Satellite System Design Examples for Maximum
MIMO Spectral Efficiency in LOS Channels
A. Knopp and R.T. Schwarz
D. Ogermann, C.A. Hofmann, and B. Lankl
Fed. Office of the German Armed Forces (Bundeswehr)
Munich Univ. of the German Armed Forces (Bundeswehr)
for Information Technology, Dept. for Satellite
Institute for Communications Engineering
Communications, 56076 Koblenz, Germany
85579 Neubiberg, Germany
Email: {andreas1knopp, robert7schwarz}@bundeswehr.org Email: {andreas.knopp, r.schwarz, berthold.lankl}@unibw.de
Abstract—MIMO satellite links have recently attracted a high
interest with respect to possible link capacity enhancements. In
[1] it has been shown that especially in Line-of-Sight (LOS) satellite channels maximum multiplexing gain can be achieved via the
construction of orthogonal channels by means of the geometrical
arrangement of the ground terminal antennae in relation to the
antennae in orbit. Based on these theoretical results, we present
practically relevant design and configuration examples for satellite communication systems, involving transparent payloads for the
first time. Thus, we significantly extend the results of [1], [2] that
have been limited to regenerative payloads. The examples cover
multiple-satellite and single-satellite MIMO scenarios. The assets
and drawbacks of the applications are investigated, especially
highlighting system-inherent design uncertainties.
I. I NTRODUCTION
Due to their potentially high bandwidth efficiency [3],
multiple-input, multiple-output (MIMO) systems are very
promising especially with regard to satellite communications.
This is mainly for the reasons that on the one hand there is a
rising demand for higher data rates, while on the other hand the
usable frequency spectrum becomes short of available bandwidth, which has, therefore, become particularly expensive.
However, in MIMO satellite communications (SatCom) it is
nowadays widely believed that a high MIMO capacity gain can
only be achieved in the rich scattering multipath propagation
channel [4], but not in the LOS channel due to its high
correlation coefficients [5]. Contrarily to this impression, early
results of [6] indicated that in principle it is possible to
form a LOS channel with maximum MIMO multiplexing gain
by smart geometrical arrangements of the antenna elements
at both link ends. Despite, as typical examples reveal [4],
[5], [7], in SatCom the possibility of enhancing the MIMO
spectral efficiency via smart arrangements of the satellite and
ground terminal antennae has hardly been realized so far.
Recently, in [1] a theoretical derivation was presented that
firstly provided a design prescript for satellite links with
optimum MIMO capacity. For mathematically manageable
results, the analyses are limited to a satellite MIMO system
with two satellite antenna elements only, which have to be
placed on a single satellite each. Keeping the number of
satellites and satellite antennae low seems to be demanded
also for economical reasons. The number of ground terminal
antennae can be chosen arbitrarily. Finally, the results ended
up in tractable prescripts for the positioning of the satellites
in the geostationary orbit in relation to the placement of
the corresponding ground terminals on earth. Illustrating the
optimization procedure, in [1] the theory was applied to a
number of practical systems consisting of two satellites and
fixed or mobile receivers. Further examples were added in [2]
indicating the applicability also in satellite broadcast systems.
Extending the examples in [1] and [2] we do not limit
ourselves to communications satellites carrying a regenerative
payload, but we focus on the practically much more relevant
case of transparent satellites only working as wireless relays.
We include the cases of a single-satellite MIMO link as
well as a multiple-satellite MIMO link covering the most
important applications. Based on the design examples, the
paper also addresses assets and drawbacks of the different
system configurations, and the impact of positioning accuracy.
II. T HEORETICAL BASICS AND C ONVENTIONS
A. Characteristics of the Satellite MIMO Channel
1) Channel Model: We presume a flat-fading channel
model, only taking into account the LOS signal path as the
dominating wave propagation mechanism. This is a reasonable assumption because in SatCom multipath signals play
a secondary role. The power of the LOS signal component outvalues the power of reflections and scattered waves
[8]. Denoting the carrier frequency by fc , the frequency-flat
MIMO channel is described by its channel transfer matrix
H(fc ) = H ∈ CM ×N for a MIMO system consisting of
N transmit and M receive antennae. The element [H]mn of
the channel transfer matrix at the position mn in equivalent
baseband notation is described by the mechanism of free-space
propagation according to
2πfc
(1)
rmn
Hmn = amn · exp −j
c0
where rmn is the distance between the m-th transmitter
(Tx) antenna and the n-th receiver (Rx) antenna. c0 is the
speed of light in free space. amn is the complex envelope
that is calculated by: amn = c0 · ejϑ0 /(4πfc rmn ) with ϑ0
marking the carrier phase angle at the time of observation.
Observing the channel path gain for all M × N pairs of TxRx antenna combinations, they are found to be approximately
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parameter
EIRPu
(G−T )u
Bu
Lu
EIRPd
(G−T )d
Bd
Ld
value
69.3dBW
2.4dBK
5MHz
209.3dB
35.4dBW
16.4dBK
1MHz
207dB
explanation
uplink effective isotropic radiated power
logarithm. ratio of uplink fig. of merit [8]
transmission bandwidth for uplink
path loss uplink, carrier fc,u = 18GHz
downlink effective isotropic radiated power
logarithm. ratio of downlink fig. of merit [8]
transmission bandwidth for downlink
path loss downlink, carrier fc,d = 14GHz
TABLE I
MIMO CAPACITY CALCULATION
PARAMETERS FOR THE EXEMPLARY
identical due to the large distance between Tx and Rx, i.e.
|amn | ≈ const. ∀ m, n with |.| denoting the absolute value. A
further, important property of the LOS MIMO channel is its
high spatial correlation. As a novelty compared to previous
publications (e.g. [5]), this correlation is used to construct
high-capacity channel transfer matrices.
2) Assumptions for the System Parameters: In order to
illustrate the capacity gains possible with SatCom MIMO
systems, we have to define a set of characteristic figures
describing the satellite link, the link budget, and the signal-tonoise ratio (SNR) for the MIMO capacity calculation.
Table I summarizes the characteristic figures for the uplink
and downlink channels. In general, the transmission channel
from the transmitting ground terminal to the satellite is denoted
by uplink channel (formula index u) and the corresponding
propagation path from the satellite down to the Rx ground
station is denoted by downlink channel (formula index d).
Using the figures of table I, for the SNRu at the satellite and
the SNRd at the Rx on earth ground we obtain
SNRu = EIRPu + (G−T )u − Lu − K − Bu = 24dB, (2)
SNRd = EIRPd + (G−T )d − Ld − K − Bd = 13.4dB, (3)
respectively, where K = 10 log10 (k) is the logarithmic
value of the Boltzmann constant k. Analogously, Bu/d =
10 log10 (Bu/d ) denote the logarithmic values of the up- and
downlink bandwidth. This way, the transmit and receive antennae are not counted among the propagation channel, they
are part of the Tx and the Rx, respectively. Furthermore, it has
to be pointed out that in a MIMO system the SNR calculated
above is measured at each combination of Tx and Rx antennae
that is possible, i.e. each antenna radiates the full EIRPu or
EIRPd as it is connected to an own amplifier.
B. MIMO Channel Capacity
Basically, we have to distinguish between a regenerative
payload, where at satellite level the signal is completely
regenerated in the baseband, and a satellite with transparent
payload acting as an amplifying relay while converting the
signal directly from the uplink to the downlink carrier.
1) Regenerative Payload: If the satellites are endowed with
a regenerative payload, we treat the uplink and the downlink as
two different end-to-end communication links. Consequently,
the uplink and the downlink channels provide different channel
capacities, where the smaller capacity determines the overall
data rate of the system. In the light of the MIMO capacity
optimization, it is most important that both, the uplink and
the downlink, have to be optimized separately, which is done
by means of the respective placement of the antennae [1].
For a MIMO channel without a relay, for a frequencyflat transmission channel the time invariant MIMO spectral
efficiency without channel knowledge at the transmitter is
calculated according to Telatar’s well-known eq. [3]
(4)
C = log2 det I M + ρ · HH H .
Here, the transmit symbols are realizations of uncorrelated
independent, identically distributed (i.i.d.) Gaussian random
variables. Furthermore, {.}H denotes the complex conjugate
transpose. In this equation, the channel path loss for each
mn-th combination of the m-th out of M receive antennae
and the n-th out of N transmit antennae is included in the
corresponding entry [H]mn of the channel transfer matrix.
Thus, ρ is the (linear) ratio of the transmit power at each
transmit antenna and the noise power at each receive antenna.
With eq. (2) and (3) it is calculated (up-/downlink)
ρu = 10(SNRu +Lu )/10 , and ρd = 10(SNRd +Ld )/10 .
(5)
As stated in the introduction, in this paper we aim at a
maximum multiplexing gain of each MIMO channel described
by the channel matrix H. In the following, w.l.o.g. we use the
uplink channel H u to recapitulate the optimization criterion
according to [1]. The procedure for the downlink is identical.
Presuming M antenna elements in the geostationary orbit and
N ground terminal antennae, maximum multiplexing gain is
achieved if the eigenvalues γi , i ∈ {1, ..., min(M, N )} of the
matrix Q are given equally by γi = |a|2 · max(M, N ), where
H
Q = H uH H
u for M < N, Q = H u H u for M ≥ N (6)
In this case we obtain an optimum eigenvalue profile and an
orthogonal channel transfer matrix. Then the channel capacity
also obtains its optimum value which is given by
Cu,opt = min(M, N ) · log2 1 + ρu |a|2 max(M, N ) . (7)
2) Transparent Payload: If we are dealing with a transparent payload instead, the channel comprises both, the uplink as
well as the downlink. Furthermore, the satellite payload has
to be incorporated as a relay that introduces an amplification
and a phase shift to the signal. In the following, we presume
N transmit antennae for the uplink ground station, M receive
antennae and M transmit antennae at the satellite and, finally,
Z receive antennae at the downlink ground terminal. In this
case, the MIMO channel capacity for given channel transfer
matrices H u ∈ CM ×N and H d ∈ CZ×M is calculated
according to [9]
H −1
(8)
C = log2 det I M + ρu H u H H
−
ρ
H
H
S
u
u
u
u
with the matrix S given by
S = I M + σu2 /σd2 · F H H H
d H dF .
(9)
The Matrix F ∈ CM ×M describes the transfer matrix of the
relay, i.e. the satellite payload, and σu2 and σd2 denote the noise
power at the receiver in the up- and downlink, respectively. For
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ρd > 0 and any H u , H d , it becomes optimal with respect to
the overall channel capacity if we choose [9]
parameter
RS
F = V d ΛF U H
u.
RE
θT
φT
θS,m
(10)
V d and U u are unitary matrices that result from the eigenvalue
decompositions of the down- and uplink MIMO channels, i.e.
H
H
H
H dH H
d = V d Λd V d and H u H u = U u Λu U u . Finally, ΛF
is a diagonal matrix that describes the power allocation within
the relay. Hence, the matrix F can be considered as a matched
filter along the singular vectors of the channel matrices [9].
Now, a special case is obtained if the uplink and downlink
channel transfer matrices H u and H d are both optimized for
optimal capacity through the geometrical arrangement of the
antennae according to [1]. This way, the matrices V d and
U u become identity matrices and, consequentially, the MIMO
relay in orbit obtains a very simple structure: Its transfer
matrix F is solely formed by the diagonal matrix ΛF , which
means that there are M independent propagation paths through
the satellite payload without any cross-couplings. A second,
very important advantage for the system design, which only
applies to optimized links with orthogonal channel transfer
matrices H d and H u , is revealed if the entries of F = ΛF
are further analyzed. As the matrix entry [F ]mm describes
the transfer function of the m-th MIMO branch through the
satellite payload, it consists of a power amplification and
a phase shift. Normally, the satellite will contain travelling
wave tube amplifiers (TWTAs) or solid state power amplifiers
(SSPAs) that are driven in saturation. Thus, we presume
|[F ]mm | ≈ const ∀ m ∈ {1, ..., M } with good accuracy.
Contrarily, each phase angle arg{[F ]mm } will usually be
different for the M signal paths because each path converts
the carrier frequency from uplink to downlink separately. Even
if the same local oscillator is used for all paths, there will
be a non-constant phase offset between them. Only in case
of orthogonal channels the matrix S remains optimal with
respect to the channel capacity, independently from the phase
angles arg{[F ]mm }. Of course F can be optimized also for
non-orthogonal channels H d and H u , but in this case also
the phase shifts become crucial and must be optimized. As a
consequence, keeping them constant over time is a very hard
requirement for the hardware used. If the antennae in orbit are
mounted on different satellites, constant phase shifts cannot be
guaranteed at all. Contrarily, this requirement can be omitted
completely for orthogonal channels.
C. Capacity Optimization for Practical Satellite Scenarios
The strategy for the MIMO capacity optimization in SatCom
scenarios was derived and thoroughly explained in [1] and [2].
Thus, at this point we only recapitulate the most important
results, for a deeper insight into the mathematical derivations
we particularly recommend a study of [1].
As we have stated before, we focus on a satellite scenario
consisting of only 2 satellite antennae that are either mounted
both on the same satellite or on two different satellites. Hence,
we deal with an 2 × N system for the uplink and a Z × 2
system for the downlink with N and Z arbitrary. In the
δT
dT
explanation
radius of satellite orbit, measured from earth center
(42.164km)
mean earth radius (6.378km)
geographical latitude of ground terminal (ULA center)
geographical longitude of ground terminal (ULA center)
geographical latitude of m-th satellite antenna, convention for nomenclature: θS,1 < θS,2
tilting angle of earth terminal antenna ULA, measured
relative to the connecting line of the satellite antennae, i.e.
parallel satellite and ground antennae arrays for δT = 0
antenna element separation within earth terminal ULA
TABLE II
FORMULA SYMBOLS FOR THE SCENARIO DESCRIPTION
previous section it was shown that in all cases, transparent
and regenerative satellite payloads, we have to form both
an orthogonal uplink as well as an orthogonal downlink
channel. Thus, H u and H d must be optimized separately
via the antenna arrangement to fulfill a particular, geometrical
condition that is outlined exemplarily for the downlink H d
subsequently, for the uplink the result is valid accordingly. In
[1] it is basically shown that an orthogonal MIMO channel is
obtained for M = 2 if the very general condition
c0
,
rk1 − rk2 + rl2 − rl1 = v · (k − l)
Z · fc
(11)
k, l = 1 . . . Z, v ∈ Z, v Z
is fulfilled. Here, rμν , μ, ν ∈ Z basically denotes the geometrical distance between the ν-th satellite antenna and the
μ-th ground terminal antenna and v Z means that v is
indivisible by Z. The parameter v ∈ Z is a constant that
regards the periodicity of the capacity optima in multiples
of the wavelength. Equation (11) shows that the antennae on
earth ground and on the satellite platform have to be arranged
accordingly to ensure very distinct LOS path lengths. Now,
the positions of the antennae have to be formulated appropriately to calculate all distances rμν . As usual in SatCom, the
principle of the Mercator projection [8] is used to describe the
positions of the ground terminals and the satellites. Applying
the nomenclature1 provided in table II and the substitutions
s2m =
cm =
2
RS2 + RE
− 2RS RE · cos φT cos (θT − θS,m ) (12)
2RS · [cos δT sin (θT − θS,m ) +
(13)
sin δT cos (θT − θS,m ) sin φT ] ,
in [1] the practical optimization criterion for M = 2 satellite
antennae for the downlink is re-formulated as
c1
c0
c2
= v·
−
, v ∈ Z, v Z.
(14)
dT
2s1
2s2
Zfc
Here it is presumed that the ground terminal antennae are
arranged in form of a uniform linear array (ULA). They can
either be mounted in close separation on a single terminal
or on separate terminals each. In the latter case, the ULA is
formed by Z terminals, but eq. (14) can still be applied (see
again [1], [2] for a more detailed discussion of eq. (14)).
1 A thorough graphical illustration of all parameters has already been provided in [1], [2]. Studying this figure is recommended to facilitate readability.
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III. E XAMPLES FOR P RACTICAL MIMO A PPLICATIONS
For all of the scenarios presented in the following, we
exemplarily assume the subsequent ground terminal positions:
Tx position θT,u φT,u Rx position θT,d φT,d
Berlin
13◦
52◦
Munich
11◦
48◦
A. Multiple Satellites carrying Regenerative Payloads
This first type of application is mentioned here for completeness only. It was extensively studied already in [1], [2].
B. Single Satellite carrying Transparent Payload
A MIMO system requiring a single satellite only, is very
desirable in practice for economical reasons. An illustration of
the scenario involving all the necessary symbols is provided
in fig. 1. The most essential finding when optimizing the
2 Besides the universal notations d and δ describing either the up- or the
T
T
downlink or both, we additionally use the indices u and d to distinguish intentionally between up- and downlink if necessary, i.e. dT,u , dT,d , δT,u , δT,d .
dS,d = dS,u
θS,2
θS,1
MIMO Rx
M =2
MIMO Tx
Hd
F
Hu
terrestrial
network
MIMO AFE
(uplink)
dT,u
N =2
Fig. 1.
capacity degradation C/Copt [%]
D. Sources of Capacity Degradations in Practical Systems
In advance of the analyses in III, we briefly summarize the
most probable reasons for a capacity degradation encountered
in a practical MIMO system that is based on our optimization.
1) The Satellite Station Keeping Box: As it is in practice
impossible to maintain the satellite absolutely immobile at
its defined position a station keeping box is defined, which
represents the maximum permitted values of the satellite excursions. For a capacity-optimized system it is, thus, desirable
to guarantee the optimum capacity as long as the satellite
stays within this virtual box. Subsequently, we assume typical
box dimensions, which are ±37.5km in longitude and latitude
and ±17.5km for the eccentricity [8]. We have investigated
the effect of satellite drifts in all three dimensions finding
significant effects in cases of changes in longitude and latitude
with respect to the nominal satellite position. Instead, the
eccentricity remains of less relevance for the channel capacity.
2) Miss-Positioning due to Ground Terminal Movement:
Naturally, each change of the ground terminal position immediately changes the phase angle relations of the entries
within the MIMO channel transfer matrix and, thus, the
MIMO channel capacity. In this context the antenna spacing
dT is incorporated as well as the tilting angle2 δT because
both parameters have an interdependence with regard to the
capacity optimization criterion according to eq. (14). Even in
a scenario with static ground terminals or anchor stations there
are environmental conditions, e.g. wind, that slightly move the
antennae. It must be remembered that even minor changes of
the antenna positions can significantly impact on the signal
phase angles at the Rx due to the very short wavelength in
SatCom. Hence, the effect of terminal movement has to be
studied depending upon the addressed application.
3) Delay of Signals: Especially multiple-satellite MIMO
systems exhibit high delay differences of the signal portions
arriving from different satellites at the ground Rx or vice versa.
Countermeasures for this phenomenon are not tackled in the
course of the subsequent capacity analyses. However, there are
system design approaches available in the literature, e.g. [5].
terrestrial
network
MIMO AFE
(downlink)
dT,d
Z=2
AFE: antenna frontend
single satellite and dislocated ground terminal antennae
miss-pos. Tx
miss-pos. Rx
miss-pos. Tx,Rx
δT,u = δT,d = 0
θS,1 = 11◦ 59 59.9755
θS,2 = 12◦ 0 0.0245
miss-positioning ΔdT = dT − dT,opt [km]
Fig. 2. MIMO channel capacity as a function of the antenna spacing on
earth ground, example for dS,u = dS,d = 10m, N = M = Z = 2, v = 1
capacity of such a MIMO system is a straightforward result
of eq. (14): Because of the comparatively small antenna
spacing dS,u and dS,d at satellite level, the ground station
antenna spacing dT,u and dT,d has to be chosen large. If
for example a spacing of dS,u = dS,d = 10m is used at
satellite level, which already marks a very large spacing,
the minimum antenna element spacings at ground level are
calculated dT,u,opt (v = 1) = 32.1km for the uplink and
dT,d,opt (v = 1) = 40.9km for the downlink if the remaining
degrees of freedom are chosen as stated in the caption of fig.
2. One has to note, that dT,opt (v = 1) even further increases
for decreasing dS . This scenario is applicable in practice for
example if a number of anchor stations is available at several
locations of a particular territory. Such a system design is
desirable in military applications because a set of dislocated
ground stations increases the overall link availability as the
probability of simultaneous attacks is lower than for compact
conglomerates. A terrestrial backbone network is needed for
the connection of the antenna front ends to a common Tx
/ Rx. Otherwise, this application is very advantageous with
respect to the impact of a miss-positioning of the antennae
on earth ground as well as the station keeping maneuvers of
the satellite. As it can be observed from fig. 2 again, even
a miss-positioning of ΔdT = dT − dT,opt in the range of
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C. Multiple Satellites carrying Transparent Payloads
In most practical cases very compact ground terminals
are required. This means that the MIMO antenna elements
have to be placed close to each other. As a straightforward
result of eq. (14) the MIMO antennae on satellite level now
have to be displaced very far in order to obtain the optimal
MIMO spectral efficiency. Consequentially, a single satellite
platform is no longer sufficient. Instead, the M = 2 antennae
have to be placed on two separate platforms as illustrated
in fig. 3. Because it is not possible to illuminate different
ΔθS
M =2
θS2
θS1
F
MIMO receiver
Hd
MIMO transmitter
Hu
N =2
dT,d
MIMO AFE
(uplink)
dT,u
Fig. 3.
MIMO AFE
(downlink)
Z=3
AFE: antenna frontend
two satellites and compact ground terminal antennae
satellites simultaneously with a typical base station antenna
(the satellites are presumed to be separated ΔθS ≥ 0.5◦ in
latitude), we assume two parallel single-input, single-output
(SISO) uplinks to the satellites. These uplinks have to be
designed such that the available data rate is sufficiently high to
support the MIMO downlink. The scenario was also proposed
in [5], but not optimized for the channel capacity. Again, we
calculate the channel capacity according to eq. (8) because
we still assume transparent payloads. The uplink channel
transfer matrix H u in eq. (8) has nonzero values only on
its main diagonal, but the downlink channel transfer matrix
H d describes a real MIMO channel that is optimized with
respect to the multiplexing gain. At this point it is useful
to further discuss the interdependence of the free parameters
in eq. (14), which builds the basis for the construction of
orthogonal channels. Those are mainly the antenna spacing
dT,d at the ground station, the angular separation of the
satellites ΔθS = |θS,1 − θS,2 |, and the tilting angle δT,d that
provides the rotational shift of the ground antenna array in
comparison to the antenna array in orbit (see also table II).
fc,d = 14GHz, M = Z = 2, v = 1
satellite displacement
dT,d,opt [m]
kilometers only slightly degrades the capacity for a satellite
with transparent payload according to eq. (8). The satellite
station keeping has almost no impact. This effect becomes
severe in case of multiple satellites only. Further crucial
parameters, instead, remain the tilting angles δT,u and δT,v
of the ground terminal antenna arrays. In our example, those
are optimally matched to the antenna spacing in fig. 2. The
impact of deviations in δT from the optimum is highlighted
in the context of the next subsection.
ΔθS = 0.5◦
ΔθS = 1◦
ΔθS = 2◦
ΔθS = 5◦
tilting angle δT,d [◦ ]
Fig. 4. interdependence of degrees of freedom for the capacity optimization
The curves in fig. 4 show possible combinations of the four
parameters that lead to an orthogonal channel matrix H d for
the downlink. The optimal values dT,d,opt are calculated for
the case v = 1 and, thus, larger but still optimal values dT,d
can be calculated by choosing v > 1, v ∈ Z, v Z. Hence,
we observe three major effects:3
a. a larger satellite displacement requires a smaller antenna
spacing of the terminal on earth
b. an increased rotational shift of the antenna arrays on earth
and in orbit requires an increased antenna spacing of either
the ground terminal or the satellite antennae4
c. an optimal antenna spacing on earth ground occurs with a
periodicity in v (see also the discussions in [1], [2])
Unfortunately, in practice the choice of the actual parameter
combination is limited by realizability. It further determines
if the system will be resistant against capacity degradations
caused by one of the error sources mentioned in subsection
II-D. An increased tilting angle δT,d , for example, requires
a more accurate positioning of the antenna array. This is
observed from fig. 4 again, because the slope of the curves
increases for higher values of δT,d . Also the accuracy of dT,d
is important. If the capacity is drawn as a function of the
miss-positioning compared to the optimum antenna spacing,
a very similar curve as provided in fig. 2 will be obtained.
But as a crucial difference to the single-satellite case, in the
current scenario the ground terminal antenna spacing must be
adjusted with a typical accuracy in the range of few meters (no
longer kilometers) to avoid severe capacity degradations. In
general, if two satellites are used it can be summarized that the
farther the satellites are displaced, the more exactly the ground
terminal antennae have to be separated by the theoretically
optimal value of dT,d and vice versa. This fact was already
discussed and proven by simulations in [1], [2] in more detail.
3 Of course, if available, for a MIMO uplink involving two separate
satellites, these effects are found identically (see also examples in [1], [2]).
4 Basically, for any configuration there always exists a value δ
T,d for which
no optimal solution is possible because Hd generally becomes rank-deficient
and the left hand side of eq. (14) turns to zero. Then, dT,d,opt = ∞ would
be required theoretically. In the current example this occurs for δT,d = 90◦ .
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C [Bit/s/Hz]
0.9Copt = 11.6Bit/s
0.9Copt = 10.9Bit/s
0.9Copt = 9.5Bit/s
δT,d = 40◦
θS,1 = −1◦
θS,2 = 1◦
} Δθ
S
= 2◦
v · dT,d,opt [m]
practically feasible while choosing v as well as δT as small
as possible for the construction of orthogonal channels
3. higher numbers of ground terminal antennae are advantageous for stable channel capacities and high receive power
To finally provide a precise example for a possible system
design, we could choose the following set of parameters:
N = M = 2, Z = 4, δT,d = 40◦ , ΔθS = 1◦ ({θS,1 , θS2 } =
{−0.5◦ , 0.5◦ }), v = 10 ⇒ dT,d,opt (v = 10) = 4.31m. In
order to guarantee that the MIMO channel capacity degrades
less than 5% down from its optimum, we would have to adjust
dT,d with an accuracy of dT,d,opt (v = 10) ± 0.72m while
preserving a deviation of less than ±5◦ for δT,d compared
to its specified value. Such an accuracy can actually be
maintained in practice.
IV. C ONCLUSION
Fig. 5. worst-case capacity degradation as a function of the antenna spacing
on earth, two satellites performing station keeping maneuvers are considered
Finally, the mobility of the satellites caused by their independent movements within the station keeping box was
examined. It turned out that the capacity degradation, which
is encountered due to varying phase angles of the receive
signals, is strongly linked to the actual antenna spacing
dT,d = v · dT,d,opt (v = 1). Therefore, the stability of the
channel capacity depends upon the choice of v. This fact is
illustrated by fig. 5, which shows the channel capacity as a
function of the antenna spacing of the ground terminal. The
abscissa comprises discrete values that are obtained by increasing v ∈ Z. Each ordinate value C(v · dT,d,opt ) represents
the worst-case capacity value that is found if the channel
capacity is calculated for all possible pairs of latitude positions
{θS,1 , θS,2 }, which can occur due to the satellite movements.
Of course, only those latitude values falling within the station
keeping boxes of the satellites have been taken into account.
A zero but constant inclination and eccentricity was presumed
for each satellite because both parameters have a negligible
impact on the channel capacity for the actual setup. Especially
a varying inclination hardly impacts on the channel capacity
because the tilting angle was chosen δT,d = 0 and, thus, the
phase angle relations within H d are widely independent of
the inclination. Contrarily, for nonzero δT,d the inclination
becomes increasingly relevant. From fig. 5 we finally observe
that the station keeping maneuvers do not cause a significant
capacity degradation as long as the number of receive antenna
elements is high and, most important, the values of parameter v
remain low. In other words, if a large antenna spacing is chosen
on earth ground, the system becomes sensitive with respect to
even slight changes of the satellite positions. For high v this
even means that the capacity probably degrades down to its
minimum (keyhole) capacity, as the curves indicate.
Based on our investigations, the following design hints for a
system with optimum MIMO capacity can be provided, which
at the same time is resistant against degradations due to misspositioning of both, the satellites and the ground terminals:
1. the satellite displacement should usually be chosen small
2. the antenna spacing of the ground terminal should be
We optimized the MIMO channel capacity for satellite
systems that utilize transparent communication payloads. For
that purpose, in a first step it was necessary to exploit the
channel correlation to construct orthogonal and, thus, optimal
MIMO channel transfer matrices for both, the up- and the
downlink channel of the satellite link. Extending results of [1],
such orthogonal channels have been realized by an adequate
positioning of all transmitter and receiver antenna elements.
Secondly, the optimum transfer function for the MIMO relay
in the geostationary orbit had to be chosen. Basically, this
was achieved applying the very general, mathematical results
of [9]. In the course of this, we have shown that only
for orthogonal MIMO channels the system design for the
satellite payload becomes practically feasible without causing
capacity degradations due to phase uncertainties within the
MIMO system branches. This further outlines the importance
of orthogonal MIMO channels. In practice, for the construction
of capacity-optimal MIMO channel transfer matrices, the user
is provided with several degrees of freedom for the system
design. The interdependence and optimization of the most
important design parameters was illustrated, also highlighting
practical limitations and capacity-degrading error sources.
R EFERENCES
[1] R.T. Schwarz et al., Optimum-Capacity MIMO Satellite Link for Fixed
and Mobile Services, Proc. 2008 ITG Workshop on Smart Antennas
(WSA’08), Feb. 2008, pp. 209-216.
[2] R.T. Schwarz et al., Optimum-Capacity MIMO Satellite Broadcast
System: Conceptual Design for LOS Channels, Proc. Advanced Satellite
Mobile Systems Conference (ASMS’08), Aug. 2008.
[3] E. Telatar et al., Capacity of multi-antenna gaussian channels. AT&TBell Technical Memorandum, 1995.
[4] G.E. Corazza (ed.), Digital Satellite Communications, Springer Series
on Information Technology, 2007.
[5] F. Yamashita et al., Broadband multiple satellite MIMO system. Proc.
VTC-2005-Fall, pp. 2632-2636.
[6] P.F. Driessen and G.J. Foschini, On the Capacity Formula for Multiple
Input-Multiple Output Wireless Channels: A Geometric Interpretation,
IEEE Trans. on Communications, vol. 47, No. 2, Feb. 1999, pp. 173-176.
[7] Cheon-In Oh et al., Analysis of the Rain fading channel and the system
applying MIMO. Proc. ISCIT 2006, pp. 507-510.
[8] G. Maral et al., Satellite Communications Systems. Wiley&Sons, 2006.
[9] X. Tang and Y. Hua, Optimal Design of Non-Regenerative MIMO
Wireless Relays, IEEE Transactions on Wireless Communications, vol.
6, no. 4, Apr. 2007, pp. 1398-1407.
978-1-4244-2324-8/08/$25.00 © 2008 IEEE.
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.
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