Cooperative Transmission for Satellite Wireless Sensor Networks

Abstract

This paper presents the performance analysis of wireless sensor networks- (WSNs-) based satellite collocation system as cooperative transmission is applied. A scenario of satellite cluster is considered, where multiple satellites are collocated with each other. Considering that the performance of satellite systems can be improved as the spot beam is split into virtual beams, the optimized system capacity can be expected by applying the geometrical arrangement of the antennas of transmitters and receivers. In the following simulation experiments, the system capacity is investigated as various system configurations are applied. It is found that the optimal value of system capacity exists as expected, and the simulation observations are illustrated as practical limitations are considered.

1. Introduction

Technological progress in wireless communications and microelectromechanical systems (MEMSs) have brought about the opportunity of the development of low-power, multifunctional, low-cost, and tiny sensor nodes which can communicate to each other directly. This progress, together with marked advances in the area of microsensors, has allowed inexpensive, energy-efficient, and reliable sensors with wireless networking capabilities to become a reality. The development of such devices has given rise to the increasingly popular concept of wireless sensor networks (WSNs) [1]. This network consists of a large number of nodes, which have sensing, processing, and communicating capabilities and perform human-unattended information collection [2], which has been the subject of extensive studies [3–5].

Nowadays, various kinds of WSNs-based application networks are attracting more and more attentions, such as smart grids [6], smart home [7], and e-health [8]. recently, wireless sensor network market also represents an interesting and potentially huge revenue stream for the satellite industry. The reason is that terrestrial network currently covers only 10% of the world and has created a need to switch toward satellite networks which would help to cover the remaining 90% of the world, making it the most attractive aspect of satellite sensor connectivity. In recent years, the research of combing wireless sensor network with satellite communication represents a good opportunity [9–11]. The wireless sensor satellite communication market comprises services such as environmental and habitat monitoring, satellite remote sensing for ocean research, and structural health monitoring. These services can make use of the backward satellite link capacity for the transmission of telemetry data, requiring only a small fraction of capacity in the forward direction.

In order to improve the capacity and spectral efficiency of satellite sensor network, the construction of multiple input multiple output (MIMO) systems is very promising especially with regard to satellite transmission. This is mainly due to two reasons: on the one hand, there is a rising demand for transmission bandwidth; on the other hand, the usable frequency spectrum of available bandwidth becomes short and particularly expensive. However, it is difficult for satellites to employ MIMO techniques directly on the existing satellite platform, due to the constraints from satellite's size or hardware implementation.

Compared to traditional MIMO systems, cooperative communication does not need multiple antennas at terminal devices and therefore significantly reduces the implemental complexity and cost, which is widely studied in wireless networks [12, 13]. The performance of cooperative transmission in satellite network is carried out in [14, 15], but only the low earth orbiting satellites scenario is considered. For the scenario of geostationary (GEO) orbiting satellite networks, only the limited numbers of satellites can be supported by the GEO orbit as GEO orbit is one kind of scarce nature resource. Unlike traditional GEO satellite system, multisatellite collocation technology [16], which can allow several satellites running in one GEO orbit window ( ${\mathrm{0}.\mathrm{2}}^{°}$ ), is considered as an available way to use the GEO orbit efficiently. However, it is nowadays widely believed that a high MIMO capacity gain can only be achieved in the rich scattering multipath propagation channel [17]. Contrarily to this impression, in principle, it is possible to form a LOS channel with maximum MIMO multiplexing gain.

The main drawback of the satellite cooperative transmission system to be used in satellite sensor networks is found in satellite antenna element spacings, which are too small as several satellites are collocated with each other. In other words, these systems only offered low multiplexing gain coinciding with only a logarithmic rather than a linear increase of the capacity with the used number of antenna elements. In fact, there exist very distinct requirements for the positioning of both the satellite antennas and the ground stations that have to be fulfilled in order to create an orthogonal MIMO channel transfer matrix that offers optimum multiplexing gain. To overcome this problem, for terrestrial antennas system, the optimized antenna setups have been proven to deliver significantly higher channel capacities [18]. Contrarily, with respect to satellite systems, in [19–22], the maximum multiplexing gain of the line-of-sight (LOS) satellite channel is achieved by constructing an orthogonal channel matrix. However, for mathematically manageable results, the analyses are limited to a satellite cooperative transmission system with two satellite antenna elements only, which have to be placed on a single satellite each. In [23], a cooperative transmission scheme and a ground user selection criterion are proposed, but it also limited to the scenario with two satellites and two users. In the multisatellite collocation system, the use of multiple satellites can obviously improve multiplexing gain and spectral efficiency.

This paper addresses the optimization of the cooperative transmission channel capacity for a system consisting of $N_T$ ground satellite terminals and $N_S$ collocated satellites. Furthermore, we presume a regenerative payload for the satellites, and a communication link is established between each satellite, which could be realized, for example, by a high-bandwidth optical link. We proved that maximum MIMO spectral efficiency can be achieved by proper construction of the positions of each antenna on both sides, for satellite terminals and collocated satellites. We also verified that when multiple satellites are used, system capacity can be further improved.

The remainder of this paper is organized as follows. The system model and channel model are given in Section 2. In Section 3, we describe a capacity calculation and optimum capacity. The optimization criterion is detailed in Section 4, and the performance analysis is given in Section 5. We conclude the paper in Section 6.

The superscripts H stand for the Hermitian transpose. Upper and lower boldfaced letters are used for matrices and column vectors, respectively. Denote by $|x|$ the absolute value of a scalar x or cardinality of x if x is a set. I is the identity matrix of a certain size implicitly given by the context. We denote by $\left[{\mathbf{w}}_{\mathrm{1}},{\mathbf{w}}_{\mathrm{2}},\dots ,{\mathbf{w}}_N\right]$ the concatenation of N column vectors; ${\log }_{\mathrm{2}}$ represents the binary logarithm.

2. System and Channel Model

2.1. System Model

A scenario of cooperative collocated satellite system is given in Figure 1, where at least two satellites collocated in one GEO orbit window and multiple users within one spot beam. Here, we suppose that there are $N_S$ satellites collocated in the GEO orbit and $N_T$ satellite terminals within one spot beam, each of which is equipped with single antenna. Each satellite is able to generate multibeam. However, for satellite terminals within a spot beam, it can be equivalent to a satellite equipped with only one antenna. Since downlink is quite reasonable and straightforward for GEO satellites to broadcast, in this paper, we mainly focus on uplinks with multiple satellites.

Figure 1

System scenario.

The system model is given in Figure 2. As is shown in Figure 2, selected terminals within one beam transmit data to multiple satellites simultaneously with the same frequency and time slot. After receiving the uplink data, satellites forward them to a ground control station. When a satellite terminal has a request for transmitting, the control station allocate a specific frequency and time slot to multiple selected users (for the sake of simplicity, only two satellite terminals are considered for cooperation).

Figure 2

System model.

The system characteristics and parameters are given in Table 1. Where EIRP is the effective isotropic radiated power, $G-T$ is the logarithm ratio of figure of merit, B is the transmission bandwidth, and L is path loss, with carrier frequency $f=\mathrm{18}$  GHz. Using the parameters of Table 1, for the SNR at the satellite, we obtain

$\begin{array}{cc}& \text{SNR}=\text{EIRP}+\left(G-T\right)-L-K-B=\mathrm{24}\hspace{0.17em}\hspace{0.17em}\text{dB},\end{array}$

(1)

where K and B are the logarithmic value of the Boltzmann constant and uplink bandwidth, respectively. It is assumed here that SNR is constant and identical for each pair of Tx-Rx antennae, and it includes all the gains of the link budget, except for the LOS path loss, which has been incorporated into the channel matrix.

Table 1

Parameters of satellite system.

Parameter	EIRP	$$(G-T)$$	B	L
Value	69.3 dBW	2.4 dBK	5 MHz	209.3 dB

2.2. Channel Model

The frequency selective MIMO satelite channel $\mathbf{H}\left(f\right)$ consists of a LOS signal part ${\mathbf{H}}_{\text{LOS}}\left(f\right)$ and a second part ${\mathbf{H}}_{N\text{LOS}}\left(f\right)$ , which is formed by multipath signal. The channel model is described as [24]

$\begin{array}{cc}& \sqrt{\frac{K}{K+\mathrm{1}}}{\mathbf{H}}_{\text{LOS}}\left(f\right)+\sqrt{\frac{\mathrm{1}}{K+\mathrm{1}}}{\mathbf{H}}_{N\text{LOS}}\left(f\right),\end{array}$

(2)

where K denotes the Ricean K-factor, which is defined as the power of the LOS signal component divided by the power of the multipath parts. Unlike terrestrial mobile scenario, the dominating radio wave propagation mechanism in a satellite channel is formed by the LOS signal component due to its low signal attenuation and multipath signals play a subordinated role. In urban environments, shadowing and blocking cannot be avoided, so, in principle, NLOS signals cannot be ignored. However, an antenna separation of at least half wavelength is commonly satisfied in pure NLOS scenario. Therefore, we restrict ourselves to only consider the LOS signal part and assume in the following a MIMO LOS channel, which is given by

$\begin{array}{cc}& \mathbf{H}\left(f\right)={\mathbf{H}}_{\text{LOS}}\left(f\right).\end{array}$

(3)

Moreover, we assume the channel to be frequency flat if the bandwidth is much smaller than the carrier frequency; that is, $B\ll f$ . In this case the frequency dependence can be omitted and we have $\mathbf{H}\left(f\right)=\mathbf{H}$ . With the help of the free-space propagation model, each element of channel matrix H is given by [18]

$\begin{array}{cc}& {\mathbf{H}}_{mn}=a_{mn}\exp \left(-j\mathrm{2}\pi f\frac{r_{mn}}{c_{\mathrm{0}}}\right),\end{array}$

(4)

where $c_{\mathrm{0}}$ is being the speed of light in free space and $r_{mn}$ is being the distance from the mth satellite terminal to the nth satellite antenna. The term $a_{mn}=\left(c_{\mathrm{0}}/\mathrm{4}\pi f{r}_{mn}\right)\exp \left(j\varphi \right)$ is the complex envelope, for which $\varphi =\mathrm{0}$ will be assumed in the following, as it is a phase angle common to every channel entry. We also assume that the magnitude of the channel gain is approximately constant for each couple of transmit and receive antenna, as the distance from the user antenna to the satellite is much larger than the distance between each terminal; that is, $a_{mn}\approx a$ .

3. MIMO Capacity

3.1. Channel Capacity

In the scenario of conventional satellite communication networks, with the help of Shannon theory, the capacity of a single-input-single-output (SISO) system is given by

$\begin{array}{cc}& C_{\text{SISO}}={\log }_{\mathrm{2}}\left(\mathrm{1}+\rho {\left|h_{\text{SISO}}\right|}^{\mathrm{2}}\right),\end{array}$

(5)

where $h_{\text{SISO}}$ is the channel gain between the satellite terminal and the satellite and ρ denotes the SNR at the transmitter. However, in a scenario of cooperative transmission, the system can be treated as a virtual distributed MIMO. If the transmit symbols are independent, identically distributed (i.i.d.) Gaussion random variables and the channel state information (CSI) areunknown to the transmitter; the time invariance spectral efficiency of a frequency-flat time invariance MIMO transmission channel without a relay is calculated according to the Telatar's well-known equation [25]

$\begin{array}{cc}& C_{\text{MIMO}}={\log }_{\mathrm{2}}\left[\det \left({\mathbf{I}}_{N_T}+\rho \mathbf{H}{\mathbf{H}}^H\right)\right],\end{array}$

(6)

where the transmit symbols are realizations of uncorrelated independent, identically distributed (i.i.d) Gaussian random variables. The ratio ρ, which incorporates all the gains of the link budget except the LOS path loss, is calculated as

$\begin{array}{cc}& \rho =\mathrm{1}{\mathrm{0}}^{\left(\text{SNR}+L\right)/\mathrm{10}}.\end{array}$

(7)

3.2. Optimum Capacity

We define the square matrix Q with eigenvalues ${\gamma }_i$ as follows [20]:

$\begin{array}{cc}& \mathbf{Q}=\left\{\begin{array}{cc}{\mathbf{H}}^H\mathbf{H}\mathbf{,}\hfill & N_S\ge N_T,\hfill \\ \mathbf{H}{\mathbf{H}}^H,\hfill & N_S<N_T.\hfill \end{array}\right.\end{array}$

(8)

Then, the maximum spectral efficiency is achieved when all eigenvalues of Q are identical; that is, ${\gamma }_i=|a{|}^{\mathrm{2}}\max \left\{N_S,N_T\right\}\left(i=\mathrm{1,2},\dots ,\min \left\{N_S,N_T\right\}\right)$ . In this case we obtain an optimum eigenvalue profile and an orthogonal channel transfer matrix. Then, the spectral efficiency also obtains its optimum value which is given by

$\begin{array}{cc}& C_{\text{opt}}=\min \left\{N_S,N_T\right\}{\log }_{\mathrm{2}}\left(\mathrm{1}+\rho {\left|a\right|}^{\mathrm{2}}\max \left\{N_S,N_T\right\}\right).\end{array}$

(9)

This optimal capacity is also the upper bound we try to obtain in the following. Considering that the channel matrix is related with the distances between the satellite terminals and the satellites, so those path lengths are the degrees of freedom, which can be exploited to adjust the channel entries so that the eigenvalues of Q are equal, and thus $\min \left\{N_S,N_T\right\}$ identical eigenvalues are generated. On the contrary, if the channel matrix is rank deficient, the so-called keyhole effect and the lower bound of the spectral efficiency are achieved.

3.3. Cooperative Gain

In practice, if the LOS channel is not orthogonal for all satellites and satellite terminals, the capacity of cooperative system may not be better than that of noncooperative system. Therefore, following the definition given in [23], we define the cooperative gain generated by the cooperative transmission as follows:

$\begin{array}{cc}& G=\frac{R_{\mathrm{co}}}{R_{\text{nco}}},\end{array}$

(10)

where $R_{\mathrm{co}}$ and $R_{\text{nco}}$ are the achievable transmission rate of the cooperative and noncooperative system, respectively. It is well known that the capacity denotes the spectral efficiency of a system, so the achievable transmission rate of cooperative and noncooperative system can be given by

$\begin{array}{cc}& R_{\text{nco}}=N_T{f}_{\text{nco}}{C}_{\text{SISO}},\\ R_{\mathrm{co}}=f_{\mathrm{co}}{C}_{\text{MIMO}},\end{array}$

(11)

where $f_{\text{nco}}$ and $f_{\mathrm{co}}$ are the bandwidth of the noncooperative and the cooperative system, respectively. $N_T$ is the number of satellite terminals. From (5), (6), and (11), we can obtain

$\begin{array}{l}G=\frac{R_{\mathrm{co}}}{R_{\text{nco}}}=\frac{f_{\mathrm{co}}{C}_{\text{MIMO}}}{N_T{f}_{\text{nco}}{C}_{\text{SISO}}}\\ =\frac{f_{\mathrm{co}}\text{lo}{\text{g}}_{\mathrm{2}}\left\{\det \left({\mathbf{I}}_{N_T}+\rho \mathbf{H}{\mathbf{H}}^H\right)\right\}}{N_T{f}_{\text{nco}}\text{lo}{\text{g}}_{\mathrm{2}}\left(\mathrm{1}+\rho {\left|h_{\text{SISO}}\right|}^{\mathrm{2}}\right)}.\end{array}$

(12)

4. Optimum Criterion

4.1. Geometrical Description

The principle of Mercator projection is used to obtain the positions of the satellites and satellite terminals. The geometrical description of satellite terminals is given in Figure 3, where the center of the satellite terminals has longitude ${\theta }_T$ and latitude ${\varphi }_T$ on the Earth surface, while the position of each satellite is given by longitude ${\theta }_{S,n_S}$ , for $n_S=\mathrm{1,2},\dots ,N_S$ . Define $R_T$ and $R_S$ as the mean Earth radius and the radius of the geostationary orbit, respectively. Then, the position vector of the center of two satellite terminals in Cartesian coordinates is given by

$\begin{array}{cc}& {\alpha }_T=\left(\begin{array}{c}{x}_T=R_T\cos {\varphi }_T\cos {\theta }_T\\ y_T=R_T\cos {\varphi }_T\sin {\theta }_T\\ z_T=R_T\sin {\varphi }_T\end{array}\right).\end{array}$

(13)

Figure 3

Geometrical description of two satellite terminals.

Then, we define the angle between a tangent in the east-west direction and the direction of two arbitrary satellite terminals is denoted by $u_T\in \left\{-\mathrm{18}{\mathrm{0}}^{°}\le {\delta }_T\le \mathrm{18}{\mathrm{0}}^{°}\right\}$ , and the position vectors of two satellite terminals are given by

$$\begin{array}{l}{\alpha }_{T,n_T}\hfill \\ =\left(\begin{array}{c}{x}_{T,n_T}=x_T-p_{n_T}\left(sin{\theta }_{T}cos{u}_T+sin{\varphi }_{T}cos{\theta }_{T}sin{u}_T\right)\\ y_{T,n_T}=y_T-p_{n_T}\left(cos{\theta }_{T}cos{u}_T-sin{\varphi }_{T}sin{\theta }_{T}sin{u}_T\right)\\ z_{T,n_T}=z_T+p_{n_T}cos{\varphi }_{T}sin{u}_T\end{array}\right)\hfill \end{array},$$

(14)

where $p_{n_T}=\pm \left(\mathrm{1}/\mathrm{2}\right)d_T$ .

For each of the satellites, we may define

$\begin{array}{cc}& {\beta }_{S,n_S}=\left(\begin{array}{c}{x}_{S,n_S}=R_S\cos {\theta }_{S,n_S}\\ y_{S,n_S}=R_S\sin {\theta }_{S,n_s}\\ z_{S,n_S}=\mathrm{0}\end{array}\right).\end{array}$

(15)

Thus, the distance between the nth terminal and the mth satellite is given as $r_{mn}=‖{\alpha }_{T,n}-{\beta }_{S,m}‖$ .

Thanks to the approximation on the magnitude of the channel gain given by (4), it is actually shown that the phase relations between the channel matrix entries are the only degrees of freedom to modify the properties of the channel. As explained in Section 3.2, by imposing the eigenvalues of Q to be equal, a criterion on these phase relations which allows maximizing the system capacity for a $N_T\times 2$ MIMO channel was proposed in [20]. The most general formulation of this criterion, which can be used as a starting point for the present work, can be described as

$\begin{array}{cc}& r_{k,\mathrm{1}}-r_{k,\mathrm{2}}+r_{l,\mathrm{2}}-r_{l,\mathrm{1}}=v\left(k-l\right)\frac{c_{\mathrm{0}}}{N_{S}f},\end{array}$

(16)

where $k,l\in \left\{\mathrm{1,2},\dots ,N_S\right\}$ and $r_{n_S,n_T}$ is the distance between the $n_T$ th transmit antenna to the $n_S$ th satellite receive antenna. The parameter $v\in Z$ is indivisible by $N_S$ , and the possible configurations satisfying the condition have actually an angle periodicity of $\mathrm{2}\pi$ .

As a second step, we look for the expression of the length $r_{n_S,n_T}$ . We first express the distance vector between the terminal antenna $n_T$ and the satellite antenna $n_S$ as a Euclidean norm, following the notation introduced before as

$\begin{array}{cc}& r_{n_S,n_T}=\sqrt{{\left(x_{T,n_T}-x_{S,n_T}\right)}^{\mathrm{2}}+{\left(y_{T,n_T}-y_{S,n_T}\right)}^{\mathrm{2}}+z_{T,n_T}^{\mathrm{2}}}.\end{array}$

(17)

Inserting the position vector of ${\alpha }_{T,n_T}$ and ${\beta }_{S,n_S}$ and after a number of mathematical manipulations involving trigonometrical theorems, we obtain

$$\begin{array}{l}{r}_{n_S,n_T}=\left[R_S^{\mathrm{2}}+R_T^{\mathrm{2}}-\mathrm{2}{R}_S{R}_T\cos {\varphi }_T\cos \left({\theta }_T-{\theta }_{S,n_S}\right)\right.\\ \phantom{Ww}+2R_S{p}_{n_T}\left(\cos u_T\sin \left({\theta }_T-{\theta }_{S,n_S}\right)\right)\\ \phantom{Ww}+{\left.\sin u_T\cos \left({\theta }_T-{\theta }_{S,n_S}\right)\sin {\varphi }_T+p_{n_T}^{\mathrm{2}}\right]}^{1/2}.\end{array}$$

(18)

Two auxiliary quantities are now introduced as

$$\begin{array}{l}{s}_{n_S}=\sqrt{R_S^{\mathrm{2}}+R_T^{\mathrm{2}}-\mathrm{2}{R}_S{R}_T\cos {\varphi }_T\cos \left({\theta }_T-{\theta }_{S,n_S}\right)},\\ c_{n_S}=\mathrm{2}{R}_S\left[\cos u_T\sin \left({\theta }_T-{\theta }_{S,n_S}\right)\right.\\ +\left.\sin u_T\cos \left({\theta }_T-{\theta }_{S,n_S}\right)\sin {\varphi }_T\right].\end{array}$$

(19)

Now, we apply a first order Taylor series expansion as a good approximation of the square root $\sqrt{\mathrm{1}+a}\approx \mathrm{1}+a/\mathrm{2}$ ; we have [20]

$\begin{array}{cc}& r_{n_S,n_T}\approx s_{n_S}+\frac{p_{n_T}{c}_{n_S}+p_{n_T}^{\mathrm{2}}}{\mathrm{2}{s}_{n_S}}\approx s_{n_S}+p_{n_T}\frac{c_{n_S}}{\mathrm{2}{s}_{n_S}}.\end{array}$

(20)

This result may be inserted into (16), and we have

$\begin{array}{cc}& d_T\left(\frac{c_l}{\mathrm{2}{s}_l}-\frac{c_k}{\mathrm{2}{s}_k}\right)=v\frac{\left(k-l\right)c_{\mathrm{0}}}{N_{S}f}.\end{array}$

(21)

A comparison with this result for the equation obtained in [20] clearly shows that this time the optimization criterion keeps the dependence on the difference between k and l, while, in [20], this dependence disappeared; thanks to it, only consider two satellites, and the optimization of the positions of both satellites and terminals became independent of the choice of k and l. However, the term $\left(k-l\right)$ shows that different solutions exist, considering the particular couple of antenna elements chosen for the optimization. In other words, each couple of $\left(k,l\right)$ can be treated as 2-satellite pairs, and optimizing this result is equivalent to the joint optimization of these $N_S\left(N_S-\mathrm{1}\right)$ satellite pairs which compose the chosen ones from collocated satellite system.

Assume that the positions of satellites are given; in order to simplify (21), we first review the degree of freedom in the optimization of the position of two satellite terminals:

• (1)

the number of transmit antenna $N_T$ ,

(2)

the distance $d_T$ between two cooperative satellite terminals within one spot beam,

(3)

the direction angle $u_T$ .

Assuming that $N_T$ is given, then we try to find the values of distance $d_T$ and angle ${\delta }_T$ to maximize the system capacity. Next, we reformulate (21) to separate the dependency on those two parameters. Whereas the dependency on distance $d_T$ is already evident, we reformulate $c_{n_S}$ as

$\begin{array}{cc}& c_{n_S}=R_S\left(G_{n_S}^{\mathrm{1}}\cos u_T+G_{n_S}^{\mathrm{2}}\sin u_T\right),\end{array}$

(22)

where the coefficients $G_{n_S}^{\mathrm{1}}=\mathrm{2}\sin \left({\theta }_T-{\theta }_{S,n_S}\right)$ and $G_{n_S}^{\mathrm{2}}=\mathrm{2}\cos \left({\theta }_T-{\theta }_{S,n_S}\right)\sin {\varphi }_T$ include parameters related to the positions of each satellite and the centre of two terminals, independent of the two optimization parameters. We then insert (22) into (21), after assuming the following definition for ease of notation as

$\begin{array}{cc}& A_{kl}=\frac{G_l^{\mathrm{1}}}{s_l}-\frac{G_k^{\mathrm{1}}}{s_k},\\ B_{kl}=\frac{G_l^{\mathrm{2}}}{s_l}-\frac{G_k^{\mathrm{2}}}{s_k},\\ C_{kl}=\frac{\mathrm{2}\left(k-l\right)c_{\mathrm{0}}}{R_S{N}_{S}f}.\end{array}$

(23)

Then, the optimization equation becomes

$\begin{array}{cc}& d_T\left(A_{kl}\cos u_T+B_{kl}\sin u_T\right)=v·C_{kl}.\end{array}$

(24)

Now, we have one equation for each couple of $\left(k,l\right)$ , for $k\ne l$ . Actually, (24) describes a nonlinear system which contains $N_S\left(N_S-\mathrm{1}\right)$ equations, where parameter v gives a further degree of freedom for each equation of the system. Although we can assume that approximate solutions may be found in the least squares sense to deal with an overdetermined system, it is still impossible to give a closed-form expression for an exact solution of the system, for the chosen values of distance $d_T$ and angle $u_T$ that would ensure full achievement of the optimal system capacity. For this reason, we continue our analysis through numerical simulations, which will be shown in the following section.

5. Simulation Results

In this section, we present performance results through numerical simulations, where two satellite terminals and multiple collocated satellites are considered.

In Figure 4, we display the normalized achievable rate of cooperative and noncooperative system. Here, we assume that three satellites are chosen to communicate with two terminals. The simulation parameters are shown in Table 2, the distance between two satellites are set to 60 km, and the angle $u_T$ is set to ${\mathrm{0}}^{°}$ . In this figure, three scenarios are taken into consideration: (1) three satellites with cooperation; (2) three satellites without cooperation; (3) one big satellite without cooperation. For cooperative system, the frequency can be reused by two satellite terminals, while, for noncooperative system, frequency division is necessary. Therefore, we set $f_{\mathrm{co}}=\mathrm{2}{f}_{\text{nco}}$ . It is shown that the achievable rate of cooperative system oscillates with a periodicity. For the scenario of three satellites without cooperation, the uplink contains two SIMO channels, while, for the scenario of one big satellite, we set the SNR as ${\rho }^{\prime }=\mathrm{3}\rho$ , and the uplink contains two SISO channels. From this figure, we can see that, with satellite collocation, the achievable rate of both cooperative and noncooperative system is much better than that of one big satellite with high power.

Table 2

Parameters of simulation.

Parameter	$${\mathrm{\theta }}_T$$	$${\mathrm{\varphi }}_T$$	$${\mathrm{\theta }}_{S,\mathrm{2}}$$
Value	$$\mathrm{1}{\mathrm{2}}^{°}$$	$$\mathrm{5}{\mathrm{2}}^{°}$$	$$\mathrm{1}{\mathrm{2}}^{°}$$

Figure 4

Achievable rate for three satellites with cooperation, noncooperative satellites and one big satellite for $N_T=\mathrm{2}$ , $u_T={\mathrm{0}}^{°}$ , and $f=\mathrm{18}$  GHz.

Figures 5 and 6 display the contour plot of the calculated system capacity as a function of $d_T$ and $u_T$ . Because the distance interval between two satellites is set identically, it is shown in Figure 5 that the near-maximum capacity is achievable in the region by choosing the direction angle near the range of $\pm \mathrm{18}{\mathrm{0}}^{°}$ and ${\mathrm{0}}^{°}$ ; that is, when $u_T={\mathrm{0}}^{°}$ and $d_T=\mathrm{1.9}$  m, we can achieve more than 98% of the optimal capacity. In Figure 6, we observe that (24) describes a straight line on the cartesian plane, whose slope depends on the parameters $A_{kl}$ and $B_{kl}$ ; it also shows that we can achieve more than 98% of the optimal capacity when $d_T\cos \left(u_T\right)>\mathrm{1.9}$  m or $d_T\cos \left(u_T\right)<-\mathrm{1.9}$  m; during this region, $d_T>\mathrm{1.9}$  m and $u_T=\pm \mathrm{18}{\mathrm{0}}^{°}$ .

Figure 5

System capacity computed by numerical simulations for $N_S=\mathrm{3}$ and $f=\mathrm{18}$  GHz.

Figure 6

Polar coordinates representation of the system capacity computed by numerical simulations for $N_S=\mathrm{3}$ and $f=\mathrm{18}$  GHz.

In Figure 7, we take into consideration to which degree any deviations from the optimum terminal distance setup will cause degradations of the system capacity. The figure shows that the system capacity C changes with the distance between two satellite terminals $d_T$ for four different directional angle $u_T$ . It is observed that capacity oscillates with a periodicity in $d_T$ . Due to the parameters setting of the simulation in which ${\theta }_T={\theta }_{S,\mathrm{2}}\approx \left({\theta }_{S,\mathrm{1}}+{\theta }_{S,\mathrm{3}}\right)/\mathrm{2}$ , given $d_T$ , system capacity is symmetrical to angle $u_T={\mathrm{0}}^{°}$ and has a capacity minimum at $u_T=\pm \mathrm{9}{\mathrm{0}}^{°}$ , so when the angle $u_T$ is large, system can achieve its optimal capacity even for small terminal distance $d_T$ and therefore has small $d_{\text{opt}}$ and small periodicity. This figure gives four group curves for $u_T={\mathrm{0}}^{°}$ , $\mathrm{3}{\mathrm{0}}^{°}$ , $\mathrm{6}{\mathrm{0}}^{°}$ , and $\mathrm{9}{\mathrm{0}}^{°}$ , respectively. It shows that when $u_T=\mathrm{9}{\mathrm{0}}^{°}$ , we can only achieve 55% as percentage of the optimum capacity $C_{\text{opt}}$ ; it is because at this point the condition $r_{n_S,\mathrm{1}}=r_{n_S,\mathrm{2}},\hspace{0.17em}\mathrm{\hspace{0.17em}\hspace{0.17em}}\hspace{0.17em}\left(n_s=\mathrm{1,2},\dots ,N_S\right)$ is fulfilled and, therefore, the MIMO channel cannot be optimized at all. This makes the worst-case type of the channel with respect to the channel multiplexing gain and it is generally referred to as the keyhole channel.

Figure 7

System capacity with different $d_T$ and $u_T$ for $N_S=\mathrm{3}$ , $d_S=\mathrm{60}$ , and $f=\mathrm{18}$  GHz.

Figure 8 shows the value of optimal distance between two satellite terminals $d_{\text{opt}}$ with different satellite interval $d_S$ and the number of cooperative satellites $N_T$ . For a collocated satellite system within one GEO orbit window, $d_S$ cannot increase infinitely, so we set $d_S$ from the range of $\mathrm{10}$  km to $\mathrm{70}$  km. It can be observed from Figure 8 that $d_{\text{opt}}$ decrease with the increase of satellite interval $d_S$ . It also shows that shorter optimal distance is available when we use more satellites, that is, when the interval between two satellites is $\mathrm{40}$  km and when we choose four and three satellites for cooperation, the optimal distance is 51% and 33% times shorter than only one satellite are chosen, respectively.

Figure 8

Optimal distance $d_{\text{opt}}$ with different $d_S$ for $f=\mathrm{18}$  GHz and $u_T={\mathrm{0}}^{°}$ .

Figure 9 shows the increase of the percentage of $C/C_{\text{opt}}$ with terminal distance $d_T$ , with different satellite interval $d_S$ and the number of cooperative satellites $N_T$ , respectively. In this figure, three collocated satellites are considered. It can be observed from the first figure on the top of Figure 9 that the achievable capacity increases with the distance of three collocated satellites. In other words, the required distance between two satellite terminals would be shorter and a better system capacity performance can be achieved with $d_S$ increases. However, $d_S$ cannot increase infinitely due to the GEO orbit window of satellite collocation. Note that, for the safety of the satellites, $d_S$ is not more than $\mathrm{70}$  km. Therefore, we can use a large $d_S$ to enhance the system capacity. Another figure which is at the bottom of Figure 9 analyzes the system capacity performance with different $d_T$ and the number of cooperative satellites $N_T$ , for $d_S$ is set to $\mathrm{20}$  km. It shows that better capacity can be achieved when we use more satellites; that is, when the distance between two terminals is $\mathrm{1.5}$  m and when we choose four satellites for cooperation, the achievable capacity is 11% and 32% times larger compared to the ones when three and two satellite are chosen, respectively.

Figure 9

System capacity with different $d_T$ and $N_S$ for $d_S=\mathrm{20}$  km, $f=\mathrm{18}$  GHz, and $u_T={\mathrm{0}}^{°}$ .

In practice system, satellite is impossible to maintain absolutely immobile at its defined position. Therefore, 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 37.5 km in longitude and latitude and 17.5 km for the eccentricity [26]. 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. The capacity degradation is less than 1% of the optimum value $C_{\text{opt}}$ . Hence, station keeping has a negligible impact on the capacity as long as the pointing of the satellite is correct.

6. Conclusions

In this paper, we propose a scheme to combine satellite cooperative transmission networks and WSNs. The two sensor devices within one spot beam access a collocated satellite system for data transmission. With the help of cooperative beamforming, each satellite spot beam can be split into multiple virtual beams, by which the capacity can be improved. We present an approach which enables the construction of uplink with two terminals and multiple satellites that can achieve the maximum MIMO spectral efficiency for the case of LOS signal propagation. Maximum multiplexing gain is achieved via an optimized positioning and displacement of the MIMO antenna elements at both sides, which results in an orthogonal MIMO channel transfer matrix. Through simulations, we have confirmed that, for the construction of capacity-optimal MIMO channel transfer matrices, the user is provided with several degrees of freedom for the system design. It shows that capacity increases with the number of cooperative satellites $N_S$ and satellite interval $d_S$ , while showing a periodical performance with terminal distance $d_T$ and angle $u_T$ . The interdependence and optimization of the design parameters are illustrated, also highlighting practical limitations.