.. ⎥⎥⎥ .
..
⎢⎢⎢ ..
.
.
. ⎥⎥⎥⎦⎣
hnR ,1 hnR ,2 . . . hnR ,nT

The results presented in [4] and [5] have contributed to create new interest in the use of spatial multiplexing in environments with a strong LOS component. Notably, results have been presented in recent
publications on fixed wireless access schemes [6], indoor communication [7], and vehicle-to-roadside communication [8]. However, to the author’s knowledge,
no one has yet looked into the use of the results foraeronautical communications. Hence, in this paper,
the theoretical results in [4] are re-visited but viewed
from an aeronautical perspective, where the range now
is variable rather than fixed. Note however that theparticular view of a variable range is not original tothis paper, since it has been addressed to some extent
in small range applications [7–9]. In particular, it isshown in [7] that a MIMO system operating in a LOS
environment may be used over a larger set of ranges
by using optimized nonuniform antenna arrays rather
than uniform antenna arrays.
In this paper, a MIMO system which involves aground terminal with 2 antennas and a ULA antenna
mounted on the aircraft wings is proposed. The channel capacity of such a system is quantified and presented as a function of range and angle between the
ground antenna array and the aircraft antenna array.
Numerical results show that the channel matrix may
be kept at full rank in a LOS environment over an
extended range, which is essential to accommodate
MIMO communication in the en-route domain.
The paper is organized as follows. In Section 2,
the system and channel model is presented. Then, forclarity, information theoretic results on the channel
capacity of a MIMO system are reviewed in Section
3. In Section 4, numerical and analytical results thatquantify the performance of a MIMO system operating in a LOS environment are presented. The conclusions of the paper are presented in Section 5.

For simplicity (and later reference), it can also be presented in terms of its individual column vectors h n as
⎤
⎡
⎢⎢⎢ |
|
| ⎥⎥⎥
⎥
⎢
H = ⎢⎢⎢⎢ h0 h1 . . . hnT −1 ⎥⎥⎥⎥ .
(3)
⎦
⎣
|
|
|
In the en-route domain, the aeronautical channel
is commonly characterized as a fast fading (rapidly
time varying) channel where the amplitude of the received signal follows a Rice distribution [11]. In thiscase, the MIMO channel matrix in (2) may be written
as a sum of two parts, a deterministic part (LOS component) and a Rayleigh fading part (Non-LOS (NLOS)
component):
H = aHLOS + bHNLOS .

(4)

The power ratio of the two matrix components defines
the Rice factor as K = a2 /b2 , whichcommonly is2expressed in decibels as K = 10 log10 ab2 dB. In thefollowing, it is assumed that the sum power of thecomponents is normalized, i.e. a 2 + b2 = 1. With thisassumption, a and b√can be expressed in √terms of the
Rice factor as a = K/(K + 1) and b = 1/(K + 1).
For simplicity, a flat fading channel is assumed,which means that the channel impulse response ofeach subchannel in the channel matrix consists of one
tap only (thus no inter-symbol interference). In addition, since the channel capacity subsequently is used
as a measure to quantify the performance, coherent
detection with perfect channel knowledge at the receiver is assumed throughout the paper. In practice,perfect channel knowledge is not possible in a rapidly
time varying channel, so the results of this paper will
only serve as performance upper bounds. 3

2. CHANNEL MODEL
Using complex baseband vector notation, the input/output relations of a narrowband single user MIMO
link with nT transmit antennas and n R receive antennas can be written as√(1)
y = ΩHx + n,
where y ∈ CnR ×1 is the received signal vector, x ∈CnT ×1 is the transmitted signal vector, H ∈ C nR ×nT is
the channel matrix, Ω is the common power attenuation factor for all the channels in the channel matrix,
and n ∈ CnR ×1 is an additive noise vector containing
i.i.d. circularly symmetric Gaussian elements withzero mean and variance σ 2 (in short CN(0, σ2 )).2 Asin [10], H represents a normalized channel matrix,
which means that all entries of the matrix have unit
2
A circularly symmetric complex Gaussian random variablewith zero mean and variance σ2 is a complex random variable x +jy, where x and y are independent and normally distributed withzero mean and variance σ2 /2.

3 In [12], performance degradation of basic modulation
schemes in a rapidly time varying channel using a first order autoregressive channel model is presented.

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ENRI Int. Workshop on ATM/CNS. Tokyo, Japan (EIWAC2010)

normalized, it is assumed that σ 2h = 1. Since the aeronautical channel is a rapidly time varying channel, it
is assumed that HNLOS is memoryless, i.e. for each useof the channel and independent realization of H NLOS isdrawn.

2.1. LOS channel
The matrix HLOS represents the transfer function
for signals that have traveled from the transmitter tothe receiver by a direct path. As a result, it is a deterministic matrix since all the entries only depend on
the distance between the transmit and receive antenna
elements. The entries of H LOS can then in general be
expressed as
hm+1,n+1 = exp ( jβrmn ) ,3. CHANNEL CAPACITY
The channel capacity is a measure of the amountof information which can be transmitted and received
with a negligible probability or error. With a uniform
power distribution among the transmit antennas, thechannel capacity of a MIMO system is equal to(5)where β = 2π/λ is the wave number, λ is the wavelength of the transmitted signal, and r mn is the direct
path length between transmit antenna n and receive
antenna m. In this paper, the expression for r mn is
identical to [4, Eq. (7)], which is based on the geometrical model depicted in Fig.1. It shows a general MIMO system with ULA antennas at both endsof the link. This means that the inter-element distance between adjacent antennas in each of the arrays is fixed. The antenna inter-element distance atthe transmitter and receiver are denoted d t and dr , respectively. With nT transmit antennas and n R receive
antennas, the total length of the transmitter array becomes (nT −1)dt , while the length of the receiver array
becomes (nR − 1)dr . The distance (or range) between
the transmitter and receiver is denoted R, and it is defined to be the distance between the lower end of thetwo arrays. Finally, the ULAs are not restricted tobe parallel, and the angles θ t , θr , and φr are used tomodel the impact of arbitrary orientations. Based on
the geometry in Fig.1, the distance r mn in (5) can be
expressed as [4, Eq. (7)]
rmn

≈
+
+

R + mdr sin θt cos φr − ndt sin θt(mdr sin θr sin φr )22R(mdr cos θr − ndt cos θt )2.
2R

C = log2 det I M +γWnTbit/s/Hz,(8)

where M = min{nR , nT }, I M is the M × M identityT
is the average received SNR, P T is
matrix, γ = ΩP
σ2
the total average transmit power, 4 and
HHH , nR < nTW=
(9)
HH H, nR ≥ nT
is the channel covariance matrix. The superscript Hin (9) denotes Hermitian transpose. Note that (8) is
valid only for a given channel realization. When H isstochastic, C becomes a random variable and the ergodic channel capacity (mean capacity over all channel realizations for a given average SNR) can be obtained as C = E H (C).5
Using the matrix notation introduced in (3) andassuming nR ≥ nT , the covariance matrix W can be
expressed as
⎡
⎢⎢⎢
⎢⎢⎢
⎢
W = ⎢⎢⎢⎢⎢
⎢⎢⎢
⎣

(6)

The approximation sign in (6) is used as a reminderthat the result is valid only when R is much larger
than the transmit and receive antenna dimensions.

hH
0 h0
hH
1 h0
..
.

hH
0 h1
hH
1 h1
..
.

hH
nT −1 h0hH
nT −1 h1

...
...
..
.

hH
0 hnT −1hH
1 hnT −1..
.

. . . hHnT −1 hnT −1

⎤
⎥⎥⎥
⎥⎥⎥
⎥⎥⎥
⎥⎥⎥ .
⎥⎥⎥
⎦

(10)
Using eigenvalue decomposition on W, (8) may alternatively be written as [14]2.2. NLOS channel
The matrix H NLOS represents the transfer function
for signals that are received as a result of reflection,
diﬀraction and scattering from the environment. The
combined reception of such signals are commonly modeled by a stochastic process. As in [10], the entries
of HNLOS are modeled as circular symmetric complex
Gaussian random variables with zero mean and variance σ2h . Hence,(7)
hm+1,n+1 ∼ CN 0, σ2h .
Since the individual channel gains in this case will
be Rayleigh distributed, H NLOS is commonly referred
to as a Rayleigh fading matrix. To keep the matrix

303

C

=

rω

i=1

=

log2 1 +

γ
ωi ,
nT

⎞
⎛ rω
⎟⎟
⎜⎜⎜
γ
log2 ⎜⎜⎝
1 + ωi ⎟⎟⎟⎠ ,
nT
i=1

(11)

M
are the eigenvalues of W, and r ω ≤ M
where {ωi }i=1
is the rank of W (or H).
4 For a pure Rayleigh fading channel, an uniform power distribution is optimal in the sense that it maximizes the capacity, but
this is generally not the case for a Ricean fading channel. However,a uniform power distribution is asymptotically optimal as the SNR
tends to infinity if nR ≥ nT [13].
5 E (·) denotes the expectation over all channel realizations.
H

B. Holter, J. E. Håkegård, T. A. Myrvoll

z

z



Tx

Rx
θt

θr
(nR − 1) · dr(nT − 1) · dt
origo

φr

x

R

y



x



Figure 1 A general MIMO system with uniform linear arrays at both the transmitter and the receiver [4, Fig. 1].

channel. Hence, in order to exploit MIMO spatial
multiplexing techniques in a LOS channel, methods
that can contribute to increase the rank of M must
be employed. Note that the rank of a matrix usually is defined as the number of non-zero singularvalues. However, when the rank of M is maximized,
the rank of H LOS is also maximized since the squared
singular values of H LOS are equal to the eigenvaluesof M. As mentioned in the introduction, the authorsin [4] and [5] have proposed a design methodology toachieve a full rank MIMO channel matrix in a LOS
environment. With the assumption of ULA antennas at both ends, the technique is to use an optimized
inter-element distance at both the transmitter and thereceiver to obtain a full rank channel matrix at a given
fixed range R. In [4], the key design parameter ispresented as the product of d t and dr , referred to as
the antenna separation product (ASP). The optimal
ASP which maximizes (13) for a pure LOS channel isequal to [4, Eq. (12)]Table 1 List of the eight first Chebyshev polynomialsof the second kind.
nR

UnR −1 (cos x)

1

1

2

2 cos x3

4 cos2 x − 1

48 cos3 x − 4 cos x

5

16 cos4 x − 12 cos2 x + 1

632 cos5 x − 32 cos3 x + 6 cos x

7

64 cos6 x − 80 cos4 x + 24 cos2 x − 1

8128 cos7 x − 192 cos5 x + 80 cos3 x − 8 cos x

In the following, it will be focused on the matrix
HLOS since the overall rank of H in a strong LOS environment in general will be dominated by the rank of
HLOS . As such, it is convenient to define the associated
LOS version of (9) as K → ∞ (pure LOS channel).
In this case, the channel covariance matrix can be defined as, nR < nT
HLOS HH
LOS
M=
.
(12)
HH
nR ≥ nTLOS HLOS ,The capacity may then be expressed as
⎞
⎛ rλ
⎟⎟
⎜⎜⎜
γ
1 + λi ⎟⎟⎟⎠ ,C = log2 ⎜⎜⎝nT
i=1

dt dr =

(13)M
are the eigenvalues of M, and r λ ≤ M iswhere {λi }i=1
the rank of M (or H LOS ).
From (13), it is observed that the rank of M plays
an important part in maximizing the capacity of a MIMO
system. A MIMO system thrives in a rich multipath
environment since it contributes to realize a high rank
channel matrix. However, when a strong LOS component is present, H LOS becomes rank deficient since
all the signals then are conveyed through the same

304

λR
,
N cos θt cos θr(14)

where N = max(n R , nT ). When the ASP is equal to(14), the rank of M (and H LOS ) is maximized and equal
to rλ = M.
In the following, a MIMO system with n T = 2 andnR ≥ nT will be assumed. In this case, the maximum
rank of M (or H LOS ) is rλ = 2. The eigenvalues of
M for a nR × 2 MIMO system can then be expressedcompactly as (see Appendix)
λ1=

nR + UnR −1 (cos x),

(15)

λ2

=

nR − UnR −1 (cos x),(16)

where x = nπR η , η = N cos θtλR
cos θr dt dr , and U nR −1 (·) is a
Chebyshev polynomial of the second kind. The eight
first Chebyshev polynomials of the second kind as afunction of n R are listed in Table 1.

ENRI Int. Workshop on ATM/CNS. Tokyo, Japan (EIWAC2010)

The symbol η is called the deviation factor, and it
is defined in [4, Eq. (13)] as the ratio between the optimal ASP in (14) (in the following denoted ASP opt )and the actual ASP, i.e.
η=

ASPoptλR=.
ASP
N cos θt cos θr dt dr

distance dr . In the aircraft, the antennas are assumed
to be conformal antennas evenly distributed along theaircraft wings.
For a given channel realization and a fixed number of antennas, the capacity is a function of the SNR.
The SNR is again a function of the range, since thesignal level naturally decreases as a function of range
which then eﬀectively also reduces the SNR. However, in all the numerical results presented in this paper, the SNR is kept fixed as a function of range inorder to isolate the impact of the range from the impact of the SNR. Otherwise, it would be diﬃcult toknow whether a change in the capacity is caused by achange in the range or by a change in the SNR if theyboth vary at the same time. Hence, to better visualize
the impact of a variable range on the capacity of thesystem, the SNR is kept fixed as a function of range.

(17)

In [4], η is used as a measure (in dB) for how far theactual ASP is from the optimal ASP for a fixed range
R. Hence, if η > 1, the actual ASP is too small compared to the optimal value in (14). If η < 1, the actual
ASP is too large. However, once the optimal d t and drhave been established for a fixed range denoted R opt ,deviations from R opt will cause η to deviate from its
optimal value of one as well. Hence, in this paper,η > 1 is used to signify a system that operates at
R > Ropt . Similarly, η < 1 signifies a system that
operates at R < Ropt . As such, η is in this paper used
as a measure for the performance of a MIMO system
as the range deviates from its optimal fixed value.
Finally, ρ is defined as the normalized correlationcoeﬃcient between the receive array responses fromthe lth and kth transmit element (l,k ∈ [0, 1])UnR −1 (cos x)|hH
l hk |
ρ=.
(18)hl · hknR

4.1. Part 1 - Deterministic channel
4.1.1. 2 × 2
As a reference, the capacity of a 2 × 2 MIMO system is used. In Fig.2, (19) is depicted as a function of
R when dt and dr are selected to maximize the capacity at Ropt = 150km. In order to maximize the capacity at such a long range, either d t , dr or both must bequite large. For this reason, the inter-element distanceat the ground terminal is in this paper selected to be
fixed at dt = 1500m. This makes it possible to maximize the capacity using a comparatively small interelement distance in the aircraft, which is assumed to
be a commercial passenger jet. With d t = 1500m,
the capacity is maximized at R opt = 150km with d r =15m.

With the aid of (15), (16) and (18), the capacity in(13) for an n R ×2 MIMO system can then be expressedcompactly in closed form as
⎛
⎞
2
⎜⎜⎜
⎟
γnR 
2 ⎟
⎜
C = log2 ⎜⎝1 + γnR +1 − ρ ⎟⎟⎟⎠ .
(19)
2
For nR = 2, (19) reduces to [15, Eq. 20].

4. GROUND-TO-AIRCOMMUNICATION

1×1

10Capacity [bit/s/Hz]

This section is divided into two parts. In the firstpart, numerical and analytical results of a ground-toair MIMO communication system are presented fora pure LOS channel, i.e. for H LOS only. In the second part, similar results are presented for the complete channel matrix H, which is a stochastic channel.
The numerical results in the second part are therefore
obtained by averaging over many channels realizations. In both parts, it is assumed that the transmitter(ground terminal) is equipped with n T = 2 antennas
and the receiver (aircraft) is equipped with n R ≥ 2 antennas. The distance between the two ground terminal
antennas is denoted d t , whereas the aircraft antenna
array is assumed to be an ULA with inter-element

2×2
8
6
4
2
50

100

150
R [km]

200

250

Figure 2 Capacity of a 2 × 2 system as a function of R
when λ = 0.3m (f=1.0GHz), d t = 1500m, d r = 15m,
K → ∞, γ = 10dB, and θt = θr = 0o .

305

B. Holter, J. E. Håkegård, T. A. Myrvoll

is observed in the area 100km ≤ R ≤ 250km and−50o ≤ θr ≤ 50o . For distances closer than 100km,
it is necessary to change the inter-element distancein the aircraft in order exploit MIMO spatial multiplexing techniques, since the correlation between thesubchannels of the channel matrix becomes too high(dark areas). 7

In Fig.2, it is observed that the capacity indeed
is maximized at Ropt = 150km, but it oscillates between two extremes for R < R opt .6 The oscillation
stems from ρ in (19), which for n R = 2 is equal toπρ = | cos 2η
|. Hence, for R ≤ R opt (η ≤ 1), it can be
deduced that ρ is equal to⎧1⎪⎨ 0 for η = (2q−1), q ∈ Z+ , q  0.
(20)
ρ=⎪
⎩ 1 for η = 1
2q

An option is to mount more than two antennas at
the aircraft and use a switch-based system to selectthe optimal two antennas for a given range and angle.
Assuming that dr is optimal (maximizes the capacity)
at Ropt and θr = θ1 , the new optimal inter-element
distance dr∗ at range R and θ r = θ2 is equal to

At maximum capacity, the channel matrix is a fullrank matrix (r λ = 2) since the columns of the channel
matrix are uncorrelated (ρ = 0). The capacity is thenexactly twice the capacity of a 1 × 1 system. Denoting
the distances where the channel matrix is full rank as
Rρ=0 , it occurs at

dr∗ =

(21)

cos θr cos θt dt dr, q ∈ Z+ , q  0.
q·λ

(22)

250

In general, the distances related to a given correlation
value between zero and one can be expressed as200π
π cos θr cos θt dt dr=
Ropt ,λ(θ + πq)2(θ + πq)

where q ∈ Z+ , ρ = | cos(θ + πq)|, and
0 < θ < π2 R > Ropt
θ= π
.
3π
R ≤ Ropt
2 ≤θ ≤ 2

6.5

(23)

6
Range [km]

Rρ =

(25)

By adjusting the inter-element distance close to d r∗ ina switch-based fashion as the aircraft closes in on the
ground terminal, it is possible to stay in the vicinity

At minimum capacity, the channel matrix is rank
deficient (rλ = 1) since the columns of the channel
matrix are completed correlated (ρ = 1). Denoting the
distances where the channel matrix is rank deficient as
Rρ=1 , it occurs at
Rρ=1 =

R cos θ1dr .
Ropt cos θ2

150
5.5
100

(24)

5

In Fig.3, the result of Fig.2 is reproduced but presented as function of both θ r and R to visualize how
the capacity varies with respect to range and misalignment between the ground and aircraft antenna. As
such, Fig.3 can be viewed as an illustration of the capacity region covered by the ground antenna for an
incoming aircraft at range R and angle θ r relative tothe ground antenna. It is assumed that θ t = 0o . Onceagain, it is observed that the capacity fluctuates between two extremes given by either no correlation(ρ = 0) or complete correlation (ρ = 1). A relatively large area with a full rank matrix (white areas)

50
4.5
−50

0
θr [deg]

50

Figure 3 Capacity region of a 2 × 2 in a deterministic
LOS channel (K → ∞) as a function of θ r and R.
7 An additional ground terminal with a diﬀerent coverage area
could be used to give coverage for θr > ±50o if antennas also could
be placed along the fuselage of the aircraft. The combined coverageof two ground terminals will then ensure that θr in any case will beless than ±50o .

6 A similar result is presented in [5, Fig. 2], but then as a func-tion of dt and dr when R is fixed.

306

Capacity [bit/s/Hz]

2 cos θr cos θt dt dr, q ∈ Z+ , q  0.
Rρ=0 =(2q − 1) · λENRI Int. Workshop on ATM/CNS. Tokyo, Japan (EIWAC2010)

4.1.2. n R × 2

Capacity [bit/s/Hz]

10
8
1×1

6

8×2

4
2
50

100

200

150
R [km]

250

Figure 4 Capacity of a 8 × 2 system as a function of
R when λ = 0.3m (f=1.0GHz), d t = 1500m, d r =2.14m, K → ∞, γ = 10dB, and θ t = θr = 0o .

250
10.5
10
200

9

150

8.5
8

100

Capacity [bit/s/Hz]

Range [km]

9.5

7.5
50

7
6.5
−50

0
θr [deg]

50

Figure 5 Capacity region of a 8 × 2 system in a deterministic LOS channel (K → ∞) as a function of θ r
and R.

A simple yet eﬀective way to increase the robustness of the link in a strong LOS environment is simply to use more antennas at the receiver than at thetransmitter and exploit all available antennas at the receiver at the same time rather than using a switchedbased approach between a small subset of antennas. 8
To illustrate this fact, Fig.4 shows the capacity of a
8 × 2 MIMO system. Compared to the 2 × 2 case,
it is observed that the capacity has increased as a result of the increased number of antennas in the system. However, more importantly, the capacity curve
does not contain the oscillations observed in the 2 × 2
system. Hence, the general condition n R > nT has astabilizing eﬀect on the capacity curve as a functionof range, and the eﬀect resembles the stabilizing effect which antenna diversity has on the SNR level in afading channel. Note that the receiver antenna length
is assumed to be fixed at the reference length of 15m.
Hence, in the 2×2 case, d t = 15m, whereas in the 8×2
case, dt ≈ 2.14m. As a result of the increased numberof antennas and the reduced inter-element distance,
Ropt is reduced to approximately 86km in the 8 × 2
case compared to 150km in the 2 × 2 case.
In Fig.5, the capacity region of the 8 × 2 system is
depicted. It clearly shows the improvement from the2 × 2 case, as there are no fluctuations of the capacity
curve in the entire range 10km ≤ R ≤ 250km. The
angle range is almost unchanged from the 2 × 2 case,i.e. −50o ≤ θr ≤ 50o .
The increased stability of the capacity curve as afunction of range can also be observed by examining
the condition number κ M of M (and HLOS ). The condition number is a measure of stability or sensitivity ofa matrix (or the linear system it represents) to numerical operations, and matrices with a condition number
close to one is said to be well-conditioned. The condition number of M is equal tonR + UnR −1 (cos x) 1 + ρ
λmax

=
κM =
. (26)
λminnR − UnR −1 (cos x) 1 − ρThe condition number of H LOS may be obtained fromκM by the following relationof the maximum of the capacity curve over an extended range. However, as the aircraft comes closer
and closer to the ground terminal, the switching mustbe performed faster and faster to be in correlation withthe fluctuation of the capacity curve. Hence, such an
approach is not well suited when the aircraft comes atclose range. A remedy to this challenge is explored in
the next subsection.

κM =λmax σ2max
2= 2 = κH
,
LOS
λmin
σmin

(27)

8 The motivation for this particular approach comes from [7],where numerical results illustrate the relationship between thechannel quality and the relative positions of a transmit and receiver
node in a mm-wave MIMO architecture.

307

B. Holter, J. E. Håkegård, T. A. Myrvoll

5

250

2×2
4×2

4

6.5
200

6
Range [km]

2
1
50

100

150
R [km]

200

250

150

5.5

100

Figure 6 Condition number κ M as a function of R andnR ∈ {2, 4, 8} with θt = θr = 0o .

550where σmax and σmin denote the maximum and minimum singular values of H LOS , respectively. Hence,
when M is well-conditioned, H LOS is well-conditioned.
Using (15) and (16), the normalized correlation coefficient ρ can be expressed in terms of the condition
number asκM − 1.
(28)ρ=κM + 1

Capacity [bit/s/Hz]

3

−50

0
θr [deg]

50

Figure 7 Capacity region of a 2 × 2 system in a Ricefading channel (averaged of 1000 channel realizations) as a function of θ r and R with γ = 10dB and
K = 15dB.

From (28), it can be seen that a condition number
close to one ensures that ρ is close to zero. In Fig.6,(26) is depicted as a function of n R and R. It is observed that the range over which the condition number of M (and hence H LOS ) is close to one increases as
nR increases. Basically, this means that by increasing
the number of receive antennas, H LOS may be kept atfull rank over an extended range.

250
10.6

200

10.4

Range [km]10.2

4.2. Part 2 - Stochastic channel
In this part, the capacity region results of the previous section obtained with the deterministic channel matrix HLOS are compared to simulation results
obtained with the complete Rice fading channel matrix H. Since H is a stochastic matrix, the simulation
results in this part are obtained by averaging over anumber of channel realizations. According to [11], atypical Rice factor for the aeronautical channel in the
en-route domain is K = 15dB. Hence, all the simulations are obtained for a Rice factor of K = 15dB. In
Fig.7 and Fig.8, the capacity regions of a 2 × 2 anda 8 × 2 system are depicted, respectively. They bothdo not diﬀer much from Fig.3 and Fig.5 obtained in apure LOS channel.

150
10
100

9.8

9.6

50

9.4
−50

0
θr [deg]

50

Figure 8 Capacity region of a 8 × 2 system in a Ricefading channel (averaged of 1000 channel realizations) as a function of θ r and R with γ = 10dB and
K = 15dB.

308

Capacity [bit/s/Hz]

κM

8×2

ENRI Int. Workshop on ATM/CNS. Tokyo, Japan (EIWAC2010)

5. CONCLUSIONS

The inner product of the two channel vectors in (32)and (33) can in general by written as

The results of this paper indicate that it is possible
to exploit MIMO spatial multiplexing techniques inaeronautical communications over an extended range
even in the presence of a strong LOS component. Numerical results are presented for a MIMO ground-toair communication system with n T = 2 transmit antennas (ground terminal) and an ULA antenna withnR ≥ 2 receive antennas (aircraft). The results showthat a high rank channel matrix may be oﬀered over an
extended range (and area) when n R > nT , where therange improves as n R increases. For an n R × 2 MIMO
system with nR ≥ 2, the maximum number of spatial
data pipes oﬀered is two, and the n R receive antennas
contributes to stabilize the two data pipes in a LOS environment. Additional data pipes may be oﬀered byincreasing the number of transmit antennas but due
to the large inter-element distance needed to obtain alarge range, systems with more than two antennas in
the ground terminal may not be very practical.

=

hH
l hk

=
=

+
=

(dt cos θt )2R

dt dr cos θt cos θr(l − k)mR
dt dr cos θt cos θrα+(l − k)m,
R

=
=
=
=
=

M=

2

where
h0

=

h1

=




,

e jβr0,0 , . . . , e jβrnR −1,0e jβr0,1 , . . . , e jβrnR −1,1

n
R −1



e j 2πdt dr cos θt cos θr(l−k)mλRe jβα ·

n
R −1

e jmu

1 − e jnR u
1 − e ju
sin nR2ujβα j(nR −1) u2e e·sin u2e jβα ·

sin(nR x)
sin x
e jψ · UnR −1 (cos x),

e jψ ·(34)nR
e UnR −1 (cos x)jψ

e− jψ UnR −1 (cos x)nR, (35)

where x = nπR η .10 The eigenvalues of the matrix in(35) are derived from the characteristic equation, andthey are equal to

(29)

λ1

=

nR + UnR −1 (cos x),

(36)

λ2

=

nR − UnR −1 (cos x).

(37)

7. ACKNOWLEDGEMENTS

The individual terms of the matrix product M = H H
LOS HLOS
can then be expressed as
hH
0 h1
hH
1 h1

e jβα ·

where u = 2π dt dr cosλRθt cos θr (l − k), x = u/2 = nπR η (l − k),H∗and ψ = βα + (nR − 1)x.9 Since hH
0 h1 = (h1 h0 ) , thematrix M may be written compactly as

2

hH
0 h0
hH
1 h0

dt dr cos θt cos θr(l−k)mR

m=0

θt )where α = (l − k)d t sin θt − (l2 − k2 ) (dt cos. The
2Rchannel matrix in (2) is simplified to⎡⎤⎢⎢⎢ h1,1 h1,2 ⎥⎥⎥⎢⎢⎢⎢ h2,1 h2,2 ⎥⎥⎥⎥⎥
H = ⎢⎢⎢⎢⎢ .
(30).. ⎥⎥⎥⎥⎢⎢⎢ ..
⎥⎥⎥.
⎣⎦
hnR ,1 hnR ,2M=e jβ α+m=0

For a nR × 2 system, using [4, Eq. (7)], the diﬀerence in path length from transmit antennas l ∈ [0, 1]
and k ∈ [0, 1] to receive antenna m ∈ [0, 1, . . . , n R − 1]
can be expressed as(l − k)dt sin θt − (l2 − k2 )

n
R −1
m=0

6. APPENDIX=

e jβ(rm,k −rm,l )m=0

Eigenvalues of M for an n R × 2 MIMO systemrm,k − rm,l

n
R −1

This work has been funded by the Norwegian Research Council through the SECOMAS project(180018/S10).

(31)
T
T

9 For the last equal sign in (34), see [16, Identity 4].

,

(32)

.

(33)

for an nR × nT system, x can be defined as
x = nRπη |(l − k)|. The absolute value is introduced for simplicity
since cos x = cos(−x). Hence, if (l − k) < 0, UnR −1 (cos(−x)) =U nR −1 (cos x).
10 In general,309

B. Holter, J. E. Håkegård, T. A. Myrvoll

8. COPYRIGHT

in an indoor environment,” in Proc. First International Workshop on Multi-Gigabit MM-Wave
and Tera-Hz Wireless Systems, November 2009.

The authors confirm that they, and/or their company or institution, hold copyright of all original material included in their paper. They also confirm they
have obtained permission, from the copyright holder
of any third party material included in their paper, topublish it as part of their paper. The authors grantfull permission for the publication and distribution of
their paper as part of the EIWAC2010 proceedings or
as individual oﬀ-prints from the proceedings.

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