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On the capacity of a SIMO land mobile satellite system at Cband: polarized and depolarized received field
 Nektarios Moraitis^{1}Email author,
 Péter Horváth^{2},
 Philip Constantinou^{1} and
 István Frigyes^{2}
https://doi.org/10.1186/168714992012204
© Moraitis et al.; licensee Springer. 2012
 Received: 30 November 2011
 Accepted: 14 May 2012
 Published: 29 June 2012
Abstract
Land mobile satellite can exploit multiple input multiple output techniques to achieve high transmission rates. This article evaluates, theoretically, the capacity of the single input multiple output system utilizing uniform linear arrays at the receiver terminal for satellite applications. The theoretical study is performed at Cband and accounts for different shadowing conditions. Additionally, polarization effects are introduced and capacity results are presented that take into account the depolarization. For this investigation, a model for the scattering caused depolarization based on Stokes parameters is applied. Decrease of channel capacity is determined for some special cases both for Rayleigh fading and for the ULA with different number of receive antennas.
Keywords
 SIMO satellite
 Channel model
 Downlink capacity, Uniform linear arrays
 Depolarization modeling, Depolarized field
Introduction
Multiple antenna wireless systems, and particularly multiple input multiple output (MIMO) systems, yield unprecedented possibilities of innovation in wireless communications. While in principle MIMO advantages are achievable both with free space channels (e.g. [1, 2]) and multipath channels, practical reasons prefer the latter. As a consequence satellite links are not well suited for MIMO applications: the pathlength is extremely long, propagation along most of the path is free space and antennas are—nearly always—of very narrow beam. Furthermore, it is shown in the literature that the other forms of diversity (mainly satellite diversity, where two satellites orbiting far from each other serve as diversity terminals) cause severe intersymbol interference and raise synchronization issues [2]. The solution of these problems is not simple at all and details are not yet clear. Possibilities of polarization diversity, on the other hand, are more restricted than those of, e.g. space or frequency diversity. Having taken this into account, it seems reasonable to investigate what advantages (if any) of a true MIMO system can be achieved with architectures appropriate in satellite systems, these being more conservative than MIMO architecture, i.e. single input multiple output (SIMO) in the downlink. In particular, if channel capacity can significantly be increased by the application of multiantenna satellite systems. The problem is related to MIMO studies as the question itself and concepts and methods applied have existed since the advent of MIMO in the mid 1990s. There are few articles dealing with the MIMO satellite topic. For example, King et al. [3] give a physicalstatistical model and compute the capacity of a 2 × 2 MIMO system. Further articles involved with MIMO satellite measurements are [4–6], whereas [7] investigates the modelling of the satellite MIMO channel emphasizing on polarization.
The aim of this article is to achieve a step on this path. A satellite downlink is investigated and our goal is to determine the channel capacity. The investigated system is SIMO, i.e. there is one transmit antenna onboard the satellite and a vertically polarized uniform linear array (ULA) receive antenna at the receiver terminal. Although it is known from theory that this structure yields only logarithmic increase of capacity versus the element number of the antenna, significant shift in signaltonoise ratio (SNR) can be possible with appropriate environment and design. The number of applications using global navigation satellite system positioning is increasing steadily and currently the European Space Agency explores the possibility of satellite navigation signals operating in an already allocated frequency band for satellite radio navigation around 5 GHz [8]. For that reason, the study is performed at Cband (5.2 GHz), for a light and heavy shadowed environment.
Depolarization can change channel characteristics, including capacity (usually neglected in singlepolarized situations). Therefore, the second step is to examine the channel capacity introducing SIMO depolarization scenarios and compare the difference with the polarized state. The problem of depolarization is investigated in a more general framework. In that, usual channel models—statistical like Rayleigh, Rice, CorazzaVatalaro, etc., or physical, like ray tracing, fullwave electromagnetic models (or that used in this article for polarized SIMO)—are regarded as conditional models based on the loss due to polarization mismatch of the receive antenna. In order to determine statistics of the condition, a model described in [9], based on Stokesparameters, is proposed. To give some general insight into the role of depolarization the model is also applied to various single input single output (SISO) and SIMO Rayleighfading situations. In several cases closedform results were obtained and verified by simulation providing ergodic capacity results.
The remainder of the article is organized as follows. In Section 2 we describe the propagation scenario and geometry, the channel model and the capacity calculation methodology. Section 3 presents the outage capacity results for the polarized state of the channel. In Section 4, unconditional statistics of representative channel models with representative depolarization models are determined, and, based on that, ergodic capacity of some SISO and SIMO situations is calculated. Finally, Section 5 is devoted to conclusions summarizing this study.
Capacity evaluation methodology
Propagation scenario
where $\tilde{b}$ is the timevarying angulardependent complex envelope, θ_{ 0 } is the angle at which the direct (lineofsight) component arrives from the satellite to the mobile terminal, ${\tilde{P}}_{0}\left(t\right)$ is the timevarying received complex envelope of the direct component, ${\tilde{P}}_{i}(t,{\theta}_{i})$ is the timevarying received complex envelope of each multipath component initiated by the i^{th} scatterer at specific angle θ_{ i } around the array, L is the total number of the scatterers and K is the ratio between the direct and the multipath components ( Kfactor).
The lower part in Figure 2 is generating the timevarying complex envelope for each i^{th} scatterer around the array (multipath components). The Rayleigh distributed series are spectrally shaped and multiplied by a slowly varying lognormal series thus modulating the mode of the Rayleigh series. The specific circuit implements the Suzuki distributed timeseries [10]. Fast variations are ruled by Doppler spread mainly due to the terminal’s motion. The Doppler spread is envisaged as a Butterworth filter being a more realistic approach for satellite cases. The Doppler filters would be narrower than when the overall multipath is simulated. Therefore, the Doppler spread for each one of the scatterers is f_{ i } = (V/ λ)cos( θ_{ i })/L where θ_{ i } is the angle between each i^{th} scatterer and the direction of the receiver, whereas L is the total number of the scatterers.
Taking into account a GEO satellite (fixed in the sky) we assume that the Doppler shift caused by the satellite motion is zero. For the rest of the simulation procedure, we selected ϕ = 0° (since the mobile is moving along the xaxis as shown in Figure 1) with a mobile terminal speed of 30 km/h. The sampling frequency was chosen 16 times the Doppler spread each time. In the proposed model, in case of a light shadowing scenario, we select M = 1.13 dB, Σ = 1 dB and K = 6 dB, whereas in heavy shadowing scenario we select M = −9.38 dB, Σ = 2.5 dB and K = −2.22 dB, respectively [11].
where ρ is the averaged SNR at each receiver branch. The capacity is referred as the error free spectral efficiency, or the data rate per unit bandwidth that can be sustained reliably over the channel.
Capacity evaluation procedure

Initially the frequency (in GHz), the shadowing conditions (mean power M and the standard deviation S in dB), the Kfactor (in dB), the velocity of the terminal (in km/h), the element spacing ? l and the SNR are given.

Then, the angular spread a (in radians), as well as the scatter annulus with radius S_{ R } are determined. For example if a?=?2p then the number L of the multipath components within the sector would be 100.

The elevation angle between the mobile terminal and the satellite is assigned and for each angle in the interval [?_{0} – p, ?_{0}?+?p] we calculate the timevarying complex envelope of the received signal for the direct as well as for each i^{th} multipath component according to the model in Figure 2.

According to the given elevation angle the direct component is multiplied by the factor $\sqrt{K/(K+1)}$, and each of the i^{th} multipath components by the factor $\sqrt{1/(K+1)}$ as shown by (1).

Hence, a timevarying vector $\tilde{\mathbf{b}}\left(t\right)$ is created, which contains the received complex envelope as a function of angle of arrival, for each one of the L propagation components (direct and multipath).

Then, the steering vector of the receiver antenna is calculated according to (3), and from (4) the timevarying channel vector $\tilde{\mathbf{h}}\left(t\right)$ is extracted. The direct component is in position ${\tilde{h}}_{1}\left(t\right)$.

Finally, we evaluate the outage channel capacity using (5), from which the cumulative distribution function (CDF) is extracted.
For the rest of the simulation procedure, we select the space element Δl = λ so as to have a compact receiver antenna, and for the capacity calculation we have assume an SNR of 10 dB.
Outage capacity results with polarized received field
Taking depolarization into account
Polarization effects and depolarization modelling
Multipath propagation caused by scattering along the propagation path may cause random changes in the polarization state, which is also termed depolarization of the received field. This is a result of the electromagnetic investigation of wave propagation in a random, general scattering medium (see, e.g. [13], the classical work of propagation in random media). Similar results are also found in a rigorous investigation using fullwave electromagnetic simulation in a particular environment with wellspecified scattering [14]. Experimental results are also available. In [15], satellitetoindoor propagation measurements are described; received field was measured with antennas of different polarization while transmitted field was elliptically polarized. Measured power was close to identical with receive antennas of co and counterrotating circular polarization and of arbitrary linear polarization. Nabar et al. [16] report on (terrestrial) measurements resulting in equal received power in the average of copolar and crosspolar transmitted signals if the link length is 1.6 km or more. These and other results verify that the field at the receive antenna input port is depolarized. What is of interest is the power loss caused by polarization mismatch due to depolarization of the received field. For determining this loss, we have to take into account that (i) one particular realization of the received field at the antenna has one particular polarization being responsible for the possible polarization mismatchloss; and (ii) this polarization is random.
The polarization state of an electromagnetic wave can be well described in the threedimensional space of Stokes parameters. Stokes parameters and Stokes space were introduced by G. G. Stokes in the mid1800 s as an appropriate mathematical tool for the description of polarization characteristics of electromagnetic radiation. It has been widely applied since then in optics; it is less frequently used in lowerthanoptical frequency electromagnetic theory. The main advantage of applying these concepts here is the greater insight gained by using them. While polarization mismatchloss can be determined by more direct methods, this insight, based on symmetry properties of Stokes space, leads to reasonable statistics. It would be difficult to arrive at these without this framework. A detailed description of the concept is given in [17]. Results of [17] applied in what follows are listed below. For the sake of simplicity it is assumed that polarization discrimination of the receive antenna is infinite, i.e. the transmission factor amounts to T_{ p } = 0 if the polarization of the incoming wave is orthogonal to that of the receive antenna.
where γ_{ i } is a real angle characteristic of the “distance” between the antenna polarization state and that of the incoming wave. Polarization states of waves of equal power lay in the Stokes space on the surface of a sphere called the Poincaré sphere. This definition of Cartesian coordinates in the Stokes space is described amongst others in [17]. The relationship between spherical coordinates (R γ φ) of a point in Stokes space and the physical characteristics of the (polarized) electrical field ( E_{ x }E_{ y }δ) are as follows $R={E}_{x}^{2}+{E}_{y}^{2}$$\gamma =arcsin\left(\frac{2{E}_{x}{E}_{y}sin\delta}{{E}_{x}^{2}+{E}_{y}^{2}}\right)$$\phi =arctan\left(\frac{2{E}_{x}{E}_{y}cos\delta}{{E}_{x}^{2}{E}_{y}^{2}}\right)$with E_{ x }E_{ y } and δ being the x and y components of the electric field and the phase angle difference between these two, respectively, γ is the elevation angle of any point in the Stokes space and φ is its azimuthal angle. In (7) (with some simplification), the elevation angle of the antenna polarization state is taken as π/2 and γ_{ i } is the elevation angle of the incoming wave.
If all polarization states are equally likely γ_{0} = −π/2 and the distribution is uniform between 0 and 1.
Probability density of the received field—Rayleigh fading
The effect of depolarization on the channel capacity—the Rayleigh fading SISO case
SIMO channel capacity—i.I.D. Depolarized Rayleigh fading
SIMO channel capacity—taking depolarization into account in situations of sections 2 and 3
Conclusions
This study focused on the capacity evaluation of a satellite SIMO downlink at Cband considering either a light or heavy shadowed urban environment. Polarization effects are introduced and capacity results taking depolarization into account are presented. Overall, very sufficient capacities are achieved varying between 3.8 and 8.7 b/s/Hz in comparison with a SISO link. Additionally, in the heavy shadowing scenario the capacity drops significantly comparable with the light shadowing case. In average, the capacity decreases about 1 b/s/Hz. The capacity increases as the angular spread increases since the multiple elements take advantage of the rich scattering environment and the lower spatial correlation.
The role of depolarization, causing additional loss, is usually neglected in nondualpolarized situations. In this study it is taken into account, and in order to achieve this, a model of random received polarization based on Stokes parameters is given. Applying this model to Rayleigh channels, ergodic capacity in the case of depolarization is determined. If the received field is completely depolarized and every polarization is equally likely, an additional average power loss of 3 dB is formed (being a self evident result) whereas, in order to compensate that in ergodic capacity, an increase of about 4.1 dB is needed (the difference follows from Jensen’s inequality). If the received field is completely depolarized and every linear polarization is equally likely, the difference is somewhat more (about 0.8 dB). The capacity of a depolarized 1 × 2 SIMO diversity channel is nearly the same (by 0.5 dB higher) than that of a polarized SISO channel. In case of a SISO link the cost of depolarization considering a Rayleighfading channel is about 4.3 dB. Finally, the average reduction of capacity due to depolarization in a SIMO link is approximately 1.1 bits/s/Hz. If we have four elements at the receiver and a required capacity of 6.5 bits/s/Hz, in case of depolarization, we have to increase 4 dB the SNR so as to achieve the same capacity with the polarized case (note that this result is very close to the Rayleigh case).
Endnote
^{a}As, by Jensen’s inequality${\int}_{0}^{\infty}ln(1+f\rho ){p}_{f}\left(f\right)df<ln\left(1+\rho {\int}_{0}^{\infty}f{p}_{f}\left(f\right)df\right)$; or in words: expected value of a concave real function (e.g. lnx) is less than the same function of the independent variable’s expected value.
Declarations
Acknowledgements
This study was partially carried out in the framework of the European Network of Excellence IST027393 SatNEx II, and partially in the framework of Thales “MIMOSA” project, Greek Ministry of Education.
Authors’ Affiliations
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