Mitigating inter-cluster interference on the uplink for a three-cell clustered cooperative network

3.1 Modified planar Wyner model

We modify the widely studied planar Wyner network [2, 28–32] to a three-cell clustered cooperative network. In [28], a two-dimensional hexagonal array of cells is considered, as shown in Fig. 1. This hexagonal array is represented as a checkerboard lattice, and to make it more tractable, it is converted to a rectangular array. In the planar Wyner model, the signal transmitted by a user in the home cell does not experience any attenuation. The signals from neighbouring cells interfere with the home user and the signals are attenuated by a factor of α, where α∈[0,1]. We modify this rectangular array lattice representation to factor in inter-cluster interference as well. Figure 2 shows the lattice representation for a network with a cluster size of three, and it is obtained by rotating the hexagonal lattice by 45° and scaling by a factor of \(\frac {1}{\sqrt {2}}\) [28]. Interference from only the first-tier neighbouring clusters is considered. This intra-cluster interference intensity is denoted as α, as in [28]. We introduce a new parameter for inter-cluster interference and denote it as β, where β∈[0,1]. Since the interference intensities are scalar and deterministic parameters, the scaling of the lattice does not affect the interference pattern.

Without a loss of generality, the following simplifying assumptions are made in this section. An intra-cell TDMA transmission strategy is considered where only a single user per cell is transmitting at a given time instance. Thus, the number of active users in a three-cell clustered network is K=3. The additive white Gaussian noise (AWGN) channel model is assumed with 0 mean and variance \({\sigma _{n}^{2}}\). All the BSs have perfect knowledge of the CSI as well as each other’s codebooks. The BSs within a cluster are connected using an ideal backhaul with infinite capacity. The BSs are perfectly synchronised to ensure that all signals are received at the same time. BPSK modulation is assumed and the power with which the users transmit the antipodal signals are the same, i.e. no power control is performed across cell sites. Assuming that the received amplitude of each user’s signal is A, then the received power of each user is A ². Since a cluster size of B=3 is assumed, there are nine interfering cells from the first-tier neighbouring clusters as can be seen in Fig. 2. Finally, the MSs and BSs are equipped with single, omnidirectional antennas, i.e. N _T=N _R=1.

In (9), the first term equates to the received power of the users in the neighbouring clusters at each cell in cluster c, which is \(4 \beta ^{2} A^{2} \mathbf {I}_{3}\). The second and third terms equate to 0 since the signals transmitted by the users and the noise are uncorrelated. Since the AWGN channel model with 0 mean and variance \({\sigma _{n}^{2}}\) is considered, the final term in (9) equates to \({\sigma _{n}^{2}} \mathbf {I}_{3}\). Combining all these terms gives us (10).

3.2 Capacity of ZF

Since the channel matrix is real in the modified Wyner network, we take the transpose of H ^(c) in (11) rather than the Hermitian given by (4). Step (12) is obtained since H ^(c), given by (6) is symmetric. Finally, (14) is obtained by substituting (8) in (13).

It can be observed from (15) that there is a singularity at α=1 which can be overcome by considering fading and multi-user scheduling as shown in [31].

3.3 Capacity of LMMSE

where 4β ² A ² is the interference power from the users in the neighbouring clusters.

where μ _k s are the eigenvalues of H and \({\sigma _{0}^{2}} = \frac {{\sigma _{n}^{2}}}{A^{2}}\). This equation is however not applicable in a clustered MCP system since the channel matrix is different and also since the effects of inter-cluster interference are not considered. We have to therefore re-derive the capacity for a three-cell clustered network.

where \(\mathbf {Q}^{(c)} = \left [\mathbf {q}_{1}^{(c)} \mathbf {q}_{2}^{(c)} \mathbf {q}_{3}^{(c)} \right ]\) is the 3×3 orthogonal matrix consisting of the orthonormal eigenvectors of \(\mathbf {H}^{(c)^{2}} + {\sigma _{0}^{2}} \mathbf {I}_{3}\) and \(\mathbf {\Lambda }^{(c)} = \text {diag} \left \{ \lambda _{1}^{(c)}, \lambda _{2}^{(c)}, \lambda _{3}^{(c)} \right \}\) is the diagonal matrix containing the eigenvalues of \(\mathbf {H}^{(c)^{2}} + {\sigma _{0}^{2}} \mathbf {I}_{3}\).

On the other hand, the eigenvectors of \(\mathbf {H}^{(c)^{2}} + {\sigma _{0}^{2}} \mathbf {I}_{3}\), denoted as \(\mathbf {q}_{k}^{(c)}\), are also the eigenvectors of H ^(c).

3.4 Performance analysis

We plot the capacities of ZF and LMMSE for various interference intensities in Fig. 3 using the equations derived earlier. Two intra-cluster and four different inter-cluster interference intensities are considered, i.e. α∈{0.5,0.8} and β∈{0,0.005,0.1,0.2}. It must also be noted that these results have been validated using Monte Carlo simulations.

It can be observed that for all inter-cluster interference intensities, the performances of ZF and LMMSE are better for a lower intra-cluster interference intensity. For instance, for β=0 and a very high SNR of 60 dB, the performance of ZF and LMMSE for α=0.5 is 12.5 % higher than that of α=0.8. The same is true even for low to medium SNRs. Also, the gap between ZF and LMMSE for a given α is negligible for β=0 and β=0.005. It can be observed that for a small increase in the intensity of inter-cluster interference, the performance of ZF and LMMSE deteriorate significantly only at very high SNRs. For instance, when β increases from 0 to 0.005, the performance of ZF and LMMSE deteriorate by 50 % at SNR = 60 dB. When β increases further from 0.005 to 0.1, the performance degradation can be observed even at medium SNRs. For instance, when the SNR is 20 dB the performance degrades by about 72 %. It can also be observed that at lower SNRs and higher inter-cluster interference intensities, LMMSE decoders perform better than ZF. We can conclude through this theoretical study that even the slightest inter-cluster interference can seriously deteriorate the performance of a three-cell clustered network.

4.1 Dynamic clustering

As discussed in Section 1.1, dynamic clustering (DC) involves forming clusters of BSs in a dynamic manner based on changing channel conditions. Given a system where C clusters need to be formed with B BSs within a cluster, there are \(\frac {N!}{(B!)^{C} C!}\) different cluster realisations [25] where N=B C is the total number of BSs in the network. An exhaustive search through all these cluster realisations for an optimum solution is therefore highly complex. Various efficient algorithms have been proposed in the literature for DC. We will now discuss each of them and will pick the best algorithm to be considered in this paper.

In [18], each BS in the network collects the instantaneous CSI from all its users and performs scheduling in a decentralised manner. The CSI from all the BSs are then collected by a central server where a greedy clustering algorithm is run. The algorithm picks a random cell and chooses BSs that maximise the joint capacity to form a cluster. This is done in an iterative manner until all the clusters are formed. It must be noted that the cluster size is fixed and remains constant throughout the algorithm. The system in [18] assumes MSs equipped with a single antenna. The algorithm is extended to multi-antenna MSs in [20]. It has also been shown in [21] that CSI from a limited number of BSs is sufficient to guarantee performance.

In [22] and [23], the network is modelled as a graph where the vertices represent the BSs and any two vertices are connected by an edge if the interference between the corresponding BSs is above a certain threshold. Each edge is then assigned a utility value which could be either the average increase in achievable rate or the increase in interference if the two vertices form a cluster. Once the utility values are assigned to all edges, the algorithm iteratively forms clusters so as to maximise the overall utility. The basic idea is however similar to that of [18] in the sense that the algorithm is iterative and maximises the overall sum rate. In addition, the architecture considered is centralised, the size of the cluster remains constant and instantaneous CSI is used to determine the utility values.

The DC algorithm proposed in [18] is improved on in [24] by allowing the cluster size to change dynamically in each iteration. The algorithm runs on a central server and cells that cause the highest amount of interference to each other form a cluster so as to benefit from MCP. The maximum cluster size considered in [24] was four, and the algorithm was shown to perform only slightly better than [18] at medium and high SNRs. If the cluster size was larger, then much higher performance gains would have been observed. This would however come at the cost of complexity for clustering as well as MCP.

Finally, [25] proposes a distributed architecture for DC where clustering is performed by each BS in a decentralised manner. Each BS maintains a candidate list of clusters that it would like to join based on a preference function. In [25], the preference function is the average increase in system capacity if the BSs were to form a cluster. As in [18], the size of the cluster remains constant throughout the clustering process. Invitations are then sent to the candidate BSs and each BS maintains an invitation list. Two BSs form a cluster if the pair of them maximise the preference function. A list of BSs that have not yet formed a cluster is also maintained and is expressed as ε in [25]. The iterative clustering algorithm is shown to terminate after at most C iterations, where C is the total number of clusters. The complexity of the decentralised approach is also shown to scale linearly with the network size N=B C. Although [25] claims to propose a decentralised framework, it is not entirely so as a central unit is required to maintain the list ε. In addition, the overall system performance in terms of sum rate is not better than [18].

Thus, in this paper, the DC algorithm in [18] is used to compare with our proposed algorithm. A hybrid architecture as shown in Fig. 4 is used where clustering is done by a central server, called the CPU. Each BS sends the instantaneous CSI from all its users to the CPU. The iterative greedy algorithm is run on the CPU and the decision is then fed-back to each BS. The BSs that form a cluster then work in a distributed manner to combat intra-cluster interference. This process is repeated on a per-frame or per-slot time scale.

4.2 Static clustering with adaptive FFR

The proposed static clustering with adaptive FFR scheme, referred to as CFFR, is shown in Fig. 5. For simplicity, only the home cluster and two neighbouring clusters are considered but the figure can very easily be extended to all 6 neighbouring clusters. Let W be the total available bandwidth. In DC, all users scheduled in the cluster are allowed to use the entire bandwidth. In our CFFR strategy, the bandwidth is divided into cluster-centre (W _c ) and cluster-edge bands (W _e ). No two neighbouring clusters will share the band for cluster-edge users.

We first proposed this CFFR scheme in [26]. In [26], a hybrid architecture is considered where the bandwidth partitioning is done by a central server after which MCP is done in a decentralised manner. This is to ensure that all the clusters share the same FFR scheme. The clusters share its computed bandwidth partitions with the central server and the final partitions; W _e and W _c are determined such that the largest partition is chosen for the cluster-edge users and the corresponding partition for those in the cluster-centre. This is to ensure that the performance for the cluster-edge users is improved. In this paper, we adopt an entirely distributed architecture where the bandwidth partitions and MCP are performed by each cluster in a decentralised manner. This architecture is shown in Fig. 6. Each cluster also performs intra-cluster interference mitigation independently through ZF and LMMSE decoding.

In this paper, we consider the user locations on a cluster scale and this would in-turn impact the training phase in the algorithm. The performance of LR on a cluster scale as well as the performance of CFFR with sub-optimum location classification are unknown, and these gaps are addressed in this paper.

5.1 Simulation setup

Since we assume that l _k =N _T in this paper, the maximum number of users that can be scheduled in a cluster during each TTI for DC is 6, using (3). For CFFR, on the other hand, since cluster-centre and cluster-edge users do not interfere with each other, these two types of users can be scheduled independently. Thus, for CFFR, the maximum number of users that can be scheduled in a cluster per TTI is 12 (i.e. six cluster-centre and six cluster-edge users). We employ the proportional fair (PF) scheduling algorithm where the time window over which fairness is imposed is set to 10 TTIs. Monte Carlo simulations are run over 100 different network realisations, and for each realisation, 1000 TTIs are considered.

5.2 The results

The LR algorithm performs well, and we obtain a classification accuracy between 88 and 90 %. The percentage of misclassified cluster-centre and cluster-edge MSs are also quantified in Table 2. If cluster-centre MSs are misclassified as cluster-edge, then this will impact the overall cell sum-rate since a lower bandwidth is allocated for them. On the other hand, if cluster-edge MSs are misclassified as cluster-edge, then this will result in raising the inter-cluster interference intensity.

In Fig. 7, we plot the cumulative distribution function (CDF) of the inter-cluster interference intensity, β. As mentioned in the setup, the users are distributed uniformly and we consider 100 such network realisations. For each realisation, we record β for 1000 TTIs where users are scheduled using the PF algorithm. We can observe that DC suffers from higher inter-cluster interference than our proposed CFFR technique. For instance, the probability that β is 0 (i.e. no interference), denoted as \(\mathbb {P}(\beta =0)\) is 0 for DC whereas it is 0.17 for CFFR-LR and 0.37 for CFFR-Ideal. It is therefore more likely that there is no inter-cluster interference for CFFR. Note that CFFR-LR suffers from more interference than its ideal counterpart due to sub-optimal location classification and the distributed architecture. The gap however is not that large. Similarly, it can be observed that \(\mathbb {P}(\beta \le 0.005)\) is 0 for DC, 0.53 for CFFR-LR and 0.72 for CFFR-Ideal. This CDF study therefore shows that CFFR is more superior than DC in mitigating inter-cluster interference. This is further validated by the per-cell sum rate performance shown in Fig. 8.

We can observe in Fig. 8 that the performance of DC is limited by inter-cluster interference. We have seen in the CDF of β that \(\mathbb {P}(\beta > 0.005)\) is 1, and from our theoretical study, we saw that the slightest β is detrimental to the overall cell-rate performance (see Fig. 3). CFFR, on the other hand, does a better job of lowering β and therefore shows much better sum rate performance. Since cluster-centre and cluster-edge users are allocated independent sub-bands, more users can be scheduled per TTI thereby improving the overall sum rate. It can also be observed that since the SINR is lower for DC; LMMSE performs better than ZF. Since the SINR is much better for CFFR, the simpler ZF technique can be used for decoding. In addition, the gap between CFFR-ideal and CFFR-LR is not significant. This is because the load distribution considered in this paper is symmetric. If the load distribution was asymmetric, i.e. there are different number of users in each cell, then the distributed architecture in CFFR-LR would partition the bandwidth differently for each cluster. For instance, if one cluster had all the users in the cluster-centre then it would allocate the entire bandwidth for its users. This would then increase the interference intensity to a neighbouring cluster that has users in the cluster-edge. A study of the impact of asymmetric load on the performance will be done in the future.

Finally, we compare the running times of the DC and CFFR-LR algorithms in Fig. 9. For DC, we are only interested in the running time of determining the clusters and user scheduling. For CFFR-LR, on the other hand, we are only interested in the running time of performing the location classification, scheduling and bandwidth partitioning. For DC, the greedy algorithm requires computation of the joint capacity of the cluster for each iteration. If there are K users in the cluster and B base stations forming a cluster, the complexity of computing the joint capacity can be approximated as \(O(K N_{T} B^{2} {N_{R}^{2}})\). For CFFR, on the other hand, the complexity of determining the user location given only two features (received power and SINR) can be approximated as O(K). The complexity of user scheduling should be negligible compared to this and, moreover, should be the same for both DC and CFFR. We assume that the capacity of the backhaul is infinite and ignore the latency of CSI exchange between the BSs and the central unit. We can observe from the figure that CFFR-LR runs significantly faster than DC. In addition, we can see that the running time of DC is much more dependent on the load in the cell. We see the running time flattening after a load of 6 users per cell. This is because for DC and the setup considered, we cannot schedule more than 6 users per cell per TTI.