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Multiuser Downlink Hybrid Analog-Digital

Beamforming with Individual SINR Constraints

Miguel ´

Angel V´azquez

, Luis Blanco

, Ana P´erez-Neira

⋆†

Centre Tecnol`ogic de Telecomunicacions de Catalunya (CTTC/CERCA)

Dept. of Signal Theory and Communications Universitat Polit`ecnica de Catalunya

Email:

{mavazquez, lblanco, aperez}@cttc.cat

Abstract—This paper deals with the problem of multiuser transmit hybrid analog-digital beamforming. Precisely, we tackle the problem of transmit power minimization under minimum signal-to-noise-plus-interference ratio (SINR) constraints. While the all-digital beamforming solution has been extensively studied in the recent years, little is known about the hybrid analog-digital optimization. This latter design is known to be a non-convex NP-hard problem as the analog part entails the optimization of a quadratically constraint quadratic program with several equality constraints. In order to solve this problem, we propose an alternating optimization with a successive convex approximation procedure. The conceived scheme is able to consider an arbitrary sub-array connectivity matrix (i.e. localized, interleaved, fully-connected,...). Through numerical simulations it is observed that the proposed framework yields to an efficient solution in few iterations providing a substantial gain compared to current heuristic approaches.

I. INTRODUCTION

One of the key enabling components of the fifth generation of wireless communications standards will be the transmission over millimeter wave (mm-wave) frequency bands. In these high frequency bands a very large bandwidth is available and the communication requires large antenna arrays for dealing with the huge path-loss. The spatial processing over these arrays cannot only be done through a digital processor due to the excess of cost and complexity. In order to solve this problem, academia and industry are studying hybrid analog-digital beamforming solutions which are known to offer a good cost-performance trade-off [1].

With the aim of increasing the spectral efficiency of the mentioned mm-wave bands, the hybrid analog-digital trans-mitter could simultaneously send different symbols to different receivers for taking advantage of the multiantenna interference mitigation, leading to the so-called multiuser multiple-input-multiple-output (MU-MIMO) operation. Despite MU-MIMO has been widely investigated in the sub-GHz bands [2] for all-digital beamforming schemes, the hybrid analog-all-digital MU-MIMO optimization is an open problem.

MU-MIMO hybrid analog-digital beamforming has been recently investigated in [3]–[7]. In all the mentioned works, the analog beamforming network is assumed to be fully-connected (i.e. from each radiofrequency (RF) chain there is a connection to each antenna) and equipped with phase shifters without amplitude control. The authors in [3] propose a design based on obtaining the analog part via known codebook beamformers while the digital part performs a zero forcing design. The mean square error is used as a figure of merit in [4] and

the analog beamforming part is computed by means of the orthogonal matching pursuit method. In [5] the use of zero forcing digital beamforming is again assumed but for this case the authors design the analog beamforming considering certain closed-form expressions that relate the phase-only precoder and the optimal weighted sum-rate digital design. Zero forcing digital beamforming is considered also in [6] but the analog beamforming just elects the phases equally to the channel matrix. Finally, the work in [7] the authors revisit the block-diagonal precoding design and they provide a design for analog phase-only solutions.

Among the aforementioned works, none of them consider the optimization problem of guaranteeing certain minimum SINR values to the different users with a hybrid analog-digital beamforming design. Although for all-digital transceivers the optimization problem is well investigated in [8]–[10], to the best of authors knowledge there is only the recent work in [11] that tackles the transmit power minimization with SINR constraints for hybrid analog-digital beamforming. Similarly to [6] the authors in [11] design the analog beamforming with the same phase as the channel matrix while optimizing the digital part for guaranteeing a per-user SINR.

In contrast to the previous works, this paper optimizes the problem directly without any a priori relaxation. Precisely, we propose an alternating optimization that while the digital beamforming is solved via the method described in [8], the analog part is solved via a successive convex approximation (SCA) method [12]. This analog beamforming optimization differs to preliminary work of the authors in [13] for cognitive beamforming optimization where a non-smooth method is presented.

The proposed framework is able to consider any arbitrary analog beamforming connectivity matrix. This extends the works in [3]–[7], [11] where only a fully-connected analog beamforming network can be conceived. This is of great importance as the fully-connected solution is known to be a costly and lossy alternative compared to the partially-connected schemes [1].

The rest of the paper is organized as follows. Section II presents the system and channel model. Section III describes the optimization problem and indicates how solve the analog and digital parts. Section IV presents the numerical results. Finally, Section V concludes.

Notation: Throughout this paper, the following notations are adopted. Boldface upper-case letters denote matrices and boldface lower-case letters refer to column vectors. (.)H,

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(.)T, (.) and (.)+ denote a Hermitian transpose, transpose,

conjugate and diagonal (with positive diagonal elements ) matrix, respectively. IN builds N × N identity matrix and

0K×N refers to an all-zero matrix of size K× N. If A is a N × N matrix. [X]ij represents the (i-th, j-th) element of matrix X. ⊗, ◦ and ||.|| refer to the Kronecker product, the Hadamard product and the Frobenius norm, respectively. Vector 1N is a column vector with dimension N whose entries are equal to 1. vec (·) denotes the vectorization operator.

II. SYSTEMMODEL ANDPROBLEMSTATEMENT

Let us consider a base station equipped with Q antennas transmitting K independent symbols to K receivers. The received signal by the k-th users can be modelled as

yk = hHk vksk+ Kj̸=k

hHj vjsj+ nk, (1)

where hk ∈ CQ×1 is the channel vector between the base station and the k-th receiver, vector vk ∈ CQ×1 denotes the beamforming that supports the transmission of the symbol sent to the k-th which is denoted by sk and assumed to be zero mean and unitary norm. Finally, nk is the additive white Gaussian noise with zero mean and variance equal to σk2 at the k-th receiver.

We consider a backhaul channel model scenario reported in [14] as ’above the roof top’ case. The channel vector in a backhaul scenario can be rewritten as

h =

S+1

n=1

αna(θn, ϕn), (2) where we avoid the subscribe k and S is the total number of scatters, αn is the complex gain of the n-th scatter, a∈ CQ×1 is the steering vector of the transmit antenna array. The steering vector depends on the angles of departure (AoD), θn, ϕn.

For n = 1 it is considered that the channel is deterministic and it can be obtained via a geometrical reasoning. Precisely, it is assumed that α1 = 1 and θ1, ϕ1, can be computed

by knowing the relative position of the transmitter and the receiver.

For n > 1, the channel offers a random behaviour based on the first ray (n = 1). This is, in [14] it is described that the amplitude values can be modelled as

αn = Aneψnj, (3) where An is Rayleigh distributed with mean γl/10 and ψn is uniformly distributed from 0 to 2π. Moreover, it is assumed that the number of scatters is fixed and it is S = 4. Finally, the AoDs for the different scatters n > 1 are perturbed by an additive Gaussian random variable of zero mean and 5 degrees of standard deviation.

The steering vector a depends on the antenna array struc-ture and the element spacing. Mm-wave links generally operate with planar arrays due to its high directivity. The simplest planar array representation is a uniform rectangular array (URA). These arrays are usually represented in matrix form

HPA HPA RF Chain RF Chain Baseband Processor 1 NRF 1 Q/NRF 1 Q w P HPA HPA Q/NRF

·

·

·

·

·

·

·

·

·

·

·

·

·

·

·

1 Q/NRF

Fig. 1. Localized hybrid analog-digital solution.

but, it is more convenient we consider its vector formulation as follows aURA(θ, ϕ) = vec ( u (θ, ϕ) v (θ, ϕ)T ) , (4)

where u (θ, ϕ) and v (θ, ϕ) are described in (5) and (6). Parameters dx and dy are the antenna distances in the x and y axis and Nxand Ny are the number of elements in the

x and y axis respectively. The azimuth angle is represented by ϕ∈ [0, π].

In contrast to all-digital designs where {vk}Kk=1 are de-signed to fulfil a sum-power constraint, in here we consider that each beamformer consists of an analog processing part

P∈ CQ×NRF, where N

RF is the number of RF chains, and a digital processing part wk∈ CNRF×1 so that

vk = Pwk k = 1, . . . , K. (7)

The hybrid analog-digital beamforming solutions present differences depending on how the connections between the RF chain and the antenna are performed. As a general statement, each RF chain of the digital part is connected with one or more antenna through an analog component. The most complex scheme is the one that considers an all-to-all connection (i.e. each RF chain is connected to all antennas through an analog component). For this case it is required QNRF connection lines and components.

In order to substantially reduce the number of connec-tions and analog components other connection matrices are conceived. In the rest of the paper we will consider two of them; namely, localized and interleaved. Whereas a localized architecture connects each RF chain with a subset of sequential antennas, the interleaved scheme interconnects the different RF chains with separated antennas. As the connection lines are longer in an interleaved scheme than in the localized one, the implementation complexity and losses are higher. Figures 1 and 2 describe the considered partially connected analog beamforming networks.

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u (θ, ϕ) = 1 Nx

(

1, ej2πλdxsin(θ) cos(ϕ), . . . , ej2πλ(Nx−1)dxsin(θ) cos(ϕ) )T

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v (θ, ϕ) = √1 Ny

(

1, ej2πλdysin(θ) sin(ϕ), . . . , ej2πλ(Ny−1)dysin(θ) sin(ϕ) )T (6) HPA HPA RF Chain RF Chain Baseband Processor 1 NRF 1 Q/NRF 1 Q w P HPA HPA NRF Q-NRF+1

·

·

·

·

·

·

·

·

·

·

·

·

1 Q/NRF

··

·

··

·

Fig. 2. Interleaved hybrid analog-digital solution.

Bearing in mind the above beamforming set up, this paper focuses on the following optimization problem

minimize P,{w}K k=1 Kk=1 ||Pwk||2 subject to |hH kPwk|2 ∑ j̸=k|h H k Pwj|2+ σk2 ≥ γk k = 1, . . . , K, P∈ P, (8) where {γk} K

k=1 denote the target SINR for each user k = 1, . . . , K.

We consider the following P depending on the analog beamforming connectivity matrix

Pfull:|[P]m,n|2= βfull, (9) Pinterleaved:|[P]m,n|2= [1κ⊗ βinterleavedINRF]m,n, (10) Plocalized:|[Pm,n]|2= [βlocalizedINRF ⊗ 1κ]m,n, (11) for m = 1, . . . , Q n = 1, . . . , NRF and κ = Q NRF , (12)

which is assumed to be an integer value. Moreover,

βfull, βinterleaved and βlocalized models the beamforming network

losses. For more details of these sub-array impairments, the reader can refer to [15].

Remarkably, as stated in [5] when NRF ≥ 2K and (9) is considered, the optimization problem in (8) admits a trivial solution based on the optimal all-digital design. However, either for partially-connected beamforming networks or when

NRF < 2K the optimal solution is still an open problem. For these latter cases, the following section describes a methodol-ogy for obtaining suboptimal solutions to the problem.

III. ALTERNATINGHYBRIDANALOG-DIGITAL

MULTIUSEROPTIMIZATION

As it is described in the following, a substantially lower computational complexity is required whether we consider the optimization of minimize P,{w}K k=1 ||P||2 Kk=1 ||wk||2 subject to |hH k Pwk|2 ∑ j̸=k|hHkPwj|2+ σ2k ≥ γk k = 1, . . . , K, P∈ P, (13)

instead of the original problem (8). Note that the optimization problem in (13) is an upper-bound of the original problem (8). In the next sections we describe the alternating optimiza-tion for solving the problem in (13). Firstly, the digital part and, later, the analog part optimization are addressed.

A. Digital Optimization

Given an arbitrary P, obtaining the optimal digital beam-forming entails solving

minimize {w}K k=1 Kk=1 ||wk||2 subject to |hH k Pwk|2 ∑ j̸=k|h H kPwj|2+ σ 2 k ≥ γk k = 1, . . . , K. (14)

This optimization problem has been widely studied pre-viously in [8]–[10]. In this work, we consider the solution presented in [8]. Remarkably, if we consider the original problem (8), the objective function would read∑Kk=1||Pwk||2 instead. For that case, none of the approaches in [8]–[10] can deal with the problem. Indeed, other more computationally demanding approximation methods as the one reported in the following might be used for obtaining an efficient solution. This is the main reason of the approximation in (13).

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B. Analog Optimization

Given an arbitrary set of digital beamformers {wk}Kk=1, the optimal analog beamforming design is obtained with

minimize ||P||2 subject to |hH k Pwk|2 ∑ j̸=k|h H kPwj|2+ σ2k ≥ γk k = 1, . . . , K, P∈ P. (15)

This problem can be rewritten so that

minimize p ||p|| 2 subject to |hH kWkp|2 ∑ j̸=k|h H k Wjp|2+ σ 2 k ≥ γk k = 1, . . . , K, |[p]q|2= [vec (C)]q q = 1, . . . , QNRF, (16) where p = vec (P) , (17) Wk= IQ⊗ wTk ∈ C Q×QNRF. (18)

Moreover, matrix C collapses the connectivity schemes such as

Cfull= βfull1Q⊗ 1TNRF, (19)

Cinterleaved= 1κ⊗ βinterleavedINRF, (20)

Clocalized= βlocalizedINRF ⊗ 1κ. (21) From (16) it is easy to observe that neither the SINR nor the equality constraints are convex. In order to circumvent this problem, we use a SCA technique similar to the ones reported in [12], [16]. Both methods consider the linearization of the non-convex functions of the problem and they include additional supporting slack variables.

We can re-write the problem in (16) such that

minimize p ||p|| 2 subject to pHJkk− γk  ∑K j̸=k Jkj+ σk2I     p ≥ 0 k = 1, . . . , K, |[p]q|2= [vec (C)]q q = 1, . . . , QNRF, (22) where Jkj= WjHhkhHk Wj. (23) In (22) it can be observed that all the constraints are not convex since the equality constraints are quadratic and the inequality ones could be concave. The idea of the SCA ap-proach is to linearize the aforementioned non-convex parts of the optimization problem by its linear convex approximation. According to [16], a negative definite quadratic form with matrix X∈ CM×M can be expanded by

pHXp≤ 2Re(zHXp)− zHXz, (24)

where z∈ CM×1 is an arbitrary complex vector.

With this, it is possible to optimize (22) by linearising all the constraints like in (24) and successively optimizing the linearized optimization problem. In addition to this, it is essential to obtain an initial feasible point to proceed with the SCA. Since an initial feasible point is not available, the authors in [12], [16] recommend the use of slack variables.

Bearing in mind the convex linearisation and the use of slack variable, the original problem (22) becomes (28) where

A(+)k = Jkk, (25) A(k−)= γk  ∑K j̸=k Jkj+ σ2kI , (26) c = vec (C) . (27)

Moreover, s, t, u denote the slack variables and Emis a M×

M matrix whose entries are equal to zero apart from its m-th

diagonal entry which is equal to 1.

The optimization problem in (28) requires at the (t + 1)-th step the previous solution at step t, z(t). The overall method (22) is depicted in Algorithm 1 and we shortly describe it in here.

As it can be observed, the scheme sequentially solves different second order cone programs (SOCP) based on the previous solution z. As general non-convex problems, the performance of the algorithm, strongly depends of how the initial point, z(0), is close to the optimal value. The parameter

φ balances the constraint fulfilling and the objective function.

For this case, we set an initial low value of φ and it is posteriorly increased by an additive factor ρ. The parameter

χ controls the convergence of the solution.

Data: z(0) which can be randomly obtained and φ(0) Result: p initialization ; while ||p||(t)− ||p||(t−1) ≤µ and 1T Ks + 1 T KQt + 1 T KQu≤ χ do if t < Tmax then Compute p(t) according to (28).; z(t+1)← p(t); β(t+1)← φ(t)ρ; t← t + 1; else t← 0;

Initialize with a new random value z(0);

Set up φ(0) again; end

end

Output the final solution;

Algorithm 1:Modified SCA optimization for analog beam-forming.

It is important to remark that the method described in Algorithm 1 is not ensured to converge to a feasible point as also reported in [12]. Due to that we set a maximum number of iterations and, in case they are reached, the method re-starts with a new random initial point.

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minimize p,s,t,u ||p|| 2+ φ(1T Ks + 1 T QNRFt + 1 T QNRFu ) subject to pHA(+)k p + 2Re { z(t),HA(k−)p } − z(t),HA(−) k z (t)≤ [s] k k = 1, . . . , K, pHEmp≤ [t]m+ [c]m m = 1, . . . , QNRF, pHEmp≤ [u]m+ [c]m+ 2Re { z(t),HEmp } m = 1, . . . , QNRF, [s]m≥ 0 m = 1, . . . , K [t]m≥ 0 m = 1, . . . , QNRF [u]m≥ 0 m = 1, . . . , QNRF. (28)

Bearing in mind both methods for solving the analog and digital parts respectively, we consider Algorithm 2 as a technique for obtaining efficient hybrid analog-digital beam-forming designs.

Data: P(0) randomly obtained

Result: w and P initialization ; while ||P(n)\W(n)|| − ||P(n+1)W\(n+1)|| ≤ ψ do Compute { wk(n+1) }K

k=1 according to (14) with the method described in [8] and considering P(n), ;

Compute P(n+1) according to Algorithm 1 considering { wk(n+1) }K k=1.; n← n + 1; end

Output the final solution;

Algorithm 2: Alternating analog-digital optimization. The parameter ψ controls the convergence of the solution and

c

W = (w1, . . . , wK) . (29) Note that there is no theoretical evidence that Algorithm 2 convergences due to the non-convexity of the analog beam-forming optimization problem. However, as it is reported in the following, all the considered cases lead to stable solution under the simulation set up described in the following section.

IV. NUMERICALRESULTS

This section presents the numerical evaluation of the pro-posed method. In order to properly assess its performance, we consider as a benchmark a low complex alternative as we describe in the following. Note that there is no other method in the literature for tackling this problem considering an arbitrary sub-array structure as the proposed method does.

Let us consider the optimization problem in (14) but, in-stead of proceeding with the proposed alternating optimization method, we consider a low complex closed-form approach. Similarly to [6], we consider the following analog beaforming solution

[Pbenchmark]m,n= [C]m,ne

j∠{[HH]m,n}, (30) for m = 1, . . . , Q n = 1, . . . , K and where∠ {a} denotes the angle of the complex number a. In other words, we assume

that the analog beamforming network tries to steer its beams towards the different user directions. Note that the steering is imperfect as the entries of H can have variations on its amplitude values. This approach has also been used in [6], [11].

Under this context, the proposed benchmark scheme con-sists of the digital solution obtained via solving (14) with (30) as beamforming network. As stated previously, the solution to this optimization problem is known and we consider the approach in [8] for solving it.

For the sake of simplicity, we consider that all users have the same targeted SINR:

γ1= γ2= . . . = γK = γ. (31) We have considered a channel model with ideal path loss equal to one and with θ1and ϕ1uniformly distributed between

0 and 10 degrees and -60 and 60 degrees respectively. This angle of arrival distribution is known to model the channel of rural backhauling areas [17].

In addition, we set the following parameters of the pro-posed method φ(0) = 10, ρ = 2, µ = 10−3, ψ = 10−2,

χ = 10−2, Tmax = 30. These values have been obtained

through different simulation trials and they are the ones that offer the best simulation time versus performance trade-off. As an initial point, we consider

z(0)= vec (Pbenchmark) . (32)

Figure 3 shows the transmit power for different sub-array connectivities and γ while keeping Q = 64 and NRF =

K = 4. The results are obtained via 500 Monte Carlo runs

and show that the proposed method for jointly optimizing the analog and digital parts offers a large transmit power reduction compared to the heuristic approach mentioned earlier. This gain is larger for the full connected case than the partially connected approaches.

Considering the results depicted in Figure 3, the costly fully connected beamforming network offers a large gain in terms of transmit power for both the benchmark and the proposed approach. Note that there are no notorious difference between the two considered partially connected schemes (localized and interleaved). Furthermore, in all cases and γ values the proposed alternating optimization converged more than the 95% of the times.

For the sake of completeness, we include an additional simulation set up with Q = 25 and NRF = K = 5 in

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Figure 4. Due to the decrease of antennas and the number of users to be served, in this setting we observe a larger power consumption compared to the previous one. Still the proposed method yields to a substantially lower transmit power compared to the heuristic approach.

2 4 6 8 10 −45 −40 −35 −30 −25 −20 −15 Target SINR [dB] T ra n sm it P ow er [d B W a tt s] Hybrid Full Benchmark Full Hybrid Localized Benchmark Localized Hybrid Interleaved Benchmark Interleaved

Fig. 3. Transmit power versus targeted users SINRs (γ) assumed to be the same for all cases. It is considered Q = 64 and NRF = K = 4.

2 4 6 8 10 −10 −5 0 5 10 15 20 25 30 35 Target SINR [dB] T ra n sm it P ow er [d B W a tt s] Hybrid Full Benchmark Full Hybrid Localized Benchmark Localized Benchmark Interleaved Hybrid Interleaved

Fig. 4. Transmit power versus targeted users SINRs (γ) assumed to be the same for all cases. It is considered Q = 25 and NRF = K = 5.

V. CONCLUSIONS

This paper proposes an optimization framework for opti-mizing hybrid analog-digital multiuser problem. Precisely, we focus on the problem of transmit power minimization while keeping the user SINR values above a certain threshold. An iterative two-step procedure is presented where the analog and digital parts are sequentially solved. While the digital optimization is a well known problem, the analog optimization requires novel approximation methods for dealing with the non-convex parts of the optimization problem. The resulting scheme leads to an efficient solution in few iterations and it yields to a substantial transmit power reduction compared to other heuristic approaches.

ACKNOWLEDGMENT

This work has received funding from the European Unions Horizon2020 research and innovation programme un-der grant agreement No 645047(SANSA); the Spanish Min-istry of Economy and Competitiveness (Ministerio de Econo-mia y Competitividad) under project TEC2014-59255-C3-1-R (ELISA); and from the Catalan Government (2014SGR1567 and 2014SGR1551)

REFERENCES

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[3] A. Alkhateeb, G. Leus, and R. W. Heath, “Limited Feedback Hybrid Precoding for Multi-User Millimeter Wave Systems,” IEEE Transac-tions on Wireless CommunicaTransac-tions, vol. 14, no. 11, pp. 6481–6494, Nov 2015.

[4] T. E. Bogale and L. B. Le, “Beamforming for multiuser massive MIMO systems: Digital versus hybrid analog-digital,” in 2014 IEEE Global Communications Conference, Dec 2014, pp. 4066–4071.

[5] F. Sohrabi and W. Yu, “Hybrid Digital and Analog Beamforming Design for Large-Scale Antenna Arrays,” IEEE Journal of Selected Topics in Signal Processing, vol. 10, no. 3, pp. 501–513, April 2016.

[6] L. Liang, W. Xu, and X. Dong, “Low-Complexity Hybrid Precoding in Massive Multiuser MIMO Systems,” IEEE Wireless Communications Letters, vol. 3, no. 6, pp. 653–656, Dec 2014.

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[8] M. Schubert and H. Boche, “Solution of the multiuser downlink beam-forming problem with individual SINR constraints,” IEEE Transactions on Vehicular Technology, vol. 53, no. 1, pp. 18–28, Jan 2004. [9] F. Rashid-Farrokhi, K. J. R. Liu, and L. Tassiulas, “Transmit

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[10] M. Bengtsson and B. Ottersten, “Optimal Downlink BeamformingUsing Semidefinite Optimization,” in 37th Annual Allerton Conference on Communication, Control, and Computing, 1999, pp. 987–996. [11] S. Malla and G. Abreu, “Transmit power minimization in multi-user

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[12] T. Lipp and S. Boyd, “Variations and extension of the convex– concave procedure,” Optimization and Engineering, vol. 17, no. 2, pp. 263–287, 2016. [Online]. Available: http://dx.doi.org/10.1007/ s11081-015-9294-x

[13] V´azquez, Miguel ´Angel and Blanco, Luis and Artiga, Xavier and P´erez-Neira, Ana, “Hybrid analog-digital transmit beamforming for spectrum sharing satellite-terrestrial systems,” in Signal Processing Advances in Wireless Communications (SPAWC), 2016 IEEE 17th International Workshop on. IEEE, 2016, pp. 1–5.

[14] A. Maltsev and et al, “D5.1 - Channel Modeling and Characterization,” MiWEBA Project (FP7-ICT-608637), Public Deliverable, Jan. 2014. [15] V´azquez, Miguel ´Angel and Artiga, Xavier and P´erez-Neira, Ana,

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[16] O. Mehanna, K. Huang, B. Gopalakrishnan, A. Konar, and N. D. Sidiropoulos, “Feasible point pursuit and successive approximation of non-convex QCQPs,” IEEE Signal Processing Letters, vol. 22, no. 7, pp. 804–808, 2015.

[17] G. Ziaragkas and et al, “D2.3 - Definition of reference scenarios, overall system architectures, research challenges and requirements and Key Performance Indicators,” SANSA Project (H2020-645047), Public Deliverable, Jan. 2016.

References

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