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Performance study of multiuser interference mitigation schemes for hybrid broadband multibeam satellite architectures
EURASIP Journal on Wireless Communications and Networking volume 2012, Article number: 132 (2012)
Abstract
As the demand for higher throughput satellites increases, multibeam architectures with smaller beam spots are becoming common place. If the same frequency is strongly reused, the resulting interference when serving simultaneously many users requires some sort of pre or postcancelation process. This article focuses on precoding and multiuser detection schemes for multibeam satellites, comparing hybrid onboard onground beamforming techniques with fully groundbased beamforming. Both techniques rely on the exchange of radiating element signals between the satellite and the corresponding gateway but, in the latter case, the interference mitigation process acts on all the radiating signals instead of the user beams directly, with the corresponding extra degrees of freedom for those cases for which the number of radiating elements is higher than the number of user beams. The analysis carried out in this study has shown that the potential advantage of groundbased beamforming may exceed 20% of the total throughput.
1 Introduction
The use of multiple spot beams in modern broadband satellites has increased during the last few years in an effort to serve higher throughput demands with a scalable cost, for which frequency reuse among users is required [1]. Thus, the same frequency band is shared by different beams to provide an overall higher throughput as long as the multiuser interference can be kept under control. This interference occurs due to the nonnull side lobes of the beams radiation patterns, and is related to the degree of reuse of the spectrum. A partial frequency reuse would exclude adjacent beams from using the same portion of spectrum (or color). However, more aggressive frequency reuse strategies [2, 3] can push forward the overall spectral efficiency provided the resulting interference can be efficiently managed. On the other side, the increase in the number of spot beams, in the desired capacity per beam, and in the frequency reuse might convert feeder links in a bottleneck. Higher frequency bands such as Q/V, optical communications or multigateway architectures need to be addressed to accommodate the required capacity.
Conventional beamforming techniques are spacebased (or onboard) architectures including analog or digital beamforming networks. Onboard volume and calibration requirements are perhaps their main drawbacks. In an attempt to shift the complexity to the ground segment, more recent groundbased beamforming (GBBF) techniques rely on the exchange of radiating element signals between the satellite and the gateway. The forming of beams is realized onground with all the flexibility offered by onground digital signal processing [4]. Again, at the cost of a higher feeder link bandwidth demand for those cases with more feeds or radiating elements than number of beams, more sophisticated and power consuming techniques can be implemented. Flexibility is preserved, and changes in shape, traffic and pointing direction can be accommodated. Multiuser interference mitigation schemes such as precoding or multiuser detection can be jointly designed with the beamforming process at the gateway station: this joint optimization process is expected to provide some gain in terms of capacity as we will show in this study. The interference mitigation process would act on all the radiating signals instead of the user beams directly, with the corresponding extra degrees of freedom.
More specifically, the forward link from the gateway to the users can be recast as a multipleinput multipleoutput (MIMO) broadcast channel [5] from an information theoretic perspective, whereas the return link corresponds to the multiple access channel (MAC) [6]. In the forward link, the sum rate capacity is known to coincide with the rate region on the nonlinear dirty paper coding (DPC) [7]. However, linear alternatives are more attractive in terms of complexity. For instance, zeroforcing (ZF) linear precoding performs close to DPC in case many users are available and optimal user scheduling is performed [8]. By relaxing the zero interference constraint at each user, it has been proved that the socalled regularized channel inversion (a MMSElike precoder) can significantly improve the performance [9].
As for the return link, the maximum sumrate is known to be achieved via successive interference cancelation with minimum meansquared error filtering at each stage (MMSESIC) [10]. However, much simpler, linear alternatives, such as the zeroforcing (ZF) receiver or the plain MMSE receiver ([11, 12]) are also popular because of their lower computational complexity.
Recently, the performance of the return link of a full onground architecture was investigated in [13], featuring an adaptive coding and modulation (ACM) enhanced DVBRCS physical layer. Results showed an increase in throughput at the cost of some loss in availability when linear MMSE was applied; with MMSESIC, a significant improvement in throughput and availability was reported. An equivalent analysis for the forward link was reported in [3], where again a throughput increase was achieved at the expense of a loss in availability.
Within the SatNEx III (Satellite Network of Experts) framework, funded by the European Space Agency, and building on previous results such as those above, the use of multiuser interference mitigation schemes together with GBBF was analyzed to evaluate its potential improvement with respect to more classic onboard beamforming settings. The main achievements of this research, which have been partly presented in [14, 15], are reported in detail in this article. It will be shown that full onground architectures tend to outperform hybrid architectures with fixed onboard weights and onground multiuser interference mitigation. This conclusion will be analytically supported under perfect channel state information (CSI), and a sufficient condition for the two architectures to be equivalent will be given. Moreover, detailed simulations will provide insight on the behavior of both architectures when imperfect channel knowledge is assumed, and also when the analog beamforming suffers from miscalibration issues.
The rest of the article is structured as follows: Section 2 describes the system model, Section 3 evaluates the performance of some processing techniques when the gateway has perfect CSI, Section 4 describes the modeling of nonperfect CSI, Section 5 reports simulation results and, finally, conclusions are summarized in Section 6.
Notation: Boldface uppercase letters denote matrices and boldface lowercase letters refer to column vectors. We denote by (.) ^{H} the Hermitian transpose. The N × N identity matrix is denoted by I _{ N }, and diag(a) builds a diagonal matrix from the elements of the vector a. Nonboldface lowercase letters are used to refer to the entries of a matrix: the (k, l)th entry of the matrix W is denoted by w_{ kl } .
2 System description
The object of study consists of a single satellite which gives service to a region covered by K beam spots; a single user link is active at a given time and carrier block at each beam. The satellite in Figure 1 uses a fed reflector antenna array with N feeds to exchange signals with the users. In the absence of onboard beamforming, all these signals will be relayed through a feeder link with the gateway station (GW) on Earth. In the sequel, we will assume a single gateway and neglect the possible impairments caused by the feeder link. As a more conventional option, if onboard beamforming is applied, beamforming weights will be assumed to be fixed and K signals, one per beam, will be synthesized from the combination of the N > K feed signals, with the corresponding reduction in the feeder link required capacity. For the radiation pattern which will be considered, only a small subset of the feeds will be involved in the conformation of each individual beam. As a limit case with practical application in some cases, for N = K each radiating element feeds a different beam, and the two options considered in this article collapse to the same case.
We will refer to the full onground processing as feed processing, whereas the hybrid architecture with onboard fixed beamforming will be often quoted as beam processing.
2.1 Return link
Let K be the number of users on Earth and N is the number of onboard feeds. At the feed level, the mathematical model of the return link reads
where y is an N × 1 vector that contains the symbols received at each feed, s is a K × 1 stack of the symbols transmitted by each user (see Figure 2), n is the N × 1 vector of zeromean complex white Gaussian noise, such that \text{E}\left\{\text{n}{\text{n}}^{H}\right\}={N}_{o}\cdot {\text{I}}_{N}, and H represents the N × K channel matrix. The channel flatfrequency response is parameterized as
where each matrix is described in the following paragraphs.
Feed radiation pattern and path losses. G is assumed to be an N × K matrix that accounts for the gains of the feed radiation pattern, the onboard attenuation and the free space losses; recall that the feeder link is considered transparent. Matrix G is not deterministic given the random positions of the users within their corresponding beam spots.
Atmospheric fading. The attenuation due to atmospheric phenomenaspecially the raincan be significant in bands such as the Kaband. In this study, the empirical probability density function (pdf) of this attenuation, obtained for the city of Rome, was used [16]. This pdf would define the statistics of the marginal distribution of each link's attenuation, but it reports no information about the possible spatial correlation.
This spatial correlation is of great importance for the design of multisatellite systems as the one addressed in [17], and its influence in a general random MIMO channel has been covered in [18] but, as far as the authors know, little has been said about rain correlation in multibeam satellite systems when there is only one user per beam. In [19], the concept of correlated area (CA) was introduced, defined as a spatial region in which Earth stations experiment highly correlated rain attenuation; the correlation with stations out of the CA would be considered negligible. The shape and length of a CA depends on many environmental factors, but diameters between 30 and 50 km are quoted to be frequent.
Based on these figures, it seems reasonable to assume that each beamspot belongs to a different CA when the radius of the spots is large enough. This was done, for instance, in [20], where beamspots have a diameter of 250 km and therefore users from different cells are assumed to be uncorrelated in terms of rain fading. In our case, the beam radius is also large (100 beams to cover Europe), and correlation values have been proven to be negligible, as shown in Appendix 1. As a consequence, matrix L is assumed to be diagonal and its entries are considered independent.
Further refinements of the channel model were not considered since this simple characterization has been deemed to be useful for the intended comparisons.
In those cases for which a fixed beamforming is applied onboard, the above model must be modified accordingly. Let B the K × N beamforming matrix and define {\text{H}}_{b}\doteq \text{B}\text{H} and {\text{n}}_{b}\doteq \text{B}\text{n}, then the received signal becomes
Note that \mathrm{\Sigma}\doteq \text{E}\left\{{\mathbf{n}}_{b}{\mathbf{n}}_{b}^{H}\right\}={N}_{o}\cdot \mathbf{B}{\mathbf{B}}^{H}.
2.2 Forward link
Analogously to the return link, we describe the signal model for the forward link, which reads as
where r (resp. w) is a K × 1 vector containing the stack of the received signals (resp. noise components) at each user. Similarly as before, we assume that \text{E}\left\{\mathbf{w}{\mathbf{w}}^{H}\right\}={N}_{o}\cdot {\mathbf{I}}_{K}. The N × 1 vector x is the stack of the transmitted signals at all feeds. The forward link channel matrix is simply the transpose of that of the return link: H^{T} is of size K × N. Let us stress that this reciprocity is strictly limited to the mathematical formalism of both links. In fact, the reciprocity between the forward and return links does not hold in practice, since they typically involve different frequency bands.
For a fair comparison of all forward link scenarios that will be considered in the sequel, it is critical to define a common transmit power constraint. For this, we assume the following constraint on the average power transmitted at the feed level:
where P_{ T } denotes the total transmit power.
In the case of a fixed onboard beamforming, we have that x = B^{T}x_{ b } , where x_{ b } is the stack of the onground transmitted signal in the beam space, keeping in mind that a perfectly calibrated and noiseless feeder link is assumed. The signal model (4) becomes
where {\text{H}}_{b}\doteq \text{B}\text{H} was defined above and expresses the principle of beamforming: the effect of the matrix B is essentially the linear combination of the radiation pattern of all N feeds to generate K beam radiation patterns.
3 Perfect CSI at the gateway
As a first step, let us assume that the gateway has perfect knowledge of the channel state, when acting either as transmitter or receiver. Throughout this section, we will establish measures of performance and show that, under linear combining, feed processing outperforms beam processing.
3.1 Return link
For the return link, and assuming onground feed processing, the MMSE combiner [21] that yields \widehat{\mathbf{s}}={\mathbf{W}}^{H}\mathbf{y} is
whereas the processing of the beams would entail \widehat{\mathbf{s}}={\mathbf{W}}_{b}^{H}{\mathbf{y}}_{b} with
A key objective of this study is to compare the performance of these two approaches. To accomplish this task, we will make use of the meansquared error (MSE) after combining, which is defined as \text{E}\left\{\mathbf{s}\widehat{\mathbf{s}}{}^{2}\right\}.
Let us denote Q_{ f } its covariance matrix, then it would read
for the case of feed processing. For the case of beam processing, it would be
with P = B^{H} (BB^{H} )^{1}B.
Since the SINR for the i th user is given by 1/Q_{ ii } 1[11], it makes sense to use the total MSE, given by {\sum}_{i=1}^{K}{\mathbf{Q}}_{ii}=\text{trace}\left\{\mathbf{Q}\right\}, as a performance metric. Then, it can be shown that
as follows. Let us express both traces as
and
where λ_{ i } (H^{H}H) denotes the i th largest eigenvalue of H^{H}H. We have that (11) is an immediate consequence of the following, stronger result.
Theorem 1 Let H and B^{H} be two tall matrices of the same size with full column rank. Let P be a projection matrix of B, that is, P = B^{H} (BB^{H} )^{1}B. Then, it holds that σ_{ i } (H) ≥ σ_{ i } (PH), with σ_{ i } (A) denoting the ith largest singular value of matrix A.
For the proof, see Appendix 2.
A sufficient condition for the traces to be equal is that PH = H. Since P is a projection matrix, this will happen whenever range(B^{H} ) = range(H). However, since B is fixed and H timevarying, it seems not possible to meet such condition. Recall that, even if the fading is negligible, the users are assumed to be located randomly into their beam spots. Thus, assuming constant H would require the feed pattern to be constant over each cell; it would also require users near the border of their cells to experiment almost the same interference as if they were located close to the center. On account of all these facts, ensuring range(B^{H} ) = range(H) is not realistic.
3.2 Forward link
As mentioned above, interference mitigation techniques in the forward link take the form of precoding at the gateway. In this article, we focus exclusively on linear precoding. In the case of onground feed processing, linear precoding is expressed as
with F the N × K precoding matrix, and s the K × 1 symbol vector. The k th entry of s is the unit energy constellation symbol destined to the k th user. To comply with the transmit power constraint (5), the precoding matrix F has to satisfy
In the case of adaptive linear precoding in the beam space, we write
where F_{ b } is the K × K precoding matrix, in terms of which the transmit power constraint (5) becomes
In this section, we assume a zeroforcing (ZF) precoder. The zeroforcing (ZF) criterion targets the complete cancelation of the interuser interference, by precoding with the pseudoinverse of the channel matrix. The corresponding expressions are
for the precoding in the feed space and beam space, respectively. The value of the constants β and β_{ b } has to be chosen such to comply with (15) and (17), respectively. Note that these particular versions of the ZF linear precoders are such that they equalize the signal to noise ratio (SNR) among users. The resulting SNR is given by
for the feed and beam processing, respectively. We prove here that the SNR achieved by the feed processing is always greater than or equal to that associated with the beam processing: SNR _{ f } ≥ SNR_{ b }. This is a direct consequence of the following property.
Theorem 2 Let H and B ^{H} be two tall matrices of the same size with full column rank. Then, the following inequality holds:
The equality is reached if H and B share the same left and right singular vectors, respectively.
For the proof, see Appendix 3.
As stated in the theorem, a sufficient condition for the beam processing not to suffer any performance loss with respect to the feed processing is that the matrices H and B share the same left and right singular vectors, respectively. However, in a similar way as for the return link, this condition is not likely to be met in practice since B is a fixed (non channeladaptive) beamforming matrix while H is timevarying (due to the random characteristic of the users positions).
4 Nonperfect CSI at the gateway
In a realistic scenario, the gateway does not know the actual values in the channel matrix H, but has only an estimate of them. The type and quality of these estimates, commonly based on the use of training sequences, will have an effect on the ultimate performance of the system. In this section, the estimation of the matrix H will be introduced. Some degree of uncertainty on the beamforming matrix B will also be discussed, while the noise power N_{0} will be assumed perfectly known.
4.1 Channel estimation by training sequences
For the estimation of the channel in the return link, each user employs a distinct training sequence, known as its unique word (UW). The gateway, upon reception of all the sequences, estimates the values in the channel matrix.
If perfect symbol synchronism can be assumed, then WalshHadamard sequences can be used as UWs, and it is possible to apply the pseudoinverse procedure [13] to estimate the channel matrix. However, due to the nature of the communication in the return link, symbol synchronism cannot be assumed.
In spite of this fact, the pseudoinverse procedure can still be used as long as good timing and frequency recovery is applied. Channel estimation under these circumstances would require the use of pseudorandom sequences with good crosscorrelation properties, rather than WalshHadamard sequences. Moreover, under accurate synchronization, pseudorandom sequences have been reported [13] to produce negligible correlation between the estimation errors of the different elements of the matrix.
On account of the previous statements, and in the absence of a fixed beamforming, channel estimation would be modeled as
where L is the training sequence length and W is a matrix with independent zeromean unitvariance Gaussian entries. For the case with beamforming onboard, it would read
In the forward link, the symbol synchronicity is not an issue anymore. We consider that the precoder design is now based on a feed channel estimate {\widehat{\mathbf{H}}}^{T}={\mathbf{H}}^{T}+\mathbf{E} (or {\widehat{\mathbf{H}}}_{b}^{T}={\mathbf{H}}_{b}^{T}+{\mathbf{E}}_{b} in the beam space). Each row of {\widehat{\mathbf{H}}}^{T}(or {\widehat{\mathbf{H}}}_{b}^{T}) is based on a channel estimation which is carried out separately at each user terminal and then reported to the gateway via a return channel (assumed ideal). Note that the reporting of {\widehat{\mathbf{H}}}^{T} to the gateway implies feeding back (NK )K more channel samples than for reporting {\widehat{\mathbf{H}}}_{b}^{T}. We assume Llength orthogonal training sequences, such that the entries of E (or E_{ b } ) are i.i.d. zero mean complex circular symmetric Gaussian random variables with variance inversely proportional to L.
4.2 Analog miscalibration
For the case of beam processing in the return link, the gateway must be aware of the exact beamforming weights that are set on board, since they will be necessary to compute the noise covariance matrix in the MMSE combiner (8). Even though these are subject of calibration, it is very likely that their actual values will experiment some minor changes through time, mainly because of the nonideal nature of the analog circuitry. As a consequence, the information at the gateway can be seen as an estimate of the actual beamformer. The following mathematical model is proposed:
where the entries in Δ B are of the form Δ B_{ ij } = b_{ ij } r_{ ij } and r_{ ij } are independent, real, zeromean Gaussian random variables. This models a variation, both in the real and imaginary parts of the weights, that is random with given variance, but proportional to the original value.
5 Simulation results
In order to further compare the performance of the proposed precoding and multiuser detection architectures, Monte Carlo simulations have been carried out according to the scenario described in Table 1. This scenario features K = 100 beams covering the whole Europe area. The satellite antenna pattern was provided by ESA, and corresponds to an array fed reflector antenna with N = 155 feeds. Matrix B was also provided by ESA, as a typical beamforming matrix of current systems. It was designed such as to limit the level of interference among users in a conventional system (without interference mitigation technique but with adequate frequency reuse pattern).
The user link has a total available bandwidth of 500 MHz; color schemes with frequency reuse factor equal to 3 and 1 were studied, corresponding to 166 and 500 MHz available bandwidth per beam, respectively. The reference scenario consists in a frequency reuse factor equal to 3, fixed beamforming and no processing at the gateway. Simulation results have been extracted for a number of interference mitigation techniques, both for the forward link and the return link. The purpose of this is to compare the performance of both architectures in as many different situations as possible. For illustrations purposes simulation results cover a large range of transmit powers, although it is important to stress that the most extreme values do not correspond to practical cases.
Results have been averaged for a total of 1,000 channel realizations, with the exception of those showing the average probability of nonavailability, which required 10,000 iterations to yield a reasonable confidence interval. Apart from the fading, the randomness of the channel is due to the position of the users, which are assumed to be uniformly distributed within each spot. For each realization, the SINR for each user after interference mitigation is computed, and its throughput is then inferred according to the preliminary specifications or DVBRCS2 in the RL and DVBS2 in the FL.
5.1 Return link
The user link operates at 30 GHz (Kaband), and is based on the DVBRCS2 standard [22]. The baudrate is 4 Msymb/s and the guardbands amount to the 11% of the carrier bandwidth [23]. Apart from the MMSE receiver presented in Section 3.1, the MMSESIC receiver has also been simulated, since it is known to be capacity achieving under ideal conditions, and therefore provides an upper bound on the achievable performance.
Figures 3 and 4 depict the evolution of the total average throughput as a function of the terminals EIRP. Results have been averaged only for those realizations in which the link was active. To this extent, Figure 5 shows the average probability of nonavailability for the different MUD techniques. It can be observed that a considerable increase in throughput is experimented thanks to using multiuser detection, although at the cost of some loss in availability. In fact, only SIC detection manages to reduce the outage probability with respect to the benchmark scenario.
Moreover, full onground processing reports higher throughput figures both with perfect and nonperfect CSI. To further investigate the potential advantage of this strategy, Figure 6 represents the performance gain obtained in this case with respect to the hybrid architecture, using training sequences of length 128 symbols. Let t_{ f } and ρ_{ f } be the average throughput and average availability, respectively, of the overall system when full onground feed processing is employed, and define tb and ρb as the corresponding counterparts in the beam processing case. The feed combining gain is defined as
It can be seen that, despite the existence of channel estimation errors, there are always nonnegligible improvements when choosing a full onground architecture. Moreover, results on the hybrid architecture assumed so far perfect knowledge of the fixed beamforming matrix. Recall now the error model (25) for the analog calibration, given by the error matrix Δ B ={b_{ ij }r_{ ij } } and let β be the variance of the random variables r_{ ij } . Following this model, Figure 7 depicts the evolution of the feed combining gain for different values of β, that is, for different degrees of uncertainty on the analog beamforming weights. Results account for more significant feed combining gains when some degree of uncertainty is present.
5.2 Forward link
The user link in the downstream is assumed to operate at 20 GHz (Kband), and is based on the DVBS2 standard. Besides the ZF precoder presented in Section 3.2, the following advanced precoders are considered and will be compared in the simulations:

The regularized channel inversion precoding [9].

The socalled UpConst MMSE precoder which is based on the uplinkdownlink duality [24]. This precoder solution was proposed in [3], where it is said to achieve a good compromise between throughput and availability.
Again, simulation results will use the average total throughput and availability as performance measures, but this time the total transmit power P_{ T } will be used as a parameter rather than the EIRP since the directivity of the feeds is part of the feed radiation pattern data provided by ESA.
Let us assume first that the channel is perfectly known at the gateway. Figures 8 and 9 first compare the regularized channel inversion and ZF linear precoder. The regularized channel inversion significantly outperforms the more naive ZF precoder. Most importantly, the benefit of the full onground architecture (i.e., feed processing) is apparent both in terms of throughput and availability. For instance, at P_{ T } = 30 dBW the regularized channel inversion in the feed space generates a 111% relative throughput increase with respect to the reference scenario, and 17% with respect to the same processing in the beam space. However, a slight decrease in system availability can still be observed with respect to the reference scenario. Figures 10 and 11 consider additionally the UpConst MMSE precoder, and illustrate again the benefit of the full onground architecture. It can be seen that the comparison between the regularized channel inversion and UpConst MMSE precoders depends on the value of the transmit power: in terms of availability, the regularized channel inversion outperforms the UpConst MMSE precoder for high values of the transmit power, and viceversa at low values of the transmit power.
We now disregard the assumption of perfect CSI at the gateway, and analyze the robustness of the different schemes to channel estimation errors. Figures 12 and 13 depict the achievable throughput and system availability for a training sequence length L = 256. The full onground architecture still appears beneficial, especially throughput wise. Moreover, from comparing Figures 11 and 13, it can be noticed that the regularized channel inversion appears to be more robust to imperfect CSI than the precoder based on the uplinkdownlink duality.
In Figure 14, we compare the feed combining gain (26) associated with the regularized channel inversion for different degree of CSI: perfect CSI, imperfect CSI with L = 1024, and L = 256. We can observe the robustness of the feed processing with regularized channel inversion. In fact, quite surprisingly, for moderate to high values of the transmit power the relative gain generated by the full onground architecture for this precoder increases as the degree of CSI decreases, reaching 24% with P_{ T } = 30 dBW and L = 256 symbols.
6 Conclusions
The results obtained in the previous sections show that feedlevel techniques tend to outperform beam processing ones, a fact that has been analytically proved for the case of perfect CSI. In the particular case of the return link, this gain may be rather small when the channel estimation errors are noticeable but the estimation of the onboard analog beamforming is accurate. On the contrary, if the uncertainty on the beamforming is high, then the advantage for using all the information from the feeds seems to be much higher: with the greatest level of uncertainty simulated, the feed combining gain reaches 20% at an EIRP 40 dBW. In what refers to the forward link, the uncertainty about the analog beamforming is not relevant, and the feed processing gain amounts to 24% with P_{ T } = 30 dBW and a realistic level of CSI. It is to be noticed, however, that working with the 155 feed signals would require rather more bandwidth in the feeder link. Therefore, there exists a tradeoff between performance and feeder link requirements, and the choice of the most suitable processing architecture would need to take into account all these considerations.
Appendix 1: Spatial correlation of rain fading
Next, we present a simple model for the rain fading correlation between two links to assess its potential impact on the interference mitigation schemes. To start with, we will follow the wellknown model which specifies that the joint distribution of rain attenuation on two slant paths, which we will call A_{1} and A_{2}, is lognormal and presents the following correlation factor [25]
where d is the horizontal distance between both points on Earth, {\sigma}_{i}^{2} is the variance of the marginal distribution of A_{ i } and c(d) is the correlation factor between the rain rates. According to [26], this factor may be accurately modeled as
where d_{0} is the distance at which c(d) = 1/e (usually called decorrelation distance) and s_{0} is a shape parameter. Although values for both variables must be set according to detailed environmental data, in [26] it has been shown that, for the instantaneous correlation between two points, s_{0} ≈ 1, yielding
As for d_{0}, different values have been proposed in the literature and, in any case, we must note that these will be highly dependent on the geographical area. For instance, in articles devoted to terrestrial communications like [27, 28], values around d_{0} = 0.46 km are reported, while [25] uses d_{0} = 1.844 km; [26] even reports values up to d_{0} = 7 km for the State of Oklahoma, although this value was obtained by averaging over a very long period of time.
We will now perform some simulations in order to check the evolution of r with respect to d for different values of d_{0}. Since, as stated above, we have assumed the same statistics for the marginal attenuations over each path, Equation (27) simplifies to
The results obtained are shown in Figure 15, where σ^{2} = 1.58. As we can see, the correlation factor quickly runs close to zero as the distance increases. In Figure 16, the same data is represented in log scale, in order to interpret such small values; the range of distances has also been multiplied by four. We can see that, even for the (far) most pessimistic case, the correlation values are very small, if not close to zero, after d = 60 km.
Appendix 2: Proof of Theorem 1
The goal is to prove
with σ_{ i } (A) representing the i th largest singular value of matrix A.
Consider now B = U ΣV^{H} , the singular value decomposition (SVD) of B, then
Since V is a unitary matrix, then it holds that
and
Let us define now A ≐ VH. On account of the previous statements, proving (31) is equivalent to proving
that is, each singular value of a matrix A is larger or equal to that of the same matrix after setting some rows to zero. To prove this fact, we will make use of the following property: let A be in general any tall matrix such that A = [a_{1}a_{2} ... a_{ k } ] and define A_{ r } = [a_{1}a_{2} ... a_{ r } ], then for all r from 1 to k  1 it holds that [29]
This interlacing property will prove useful for our purpose even though matrix A loses rows and not columns. Recall now that matrix A is of size N × N. If we write
with both block matrices of size K × N, and
then it is possible to define
whose singular values take the form \sigma \left(\stackrel{\u0303}{\mathbf{A}}\right)=\left[{\sigma}_{1}\left(\mathbf{A}\right){\sigma}_{2}\left(\mathbf{A}\right)\dots {\sigma}_{K}\left(\mathbf{A}\right)00\dots 0\right]. If we now remove columns from the right, the interlacing property tells us that
which implies that
and concludes the proof. □
Appendix 3: Proof of Theorem 2
In this appendix, we prove that
With the following singular value decomposition H = V Σ _{ H }U^{H} , the lefthand side in (42) can be simplified as
where {\sigma}_{k}^{2}\left(\mathbf{H}\right) denotes the k th largest singular value of H.
Similarly, denoting B = W Σ_{ B } Q^{H} , the righthand side in (42) can easily be worked out as
with the following definitions
where the submatrix Z_{1} is of size K × K , submatrices Z_{2} and {\mathbf{Z}}_{3}^{H} are K × (NK ), and submatrix Z_{4} is (NK ) × (N  K ). We denote the k th largest eigenvalue of {\mathbf{Z}}_{1}^{H}{\mathbf{Z}}_{1} by {\lambda}_{k}\left({\mathbf{Z}}_{1}^{H}{\mathbf{Z}}_{1}\right). Let us first realize that
Indeed, since Z is a unitary matrix, we have that Z^{H}Z = I_{ N }, which implies that
With the following eigenvalue decomposition {\mathbf{Z}}_{1}^{H}{\mathbf{Z}}_{1}=\mathbf{M}\phantom{\rule{2.77695pt}{0ex}}\text{diag}\left({\lambda}_{1}\left({\mathbf{Z}}_{1}^{H}{\mathbf{Z}}_{1}\right),\dots ,{\lambda}_{K}\left({\mathbf{Z}}_{1}^{H}{\mathbf{Z}}_{1}\right)\right){\mathbf{M}}^{H}, we can rewrite (48) as
Hence, the matrix {\mathbf{M}}^{H}{\mathbf{Z}}_{3}^{H}{\mathbf{Z}}_{3}\mathbf{M} has to be diagonal. Moreover, it has to have positive elements on the diagonal since {\mathbf{Z}}_{3}^{H}{\mathbf{Z}}_{3} is semi positive definite, which proves (47).
Finally, by Theorem H.1.h in [30], we have that
where (51) follows from (47), and concludes the proof.
Note that the inequality becomes an equality if Z = I_{ N } , that is, Q^{H}V = I_{ N }. In others words, the equality is reached if H and B share the same left and right singular vectors, respectively. □
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Acknowledgements
The authors would like to acknowledge the work of the University of Bologna and Technical University of Graz teams, in particular, M. Bergmann, W. Gappmair, E. A. Candreva, G.E. Corazza, F. Lombardo, I. Thibault, and A. VanelliCoralli. We are also in debt with Riccardo de Gaudenzi for his valuable comments. Research supported by ESA contract 23089/10/NL/CLP "SatNEx Network of Experts", the European Regional Development Fund (ERDF) and the Spanish Government under projects DYNACS (TEC201021245C0202/TCM) and COMONSENS (CONSOLIDERINGENIO 2010 CSD200800010), and the Galician Regional Government under projects "Consolidation of Research Units" 2009/62 and 2010/85. The work of B. Devillers was also partially supported by the Spanish Government under project TEC201017816 (JUNTOS). The work of A. PérezNeira had been supported by the Spanish Government under project TEC200806327C0301 and the Catalan Government under the grant 22009SGR0891. Some preliminary results of this study were presented at Asilomar and Globecom 2011 conferences.
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Arnau, J., Devillers, B., Mosquera, C. et al. Performance study of multiuser interference mitigation schemes for hybrid broadband multibeam satellite architectures. J Wireless Com Network 2012, 132 (2012). https://doi.org/10.1186/168714992012132
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DOI: https://doi.org/10.1186/168714992012132