NetworkMIMO for downlink inband relay transmissions
 Sandra Lagen^{1}Email author,
 Adrian Agustin^{1} and
 Josep Vidal^{1}
https://doi.org/10.1186/16871499201313
© Lagen et al.; licensee Springer. 2013
Received: 9 March 2012
Accepted: 21 December 2012
Published: 23 January 2013
Abstract
With the objective of improving the spectral efficiency and coverage homogeneity of wireless cellular systems in the downlink, we investigate how to take advantage of two promising transmission technologies envisioned in current standards: networkMIMO and relay stations (RSs). It is assumed that halfduplex RSs are deployed in a cellular system, where the duration of the relayreceive and the relaytransmit phases is fixed beforehand for all cells. In order to reduce the spectral efficiency loses associated to halfduplex relaying, we propose the use of base station (BS) cooperation under a networkMIMO precoding strategy based on blockdiagonalization zeroforcing, and optimize radio resources under the convex performance criteria constrained by the perBS power, modulation and coding schemes and the transmission rate in the relaytransmit phase. By applying convex optimization techniques, the optimal precoding strategy is derived and a suboptimal low complexity solution is also proposed. The obtained solutions are evaluated at system level and compared to other cooperative and noncooperative BSbased schemes.
Keywords
NetworkMIMO Relay transmissions Blockdiagonalization zeroforcing Convex optimization1. Introduction
Due to the availability of new sophisticated terminals and bandwidthdemanding services, future cellular network designs are expected to provide high system spectral efficiency and homogeneous coverage. The design of such advanced network architectures is a challenging task, since radio resources like power and bandwidth are often scarce and the wireless channel presents unique impairments such as fading, multipath, and interference. In this regard, next generation standards are already considering that conventional paradigms need to be rethought in a way able to exploit multiantenna capacity increase, shadowing diversity, and path loss break concept. For that reason, mature enabling technologies, like multipleinput multipleoutput (MIMO) or full frequency reuse planning, are an integral part of the system. However, more efficient wireless technologies and novel deployment concepts, like RSbased deployments or coordinated base station (BS) transmissions (often called networkMIMO or coordinated multipoint), are part of the ambitious study items [1] to deal with the demanding traffic growth expected in the near future. Leveraging on the advantages offered by the joint use of all these techniques, to improve performance and cost efficiency, is a challenge faced by IEEE 802.16 m [2] and LTEA [3].
One way to increase network capacity is by means of increasing the radio access points and reducing the cell size. However, since a regular BS is expensive, this solution is not attractive to network operators. Pursuing the same objective with lower cost, relaybased deployments is a promising approach that will have a large effect on system’s functions and architectures [4, 5]. While implementation details of fullduplex RS are still under investigation, relaybased enhancements in standards consider halfduplex relay operation [5], which incur a rate penalty as they require at least two timeslots to relay a message from source to destination [6, 7]. It is therefore crucial to enhance the capacity of the inband wireless backhaul between source and relay (in our case, the BS–RS link) to increase the information rate. One of the solutions usually assumed is that relay stations (RSs) are placed in specifically planned positions above rooftop or in lampposts, ensuring lineofsight (LOS) conditions in the BS–RS link, and hence reducing the path loss and shadowing effects. However, the price to pay is twofold: the likely LOS propagating conditions also to othercell BSs (which will inject harmful interference) and the rank deficiency of the spatial channel when both BS and RS are equipped with multiple antennas. Both effects are detrimental to MIMO channel gains [8, 9].
In this respect, networkMIMO [10] seems especially suited to address inband backhauling in relay transmissions for the downlink. While coordination may be seen as an efficient way to combat the interference from neighbor cells [11, 12], it also creates a virtual MIMO broadcast channel whose number of degrees of freedom is boosted (if compared to a conventional singleuser MIMO under TDMA) and is hardly affected by the rank deficiency of singleuser MIMO channels in LOS. It has been observed that networkMIMO based on blockdiagonalization zeroforcing (BDZF) performs closely to dirtypaper coding [10] but, although its simplicity, it requires accurate channel knowledge from all involved links. However, networkMIMO is again appropriate for our problem, thanks to the long channel coherence time of BS–RS links.
An additional way to improve the efficiency of relay transmissions is by optimizing the duration of the relayreceive and the relaytransmit phases [6, 7]. In [13], we observed that the joint optimization of coordinated BS–RS links (through networkMIMO) and transmit duration phases brings large benefits. However, this approach is not convenient when considering multiple coordinated cells: if each group of coordinated cells adapts the duration of the transmission independently, the interference power observed in each transmission slot may be timevarying, a harsh, and undesirable situation for the cellular system. Therefore, so as to guarantee the predictability of other cellclusters interference within transmission frames, the transmit duration phases should be fixed beforehand based on some criterion. For example, in [14], a statistical criterion was used and it was observed that still large gains both in terms of spectral efficiency and outage rate could be obtained.

How much a relaybased system can benefit in terms of cellular spectral efficiency and celledge spectral efficiency by BS cooperation?

How much gain can be achieved with the optimal MIMO symbol precoding compared to suboptimal and simpler precoding solutions?
This article is structured as follows. In Section 2, we state the system model. The resource allocation problem is formulated in Section 3, in which we derive the optimal precoding strategy and also a suboptimal lowcomplex precoding strategy is proposed. In Section 4, simulation results are presented. Conclusions are drawn in Section 5.
Notation: For a given matrix A, its transpose matrix is denoted as A^{T} and its transpose conjugate matrix as A^{H}; A operator is referred to the determinant of A, tr(A) to the trace, rank(A) to the rank (which is the minimum between the column and the row rank) and kernel(A) is denoting the null space of the matrix; righteigenvec(A) denotes a matrix containing the right eigenvectors of A and eigenval(A) denotes a matrix with the eigenvalues of A in its diagonal; A ≥ 0 means that matrix A is positive semidefinite; A = diag(a_{1},…,a_{ N }) describes a diagonal matrix with diagonal entries a_{1},…,a_{ N }. For given scalars a and b, min (a,b) denotes the minimum between a and b, and (a)^{+} denotes the maximum between a and 0.
2. System model
This section presents the model for a relayedbased deployment with networkMIMO applied to the inband backhaul. In Section 2.1, some necessary system assumptions are detailed. Signal model in BS–RS link and signal model in RS–UE link are devoted to Sections 2.2 and 2.3, respectively.
2.1. System assumptions
 1.
The number of antennas at BS, RS, and UE is n _{B}, n _{R}, and n _{M}, respectively.
 2.
The quality of the backhaul link supports the exchange of perfect channel state information at the transmitter side (CSIT) of the first hop among the cooperating BSs. CSIT is assumed for transmissions from BSs to RSs since long time coherence is expected in those links as both terminals have static positions.
 3.
A maximum possible MCS is assumed.
 4.
RSs are time halfduplexed terminals operating under decodeandforward. That is a suitable coding approach for BS–RS links, where high signaltonoise ratio (SNR) is expected if LOS propagation is met [16].
 5.
User equipments (UEs) do not process the signals transmitted by the BS, only those transmitted by the RS. In other words, simple forwarding relaying is assumed.
 6.
No CSIT is available at RSs. RS transmissions are not coordinated in the way BS transmissions are. Their transmissions are interfered as multiple RSs transmit simultaneously (see Figure 1).
 7.
Each RS transmits to a single associated UE and therefore it is considered as a single user MIMO link with interference. Full CSIT may be exploited at the RS if sufficient feedback rate from the UE is allowed. Otherwise, only average CSIT is assumed.
 8.
It is assumed that the duration of slots for the BS–RS links and for the RS–UE links is constant to preserve interference stationarity between cell clusters.
 9.
The scheduler implements a roundrobin TDMA algorithm to serve users in groups of R users.
 10.
The radio resource allocation (transmitted power, beamformers) is performed by prioritizing each UE selected by the scheduler, so that a certain quality of service (QoS) is provided. A reasonable criterion is that user’s priority is taken as the inverse of the capacity in the RS–UE link so as to avoid unfair service to deprived users. This way, in comparison with the sumrate (SR) criterion (equal priorities to all users), a weighted SR (WSR) criterion would perform better in terms of outage rate while losing in terms of spectral efficiency.
2.2. Signal model in BS–RS link
We shall adopt a downlink transmission setup where B BSs are coordinated and assisted by R RSs to transmit messages to R UEs. Each UE is associated to a single RS. All BSs transmit on a fixed fraction of time α_{1} on the first hop to the RSs following a networkMIMO strategy based on BDZF [17] (see Figure 1), which is appropriate for BS–RS links in LOS conditions (MMSE precoding provides improved performance at medium to low SNR [18]).
where n_{ i } is the additive complex white Gaussian noise.
It must be remarked that in the case the i th RS observes all coordinated BSs in LOS (hence BS–RS link channels are rank deficient) the rank of H_{ i } grows up to fullrows rank provided that channels to the B BSs be linearly independent.
where S_{max} is the maximum spectral efficiency that can be achieved by the LTEA MCS [3] for each transmission mode.
2.3. Signal model in RS–UE link
On the second hop, each RS transmits to its associated UE on a fixed fraction of time α_{2} = 1 – α_{1} (see Figure 1). Since R relays can transmit simultaneously, interference is created. We could also split the second hop in different time slots, reducing interference at the expenses of some loss in spectral efficiency [13].
where r_{2i} denotes the rate in the i th RS–UE link, ${P}_{i}^{R}$ is the power transmitted by the i th RS to its UE, and ${P}_{j}^{R}$ defines the power transmitted by the j th RS.
3. Resource allocation
This section is devoted to find the optimal power allocation and beamforming vectors for BS–RS links (both included in the MIMO symbol precoder S_{ i }) in order to maximize the WSR of the system, when there are perBS power constraints and maximum MCS. Following similar steps as in [12] we obtain the optimal solution in WSRbased optimal resource allocation. Similarly, in WSRbased suboptimal resource allocation, a suboptimal and simpler solution for S_{ i } is presented, which assumes the conventional structure for the beamforming in networkMIMO, [15], and then the power is allocated so as to satisfy the power and rate constraints [14].
3.1. WSRbased optimal resource allocation
where r_{1i} is the transmission rate in the BS–RS link for the i th UE and μ_{ i } the priority given to it in order to add some QoS over the served users. The first constraint accounts for the perBS power constraint (taken from (8)), where ${P}_{k}^{\mathit{\text{max}}}$ denotes the perBS available power at the k th BS. The second constraint refers to the peruser transmission rate in the BS–RS link. The third constraint contains the peruser transmission rate, which is limited by the peruser transmission rate in the RS–UE link (from Equation 12) and the peruser MCS constraint in the BS–RS link (from Equation 7). Finally, the fourth constraint imposes that the transmission rates have to be positive, and the fifth constraint imposes that the transmit covariance matrices have to be positive and semidefinite.
With the inclusion of the new variable r_{1i} in the problem, it can be verified that problem (P_{1}) turns out to be a convex optimization problem (as the objective function is convex over {r_{1i}} and all the constraints specify a convex set over T_{ i }’s and {r_{1i}} [19]). Then, problem (P_{1}) can be solved using standard convex optimization techniques, e.g., interiorpoint method [19]. However, the optimal structure for T_{ i } and r_{1i} can be found in a closedform using Lagrange duality methods [19, 20].
Proposition 1
Proof. See Appendix 1.
Once we have the optimal solution for T_{ i }’s and {r_{1i}} for a set of fixed {γ_{ k }} and {φ_{ i }}, we have to optimize {γ_{ k }} and {φ_{ i }}. The set {φ_{ i }} can be optimized applying the KKT conditions [19].
Proposition 2
where the second result is obtained by substituting the expression of the transmission rate in (15) into the third constraint of problem (P_{1}) and equaling the restriction. Notice that Equation (16) depends on the number of transmission modes per user (m_{ i }) and, in addition, we need to evaluate r_{1i} with Equation (15) which involves a waterfilling algorithm. In this regard, an algorithm to find φ_{ i }^{*}, which is based on hypothetic testing, is presented in Algorithm 1 (see Appendix 1 for details).
Algorithm 1: Efficient search of φ_{ i }for a set of fixed {γ_{ k }}
Inputs: {γ_{ k }}
 1.
Set m _{ i } = rank (H _{ i }V _{ i })
 2.
Compute p _{ ij } ({γ _{ k }}, 0) and then r _{1i} ({γ _{ k }}, 0) using (15)
 3.
if (r _{1i}  min(α _{2} r _{2i}/α _{1}, S _{max} m _{ i }) ≤ 0)
 4.
• ${\phi}_{\mathit{i}}^{*}=0$
 5.
• go to 17
 6.
else
 7.
• compute φ _{ i } using (16)
 8.
• updated p _{ ij } ({γ _{ k }}, φ _{ i }) using (15)
 9.
• ${m}_{i}=\mathit{\text{count}}\left({P}_{ij}>0,\forall \mathit{j}\right)$
 10.
$\mathbf{\text{if}}\left({m}_{i},=,{m}_{i}\right)$
 11.
◦ ${\phi}_{\mathit{i}}^{*}={\phi}_{\mathit{i}}$
 12.
◦ go to 17
 13.
else
 14.
◦ $\left({m}_{i},=,{m}_{i}\right)$
 15.
◦ go to 2
 16.
end
 17.
end
Proof. See Appendix 1.
Now, problem (P_{1}) has to be optimized in terms of {γ_{ k }}. In this regard, an effective algorithm is proposed in Algorithm 2 to compute {γ_{ k }} following the ellipsoid method [19]. For each set of {γ_{ k }} the optimal solution for T_{ i }’s, {r_{1i}}, and {φ_{ i }} are computed following (14), (15), and (16) (see details in Appendix 1).
Algorithm 2: Efficient search of {γ_{ k }} and {φ_{ i }} in problem P_{1}
 1.
Initialize: γ _{ k } k = 1,…, B
 2.
do
 3.
for i = 1,…,R
 4.
◦ ${\phi}_{\mathit{i}}^{*}$ = Algorithm 1({γ _{ k }})
 5.
◦ Compute T _{ i } using (14)
 6.
end
 7.• Compute the subgradient${d}_{k}=\text{tr}\left({\sum}_{i=1}^{R}{\mathbf{\text{B}}}_{k}{\mathbf{\text{V}}}_{i}{\mathbf{\text{T}}}_{i}{\mathbf{\text{V}}}_{i}^{H}\right){\mathit{P}}_{\mathit{k}}^{\text{max}}$
 8.
• Update ${\gamma}_{k}\mathrm{\text{'}}\text{s}$ accordingly based on the ellipsoid method [19]
 9.
until all ${\gamma}_{k}\mathrm{\text{'}}\text{s}$ converge
 10.
Compute optimal ${\mathbf{\text{T}}}_{i}^{*}$ and r _{1i} given by (14) and (15).
It is worth to remark that S_{ i }^{*} consists of nonorthogonal columns if B_{γ} is a nonidentity diagonal matrix (i.e., the optimal {γ_{ k }} are not all equal). Moreover, it can be verified that for the sumpower constraint case, S_{ i }^{*} in (17) contains orthogonal columns (i.e. orthogonal beamforming) since V_{ i }^{ H }V_{ i } = I, and hence its structure turns out to be independent of γ. For the same reason, when the optimal solution for S_{ i }^{*} (17) is used, the computational complexity is significantly higher than for the sumpower constraint case as singular value decomposition (SVD) must be computed for each algorithm’s iteration (i.e., for each value of {γ_{ k }}).
3.2. WSRbased suboptimal resource allocation
where ${\tilde{\mathbf{W}}}_{i}^{k}$ contains the n_{B} rows of ${\mathbf{V}}_{i}{\widehat{\mathbf{W}}}_{i}$ used by the k th BS in the transmission of message to the i th RS and ${\tilde{w}}_{\mathit{ij}}^{k}$ is the j th column of ${\tilde{\mathbf{W}}}_{i}^{k}$.
where ${\widehat{r}}_{1i}$ is the transmission rate in the BS–RS link for the i th UE and μ_{ i } the priority given to it. The first constraint accounts for the perBS power constraint, see (19), where P_{ k }^{max} denotes the perBS available power at the k th BS; the second constraint refers to the peruser transmission rate in the BS–RS link; the third constraint contains the peruser transmission rate in the RS–UE link (from Equation 12), and the fourth constraint is related to the peruser MCS constraint in the BS–RS link (which is here included directly to the power to be allocated). Finally, the fifth and sixth constraints account for the positiveness of the user’s rate and allocated power.
Problem (P_{2}) is convex and can be solved using standard convex optimization techniques, like interior point methods [19]. Nevertheless, in [14] we elaborate an efficient numerical algorithm based on the dual update methods [19, 20] which defines a polynomial complexity algorithm along with a reduction of the number of variables to be optimized.
Proposition 3
Proof. See [14], where also an algorithm is proposed to effectively update {ϕ_{ k }} and {κ_{ i }} in (21).
4. Evaluations and results
Overview of simulation parameters
Parameter  Value 

Central frequency  2 GHz 
Channel bandwidth and duplex method  20 MHz in TDD 
Macrocell order  B = 3 
Inter site distance (ISD)  500 m 
Number of sectors  3 
Number of relays per sector  2 (and hence R = 6) 
Sectors configurations  Convergent sectors 
Antenna height BS/RS/UE  25 m/10 m/1.5 m 
Transmit power at BS/RS  46 dBm/37 dBm 
Noise spectral density  174 dBm/Hz 
Maximum number of beams  n_{B} = 4 
Antenna elements at RS/UE  n_{R} = 2/n_{M} = 2 (and thus m_{ i } = 2) 
BS/RS antenna gain  14 dBi/5 dBi 
Noise factor at BS and RS  7 
Noise figure at UE  9 dB 
Total losses at BS/RS/UE  6 dB/2 dB/2 dB 
Duration of the relayreceive phase for simulation results
α_{1}(BS–RS link)  

Homogeneous scenario  
Relayed networkMIMO  1/8 
Relayed BDZFTDMA  2/8 
Relayed BFTDMA  3/8 
Hotspot scenario  
Relayed networkMIMO with MCS constraints  4/8 
Relayed BDZFTDMA with MCS constraints  6/8 
Two fundamental measures are adopted: cellular spectral efficiency (S_{ e }), as the sum rate of R users divided by 3 (there are R users and 3 cells deployed per macrocell) averaged over many deployments, and 95% outage rate (r_{ out }) (or celledge spectral efficiency), as the peak achievable rate of the 5percentile worst users in the cell over many deployments. Both capture most of the benefits offered by coordination and cooperation of BS and relaybased transmission.
4.1. Outdoor homogeneous scenario
In a first deployment, all RSs are at the same distance d_{BS–RS} to their associated BS, equal to 60% of the cell radius (experimentally found as the best position) and users are uniformly placed beyond 60% of the cell radius (so as to include only those users benefiting from the presence of relays). See Figure 2left for detailed deployment of the RS and users in the macrocell.
where χ_{i,k} is the shadowing between the k th transmitter and the i th receiver, and δ_{i,k}^{m,l} ∊ CN(0, 1) represents the Rayleigh fading.
In this section, the following transmission strategies are evaluated:

Relayed networkMIMO optimal: relaybased transmission with cooperative BSs using networkMIMO and the optimal precoding strategy derived in WSRbased optimal resource allocation.

Relayed networkMIMO suboptimal: relaybased transmission with cooperative BSs using networkMIMO and the suboptimal precoding strategy described in WSRbased suboptimal resource allocation.

Relayed BDZFTDMA: relaybased transmission with noncooperative BSs, where one time slot is allocated per BS transmission and each BS serves simultaneously its associated relays following a multiuser MIMO BDZF strategy [17].

Relayed BFTDMA: relaybased transmission with noncooperative BSs, where each BS serves its associated relays under roundrobin TDMA.
These transmission strategies are evaluated for the SR over 1,000 random user deployments when no MCS constraints are taken into account.
To conclude, in all studies it is observed that the suboptimal precoding strategy for a relayed networkMIMO system nearly achieves the optimum solution. Thanks to the evaluation of the BS–RS capacity in Figure 5, it is verified that the similar performance between both solutions is due to the first hop precoding strategy itself and not due to the inclusion of RS–UE link constraints or maximum MCS constraints. However, we do not have a mathematical intuition to explain the similar performance between both precoding strategies, since it is very difficult to relate the eigenvalues of N_{ i }^{1/2}H_{ i }V_{ i } in (18) with the eigenvalues of N_{ i }^{1/2}H_{ i }V_{ i }(V_{ i }^{ H }B_{ γ }V_{ i })^{1/2} in (14), whose values depend on the Lagrange multipliers associated to the power constraints {γ_{ k }} for which there is not an analytical expression.
4.2. Outdoor/indoor hotspot scenario
Overview of deployment parameters and channel models for a hotspot scenario
Parameter  Value  

Number of areas per cell  2  
Area composition  Two buildings with several floors, three streets  
Number of offices on each floor  10  
Office size  10 m × 10 m  
Office height  3 m  
Street weight  10 m  
Probability of the user to be outdoor  0.8  
Path loss models  BS to RS  B5a LOS and C1 NLOS Winner II 
BS to UE outdoor  C2 (UMa) Winner II  
BS to UE indoor  Dense urban indoor  
RS to UE outdoor  ITU UMi with 10m transmit antenna height  
RS to UE indoor  C4 Winner II  
MIMO channel  LOS and NLOS angular dispersion used  
Shadowing & fast fading  Distancedependent shadowing, from Winner  
Shadowing standard deviation  BS to RS  B5a LOS and C1 NLOS Winner II 
BS to UE  ITU UMa/C2 Winner II  
RS to UE  ITU UMi/C4 Winner II  
Antenna array/elements at BS  ULA/4–12  
Penetration loss due to buildings  10 dB 
Channel models adopted are obtained from [21]. We assume LOS conditions for all BS–RS links, distancedependent LOS/NLOS conditions for transmissions from BS and RS to outdoor UE and NLOS conditions with wall penetration losses for transmissions from BS and RS to indoor UE. See details of the models used in Table 3.
In this section, the following transmission strategies are evaluated:

Relayed networkMIMO optimal: relaybased transmission with cooperative BSs using networkMIMO and the optimal precoding strategy derived in WSRbased optimal resource allocation.

Relayed networkMIMO suboptimal: relaybased transmission with cooperative BSs using networkMIMO and the suboptimal precoding strategy described in WSRbased suboptimal resource allocation.

Relayed BDZFTDMA: relaybased transmission with noncooperative BSs, where one time slot is allocated per BS transmission and each BS serves simultaneously its associated relays following a multiuser MIMO BDZF strategy [17].

Direct networkMIMO optimal: nonrelayed transmission with cooperative BSs using networkMIMO and the optimal precoding strategy [12].

Direct networkMIMO suboptimal: nonrelayed transmission with cooperative BSs using networkMIMO and the suboptimal precoding strategy.
These transmission strategies are evaluated for the SR and for the WSR over 1,000 random user deployments when taking into account the MCS constraints (from LTEA specifications [3]: S_{max} = 4.8 bps/Hz). When using WSR, the weights are inversely proportional to the rates in the second hop (i.e., μ_{ i } = 1/r_{2i}), in order to avoid unfair service to deprived users.
When comparing the relayassisted transmissions in Figure 6 (‘relayed networkMIMO’ and ‘relayed BDZFTDMA’), networkMIMO achieves a nearly doubled cellular spectral efficiency. BDZFTDMA suffers some loses in spectral efficiency due to the time division associated to coordination. In addition, it achieves an improved celledge rate because the duration of the relayreceive and relaytransmit phases is optimized beforehand (see Table 2) on each transmission scheme to enhance the system spectral efficiency (and they differ from networkMIMO to BDZFTDMA).
If we compare the relayassisted transmissions with cooperation of BSs for BS–RS link (‘relayed networkMIMO’ in legend) when using the optimal and the suboptimal precoding structure, it can be observed that when using the optimal structure results in terms of spectral efficiency and outage rate are slightly enhanced. It has been observed that these results are achieved with a significantly increased computational complexity.
When comparing results for relayed networkMIMO with different object functions (‘SR’ or ‘WSR’ in Figure 6), it can be observed that WSR criterion performs better in terms of outage rate while it suffers some losses in terms of spectral efficiency, as it is more concerned about users with bad propagation channel conditions in the RS–UE link.
In Figure 7, we can observe the effect of the MCS constraints over the halfduplex relayassisted transmissions, which are applied to each link of the RSbased transmission. Hence, the maximum peruser transmission rate that can be achieved by a direct transmission is nearly doubled than for a relaybased transmission (even in the scenario deployed, it is achieved by less than 95% of the users).
To conclude, in a hotspot or corporate scenario, deploying relays using networkMIMO in the BS–RS link represents a competitive solution for enhanced services, since it allows improving cellular spectral efficiency and outage rate compared to a nonrelayed transmission with cooperating BSs or a relaybased transmission with no cooperating BSs.
5. Conclusions
This article derives the optimal MIMO symbol precoding matrix and power allocation for QoSconstrained relayassisted cellular systems, where cooperation between BS is appointed and the duration of the relaying phases is fixed beforehand. Results in terms of spectral efficiency and outage rate with the optimal MIMO symbol precoding matrix have been obtained and shown not to be significantly better than those obtained with the suboptimal precoding, which requires a lowcomplexity algorithm and lower computational time. Moreover, the suitability of applying networkMIMO in the BS–RS transmission has been proved in a corporate hotspot scenario. Enhanced results in cellular spectral efficiency and outage rate are obtained compared to nonrelayed transmission or relayed transmission with noncooperative BSs. Further work is oriented to study modulationconstrained resource allocation for the uplink, user’s grouping and scheduling strategies, and coordination of transmissions between cell clusters.
Appendix 1
where B_{ γ } = ∑_{k=1}^{ B }γ_{ k }B_{ k }; γ_{ k }, ψ_{ i }, and φ_{ i } denote the Lagrange multipliers or dual variables associated to the perBS power constraint, the peruser transmission rate in the BS–RS link, and the peruser transmission rate, respectively. Finally, η_{ i } is the Lagrange multiplier needed for having positive values of transmission rates.
Once we have the optimal solution for T_{ i }’s and {r_{1i}} for a set of fixed {γ_{ k }} and {φ_{ i }}, we have to optimize {γ_{ k }} and {φ_{ i }} to obtain the optimal solution to (P_{DP}).
Notice that the φ_{ i }^{*} depends on the number of transmission modes per user (m_{ i }) and, in addition, we need to evaluate r_{1i} with Equation (34) which involves a waterfilling algorithm for the power allocation. For that reason, the optimal value for φ_{ i } cannot directly be found. In this regard, an algorithm to find φ_{ i }^{*}, which is based on hypothetic testing is presented in Algorithm 1. The underlying idea is to form a hypothesis of the active and inactive transmission modes and check whether a consistent solution can be found or not. In the Algorithm 1, initially it is assumed that all transmissions modes are active (and hence m_{ i } = rank (H_{ i }V_{ i }) i = 1,…,R); then, φ_{ i }^{*} is computed following (36) and the power allocation is calculated with (33); in the case that the resulting active transmission modes due to the power allocation are lower than those initially supposed, we have to reduce the number of active transmission modes and make a new hypothesis; until consistence.
Finally, notice that once we have the optimal solution for {γ_{ k }}, we can compute the optimal solution for {φ_{ i }} following (36), and hence we can directly obtain the optimal solution to the problems (P_{DP}) and (P_{1}). In this respect, an effective algorithm is proposed in Algorithm 2 to solve the resource allocation on the BS–RS link subject to perBS power constraints, taking into account the MCS constraints and the constraints imposed by the RS–UE link. In the algorithm, {γ_{ k }} are updated following the ellipsoid method [19], and for each set of {γ_{ k }}, the optimal solution for T_{ i }’s, {r_{1i}} and φ_{ i }^{*} are computed following (34) and (36).
Declarations
Acknowledgments
This study was done in the framework of the projects TROPIC FP7 ICT20118318784 and COST Action IC0902, funded by the European Community. Also, by the Spanish Science and Technology Commissions and EC FEDER funds through projects: TEC201019171/TCM and CONSOLIDER INGENIO CSD200800010 COMONSENS, and by project 2009SGR1236 (AGAUR) of the Catalan Administration.
Authors’ Affiliations
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