Inversiones exteriores

2.3.3 Multiuser Scheduling

Multiuser scheduling can glean the multiuser gains offered by the BC channel. The transmitter, by choosing which combinations of users to simultaneously serve can maximize the system performance by reducing the interference and increasing the useful signal power at the receive side. In general, linear ZF beamforming is suboptimal compared to the ca- pacity achieving DPC. However, Yoo et al [53, 82] proved that via user selection, ZF can achieve asymptotically optimal SR performance. More- over, an iterative user selection algorithm that allows ZF to achieve the performance of non-linear precoding [55] when the number of available for selection users grows to infinity was proposed in [82]. This iterative, heuristic semi-orthogonal user selection algorithm accounts for the level of orthogonality of each user over the set of the already selected users. By rejecting users that have high cross user correlation between their vector channels a large set of random users is reduced. Subsequently, out of the users that are left, the most orthogonal ones are iteratively chosen to form a group of users equal to the number of transmit elements. Specifically, if user channels are perfectly orthogonal to each other, ZF will attain maximum performance. Under the assumption of large random user sets, the probability of orthogonal users increases. By recalling that ZF cre- ates equivalent orthogonal channels, it is intuitive to conclude that when the selected users are already orthogonal to each other, the precoding performance will be drastically enhanced. In [82] it was proven that the proposed scheduling method achieves nearly optimal performance as the number of random users grows to infinity. This result is expected since at the limit of infinite users, one can find perfectly orthogonal channels and perform ZF without any orthogonalization losses. Hence, despite their suboptimality, when compared to DPC, linear precoding methods can still achieve asymptotically optimal performance under specific con- ditions, as proven in [53, 72, 82]. More details on this topic are given in the dedicated Ch. 6.

2.4

Multibeam Satellite Channel

The differences between the satellite and the terrestrial wireless channels are fundamental. The first includes contributions from the multibeam satellite antenna, the propagation medium (rain fading) and the satellite

payload as a non linear relay. In the present section, an attempt to capture most of these effects in the channel matrix H is presented. As it be will shown, the channel matrix can be formulated as a proper combination of separate matrices, each modeling a different effect.

The most significant contribution in the total channel matrix is that of the multibeam antenna gains. The effects of the multibeam radia- tion pattern will be captured by matrix B. Beam gain for each satellite antenna–user pair, depends on the spotbeam antenna pattern and on the user position. Assuming that user position does not change considerably in the duration of a codeword, B reduces to a deterministic real positive matrix, composed of the square roots of the channel coefficients.

Based on the assumptions of each section, these coefficients can either models the position dependant beam gain, or the carrier to noise ration of each satellite antenna user pair. The latter model applies when some of the link budget parameters are included in the coefficients. To avoid any confusions, despite the fact that a single symbol will be used to describe the satellite channel matrix B, the real coefficients that compose B in each case are explicitly defined hereafter.

2.4.1 Approximated Multibeam Antenna Pattern

Bessel Function Approximations

In this case, the matrix Bb is composed of the square roots of the gain

coefficients calculated using the well accepted method of Bessel functions [83]: gij(θij) = Gmax J1(u) 2u + 36 J3(u) u3 2 , (2.18) where u = 2.07123 sin θ/ sin θ3dB, and J1, J3are the Bessel functions of the

first kind, of order one and three respectively. The jth user corresponds to an off-axis angle θijwith respect to the boresight of the ithbeam where

θi= 0◦. This beamgain is plotted versus an off axis angular distance, for

two different beam sizes in Fig. 2.2.

It should be clarified that the present section for simplicity purposes, the earth curvature and the satellite orbit geometry are not accounted for in the channel model. Subsequently, the variations in the distances of the beam centers and the distance between the satellites are not modeled.

2.4. Multibeam Satellite Channel 55 0 5 10 15 −70 −60 −50 −40 −30 −20 −10 0 off-axis angle θ B ea m G a in (No rm a li se d , d B ) R = 50km R = 200km

Figure 2.2: Main and secondary radiation lobes for different beam sizes, based on the Bessel function approximation

The assumption that the centers of all beams are equidistant from the satellite can be supported only for small coverage areas.

2.4.2 Measured Multibeam Antenna Pattern

Towards providing more accurate results a realistic antenna pattern is also employed. A 245 beam pattern that covers Europe is employed [84], as presented in Fig. 2.3. A complex channel matrix that models the link budget of each UT as well as the phase rotations induced by the signal propagation is employed [84] [85]. The real matrix B∈ RNu×Nt _models

the satellite antenna radiation pattern, the path loss, the receive antenna gain and the noise power. Its i, j-th entry is given by [84]:

bij=

pGRGij

4π(dj· λ−1)√κTcsBu

!

, (2.19) with djthe distance between the k-th UT and the satellite (slant-range),

λ the wavelength, κ the Boltzman constant, Tcsthe clear sky noise tem-

Figure 2.3: Beam pattern covering Europe, provided by [84]

antenna gain and Gijthe multibeam antenna gain between the i-th single

antenna user and the j-th on board antenna (= feed). Hence, the beam gain for each satellite antenna- user pair, depends on the antenna pat- tern and on the user position. The link budget parameters are problem dependent and will be defined explicitly for each examined scenario.

2.4.3 Correlation

An inherent characteristic of SatComs is the high correlation among the signals at the satellite side. Total absence of scatterers on the space side renders the received signals highly correlated. In addition, an adequate antenna separation is practically impossible due to on-board size limita- tions. The assumption of signal correlation, shall impose the multiplica- tion with a diagonal matrix [86, 87]. Depending on the considered link, the multiplication is over the right or the left side of the beamgain matrix.