Degrees-Of-Freedom in Multi-Cloud Based Sectored Cellular Networks

Interference is one of the fundamental obstacles for high data rate communications in current and future cellular networks because of restricting the effect on overall spectral efficiency in bits/sec/Hz/base station. Sectorization, which has been used in 4G networks, is one solution to alleviate intra-cell interference by using multiple antennas at base stations (BS) resulting in directional beams that cover an intended sector. In the literature, sectorization is often combined with hexagonal cell models, and mostly each cell is divided into three sectors [1,2]. Here, we follow the works in [3,4,5,6] that totally ignore the interference between the sectors in the same cell. In real systems, this is not the case since the side lobes of the radiation pattern cause to observe signals from adjacent inter-cell sectors. However, we ignore it in the current work since the interference is close to the noise level and our focus will be on the sum-capacity of the sectored hexagonal network in the high signal-to-noise ratio (SNR) and the degrees of freedom (DoF) per-user.

Together with sectorization, cooperation between BSs or mobile users is a well-known technique of decreasing the detrimental effect of interference in cellular networks (see e.g., [7] and references therein). In the context of cellular networks, cooperation has mostly been used to create alternate communication paths (by having mobile users or dedicated terminals relay the transmit signals of adjacent mobiles, see, e.g., [8,9]), or to provide BSs with quantized versions of the transmit/receive signals of other BSs via backhaul links (allowing for clustered decoding, see, e.g., [10,11,12,13,14,15,16]). Cooperation makes it possible for user data to be jointly processed by several BSs at both uplink and downlink, hence imitating the benefits of virtual multiple-input multiple-output (MIMO) architecture. This framework is also known as multi-cell processing (MCP) [7,16]. The study of MCP started for uplink with the works [12,13] and for downlink with the work [14] based on full cooperation assumption. In uplink, the received signals at all BSs are relayed to a central processor (CP) via perfect backhaul links (assumed to be of infinite capacity and error free). Then, the CP decodes all user messages jointly. In contrast, in downlink, the CP encodes all messages jointly and sends each transmit signal to its corresponding BS via backhaul links and each mobile user decodes the message itself. With such a full MCP through unlimited backhaul, there is no signal causing complete interference, that is, all received or transmitted signals provide useful information in decoding at the CP or mobile users. Then, interference becomes constructive rather than destructive. Thus, it is exploited.

The exploitation of interference is also possible by implementing the full MCP with limited capacity backhaul links, which is studied for uplink in [15,16] and for downlink in [17]. In [15,16], BSs share functions of their received signals using compress-and-forward protocol ([18], Theorem 6), where each BS first quantizes its received signal and then sends the quantization codeword to the CP. Then, the CP either decodes the quantization codewords and user messages jointly, or decompress quantization codewords first and then decodes the user messages. However, in the downlink [17], the CP precodes the interference across transmit signals of BSs through joint Dirty Paper Coding (DPC) [19] under individual power constraints for each BS. Therefore, each mobile user decodes its message by cancelling precoded interference. In [16,17], it is shown that the close to optimal performance can be achieved in some scenarios by full MCP with modest capacity backhaul links.

Cloud radio access network (CRAN) is a promising architecture for 5G wireless networks [20] to exploit the interference. For a single-CRAN, each BS in the network acts only as a relay and all BSs are connected to the single base-band-unit pool (BBUP) that performs encoding/decoding functionalities, which mimics the function of a CP, over dedicated rate-limited fronthaul links. Therefore, it allows for natural implementation of full MCP with limited capacity relaying links. In some works such as [3,21,22], the performance of single-CRAN is investigated. For downlink, it is shown in [3] that, when each mobile user and sector receiver have M antennas, $M/2$ DoF per-user is achievable with moderate fronthaul capacity by applying simple zero forcing scheme with smart assignment of the messages to different BSs. Similar performance can be achieved for uplink by applying the ideas in [21]. However, due to complexity, latency, connectivity, scalability, and synchronization problems, the deployment of multi-cloud radio-access-networks (M-CRAN) is recently considered in a few works such as [23,24,25,26,27,28,29,30]. For example, Reference [27] studies the problem of optimization of precoding and joint compression of baseband signals across multiple clusters of BSs in downlink. It demonstrates that the multivariate compression based solution reduces the inter-cluster interference. A similar model is studied in [28] by adding access channels from BSs to mobile users as a joint optimization parameter besides precoding and fronthaul compression optimization. Reference [29] handles the network sum power consumption minimization problem for downlink M-CRAN while fronthaul capacity, channel state information CSI error, quality of service and BS power are taken into account as optimization constraints in order to to determine the beamforming vector of each user across the network and the optimal quantization noise covariance matrix associated with each cluster. This work proposes a distributed iterative solution that achieves the performance of the case all BSs connected to a single BBUP. While the formerly mentioned works assume non-dynamic clustering for each BBUP, the authors of [30] propose and analyze dynamic clustering approach based on instantaneous CSI, where they also consider the allocation of computation resources of BBUPs as an optimization parameter.

In the present work, we consider uplink of an M-CRAN with multiple-antenna mobile users and multiple-antenna BSs. We assume $N\gg 1$ BSs, $K\le N$ BBUPs with limited processing capacity and limited fronthaul capacity. The main interest of this paper is to understand highest achievable per-DoF and sum-rate for limited fronthaul and BBUP processing capacity for given BBUP-BS ratio $K/N$. We propose two coding schemes in each of which some mobile users are deactivated to decompose the network into isolated parallelogram and hexagonal clusters, respectively. For both clustering types, the minimum number of mobile users/sectors are determined regarding a BBUP-BS ratio due to one-to-one association between BBUPs and clusters. Each BBUP collects quantized versions of the received signals of the associated cluster through fronthaul links and decodes them jointly. The considered decoding scheme is thus reminiscent of clustered decoding as performed in [10,31].

The contributions of this paper are:

The upper bound is obtained through cut-set argument. In several cases, upper and lower bounds are matched. For small and moderate fronthaul capacities, the achievability gap is given as a function of fronthaul capacity and BBUP-BS ratio, and it is shown that it decreases with the inverse of the BBUP-BS ratio irrespective of BBUP processing capacity.

In the finite SNR case, we compare the proposed coding schemes with the following schemes:

Finite SNR analysis shows that, in the strong interference regime, the proposed schemes outperform all other schemes for almost all SNR range under all scenarios except two; for the 3-sector decoding scheme, low SNR range and scarce BBUP capacities and, for non-interfering schemes, moderate SNR range and high BBUP capacities.

An interesting outcome of the finite SNR analysis is that interfering clustering schemes show either close to or better performance than proposed schemes in the finite SNR range under both weak and strong interference regimes; therefore, the interfering clusterings can be employed at finite SNR values with minor performance losses, since they may be more convenient for practical systems.

We derive two lower bounds and an upper bound on the DoF. As we will show, they match in some cases. The first and second lower bounds are achieved by the schemes described in Section 4 and Section 5, respectively. Both schemes are based on deactivating a set of mobile users. In the first scheme, the mobile users are deactivated so that the remaining active users form parallelogram-like clusters. In the second, the remaining active users form hexagon-like clusters. We name the two DoF lower bounds as parallelogram bound and hexagon bound, respectively.

where above maximizations are over all positive integers $t_1$,$t_2$ satisfying ${\displaystyle t_1{t}_2\ge ⌈\frac{1}{r}⌉}$, and where above maximizations are over all positive integers t satisfying ${\displaystyle t\ge \lceil \sqrt{\frac{1}{3r}}\rceil }$.

In the proposed uplink scheme, we deactivate a subset of mobile users so as to partition the network into non-interfering clusters of active users. These clusters have parallelogram shapes and are parametrized by positive integer pair $\left(t_1,t_2\right)$.

The same as done in the last section, we deactivate a subset of mobile users so as to partition the network into non-interfering clusters of active users and their sectors. The shape of the clusters is hexagon and the size of the hexagons are set by a positive integer t.

In the two proposed achievability schemes (“p-clustering” and “h-clustering”), we considered three non-interfering sectors in each cell. Now, if we consider cells without sectors, we can naively adapt our clustering by deactivating all users in the border cells of clusters. That is, for p-clustering, it requires deactivation of all users in the cells with one or two active mobile users and, for h-clustering, it requires deactivation of all users in the corner cells and the cells with two active users. This means that the network consists of only cells with three active users and cells with no active users for both schemes. This would again partition the network into non-interfering p-clusters and h-clusters without changing the $r_{\mathrm{p}}$ and $r_{\mathrm{h}}$ for any given $\left(t_1,t_2\right)$ pair or t, respectively.

By following the similar procedure introduced in Section 4 and Section 5, one can easily state the following result by simply distributing the available transmission resources equally among three Rxs of a given BS as long as the BBUP capacity is enough or, otherwise, distributing BBUP processing resources equally among the Rxs of a p-cluster/h-cluster. This leads to the following lemma:

Notice that the same cut-set bound, Theorem 2, applies for the naive schemes since the observation functions, Definition 1, are defined not on the sector basis but on the BS basis.

In this section, we present simulation results to evaluate the proposed coding schemes for p-clustering and h-clustering. In Figure 4a, we investigate effect of clustering size on the achievable DoF for several fronthaul capacities ${\mu }_{\mathrm{F}}=[3,7,11]$ and ${\mu }_{\mathrm{BBU}}=428$. We define size of a p-cluster as inverse of $r_{\mathrm{p}}$, i.e., $\frac{1}{r_{\mathrm{p}}}=t_1{t}_2$, and we denote it also with side length pair $\left(t_1,t_2\right)$. We define size of a h-cluster as inverse $r_{\mathrm{h}}$, i.e., $\frac{1}{r_{\mathrm{h}}}=3t^2$, and we denote it also with the parameter t. It is observed that, for p-clustering, when the fronthaul capacity is small, i.e., ${\mu }_{\mathrm{F}}\le M$, clustering size has no effect on DoF since ${\mu }_{\mathrm{F}}$ becomes a bottleneck. In general, we see that, for both p-clustering and h-clustering, the clustering size giving highest DoF decreases with ${\mu }_{\mathrm{F}}$. The figure verifies the Remark 1 since, for all $r_{\mathrm{p}}=r_{\mathrm{h}}$, p-clustering outperforms h-clustering for ${\mu }_{\mathrm{F}}=\left[3,7\right]$. It is also interesting to note that, for p-clustering, the achievable DoF is not monotonically increasing(decreasing) until(after) reaching the maximum for ${\mu }_{\mathrm{F}}=11$ (i.e., $2M<{\mu }_{\mathrm{F}}\le 3M$) since not only the clustering size but also the side length of the p-cluster is important for exploiting interference. For any $r_p$, choosing a $\left(t_1,t_2\right)$ pair that is the minimum in the sum gives the maximum DoF since it provides higher joint processing gain for a p-cluster for the given size (i.e., $\frac{1}{r_p}$), i.e., the more $t_1$ and $t_2$ becomes closer to each other the more mutual information clusters have. Therefore, larger p-cluster sizes may not result in higher DoF owing to the side length effect. However, for ${\mu }_{\mathrm{F}}\le 2M$, the side lengths of p-cluster has no effect on achievable DoF for a given cluster size.

Figure 4b shows the effect of clustering size on DoF for various values of ${\mu }_{\mathrm{BBU}}=[100,300,500]$ and ${\mu }_{\mathrm{F}}=12$. It is seen that, for each ${\mu }_{\mathrm{BBU}}$, achievable DoF increases with cluster size until it becomes a bottleneck, i.e., until ${\mu }_{\mathrm{BBU}}$ becomes active in the achievability expression. Accordingly, the results clearly indicate that having more processing power makes possible larger cluster sizes and hence larger DoF.

In Figure 5, we plot the achievable DoF and cut-set bound vs ${\mu }_{\mathrm{F}}$ for $M=4$, $r=0.025$ and ${\mu }_{\mathrm{BBU}}=428$, which refers the case BBUP processing capacity is equal to the required processing capacity when each receive signal in a p-cluster of size $\left(t_1,t_2\right)=\left(5,8\right)$ is quantized at the maximum quantization rate $R_{\mathrm{q}}=\frac{M}{2}\log (1+P)$. From the figure, we can deduce that almost upper bound for ${\mu }_{\mathrm{F}}\le 2M$ can be reached, which means that $\frac{2M}{3}$ DoF is almost achievable at ${\mu }_{\mathrm{F}}=2M$ given that processing capacity is high enough. In Figure 5, the operating points of clustering sizes is also depicted. For ${\mu }_{\mathrm{F}}\le 8$, equivalently $2M$, any p-clustering with $\frac{1}{r_{\mathrm{p}}}=40$ gives the highest achievable DoF for the given system parameters. However, for ${\mu }_{\mathrm{F}}>8$, there are several different operating points. For example, for $8<{\mu }_{\mathrm{F}}\le 9.4$, the h-clustering of size $t=4$ is the optimal clustering size, which means, for ${\mu }_{\mathrm{F}}>2M$, dividing the network into h-clusters provides higher joint processing gain than p-clustering for the same $r_{\mathrm{h}}=r_{\mathrm{p}}$ if the BBUP processing capacity is enough. For the rest, the clustering size $r_{\mathrm{p}}$ is decreasing with ${\mu }_{\mathrm{F}}$ due to the given BBUP capacity is not enough to handle the quantized data for larger cluster sizes. At the operating point ${\mu }_{\mathrm{F}}=12$, which allows maximum quantization rate for each receive signal, the p-clustering of size $\left(t_1,t_2\right)=\left(5,8\right)$ achieves capacity. This proves that the proposed scheme utilizes the system resources optimally at this operating point and almost $\frac{9M}{10}$ DoF is achievable. We plot also the lower bound on DoF achieved by the naive scheme vs. ${\mu }_{\mathrm{F}}$ for the same parameters. We can clearly see that the performance of the proposed schemes is considerably better than naive schemes due to the sectorization gain brought by nulling intra-cell interference.

In Figure 6, we plot the achievable DoF and cut-set bound as a function of processing capacity prelog ${\mu }_{\mathrm{BBU}}$, for $r=0.025$ and ${\mu }_{\mathrm{F}}=12$, which means that the fronthaul capacity has no restrictive effect on the achievable DoF. The operating points of clustering sizes regarding ${\mu }_{\mathrm{BBU}}$ is also presented. The plot clearly indicates that the cut set bound is achieved until ${\mu }_{\mathrm{BBU}}=428$, i.e., the processing resources is used efficiently even until achieving $\frac{9M}{10}$ DoF. At the rest of the ${\mu }_{\mathrm{BBU}}$ range, it is seen that the optimal clustering sizes ($\frac{1}{r_{\mathrm{p}}}$ or $\frac{1}{r_{\mathrm{h}}}$) increase with ${\mu }_{\mathrm{BBU}}$, and for most of ${\mu }_{\mathrm{BBU}}>428$, h-clustering provides highest DoF. This indicates the advantage of employing h-clustering when the processing capacity is high enough. For some range of ${\mu }_{\mathrm{BBU}}$, both h-clustering of size $t=4$ and p-clustering of size $\left(t_1,t_2\right)=\left(8,8\right)$ provide the highest DoF, which shows that h-clustering with lower clustering size provides higher joint processing gain than p-clustering with larger clustering sizes due to clustering geometry. The figure also depicts the lower bound achieved by the naive approach vs. ${\mu }_{\mathrm{BBU}}$ for the same parameters and the gain of sectorization is clearly seen for higher values of processing capacity.

In this section, we compare finite SNR performances of the proposed schemes with several other schemes, which will be introduced later on. For the finite SNR case, the quantization rates for both proposed clusterings are chosen as stated in Section 4.2 and Section 5.2, but the conditions regarding a high SNR regime are not applied, i.e., the prelog of any quantization rate is not reduced to the number of antennas M. Then, each BBUP implements joint decoding for the users of the associated cluster after reconstructing all sector receive signals of the cluster. For simplicity, we present the comparisons for $M=1$ throughout the section.

To evaluate the performance of the proposed schemes at finite SNR values, other than naive schemes, we compare our schemes with three different schemes:

In our numerical comparison, we average the rate over 5000 independent channel realizations of the channel matrices, where for each realization all channel gains are drawn independently of each other according to a Gaussian distribution, by which we aim at modeling the random location of a mobile user. The direct channel gains of intra-sector links are drawn with variance 1 and the cross channel gains of inter-sector links are drawn with variance ${\alpha }^2<1$ since any mobile user in adjacent sectors can not be closer to a sector receiver than the user in the considered sector, where $\alpha$ is the channel attenuation coefficient. Figure 9 presents the comparison of the performances of the proposed schemes with naive schemes, ${\mathrm{I}}_{\mathrm{p}}$-clustering, ${\mathrm{I}}_{\mathrm{h}}$-clustering schemes and 3-sector decoding scheme vs. SNR when $r=\frac{1}{10}$. The simulations are performed for different cluster sizes such that $t_1\ast t_2=10,11\mathrm{and}12$ and $t=2$. However, in all the subfigures of Figure 9, we present only the ones showing relatively better performance than others to make presentation better.

As seen from all the subfigures of Figure 9, the proposed schemes provides higher sum-rates than naive schemes for all SNR range and, under all scenarios, e.g., strong interference regime at low BBUP processing capacity as in Figure 9b, or low interference regime at high BBUP capacity as in Figure 9e.

By comparing the subfigures of Figure 9 for a given $\alpha$, we conclude that the proposed schemes become more efficient if the processing capacity of BBUPs increases, i.e., the allowed quantization rate increases. In addition, we can see that employing the smallest possible cluster for a given r is more advantageous for small processing capacities. For example, for $\alpha =0.9$, while the p-clustering scheme for $(t_1,t_2)=(5,2)$ shows better performance than other proposed schemes of larger cluster sizes at all SNR range for ${\mu }_{\mathrm{BBU}}=30$, it outperforms the h-clustering for $t=2$ only at low SNR values for ${\mu }_{\mathrm{BBU}}=60$, and it does not outperform either h-clustering for $t=2$ or p-clustering for $(t_1,t_2)=(5,2)$ at any SNR value for ${\mu }_{\mathrm{BBU}}=120$.

By comparing the subfigures of Figure 9 for a given ${\mu }_{\mathrm{BBU}}$, we can observe that, for each ${\mu }_{\mathrm{BBU}}$ value, the SNR range in which the performance of the 3-sector decoding scheme is superior to or close to the proposed schemes decreases when the channel attenuation coefficient is higher. In addition, we see that, for ${\mu }_{\mathrm{BBU}}=60\mathrm{and}120$, the SNR range in which the h-clustering for $t=2$ outperforms the ${\mathrm{I}}_{\mathrm{p}}$-clustering for $(t_1,t_2)=(5,2)$ and/or ${\mathrm{I}}_{\mathrm{h}}$-clustering for $t=2$ increases with the channel attenuation coefficient. We infer that the idea of isolated clustering is more advantageous at strong interference regime.

Another general conclusion that we can draw from simulation results presented in Figure 9 is that, if the processing capacity is high enough, i.e., the quantization rate is high enough, decomposing the network into hexagonal-type clusters achieves higher rates than paralellogram-type clusters especially at moderate and high SNR range even if $r_{\mathrm{h}}=r_{\mathrm{p}}$. This is due to geometrical structure of hexagonal-type clustering that includes more users for both h-cluster/${\mathrm{I}}_{\mathrm{h}}$-cluster and less interfererers for ${\mathrm{I}}_{\mathrm{h}}$-cluster in comparison with the parallelogram clusters for the same $r_{\mathrm{h}}=r_{\mathrm{p}}$.

An interesting conclusion from the finite SNR analysis is that interfering clusterings show close performance to the proposed schemes in the finite SNR range; therefore, the interfering clusterings can also be employed at finite SNR values, since it may be more convenient for practical systems.

In this paper, we analyze the uplink per-user DoF of M-CRAN based sectored cellular networks. The main features of this paper are the following: it proposes efficient ways of decomposing the network into non-interfering clusters for M-CRAN scenarios, and it characterizes per-user DoF as a function of fronthaul and processing capacity prelogs, and BBUP-BS ratio. The lower bound is obtained through two coding schemes based on decomposing the network into non-interfering parallelogram and hexagonal clusters, respectively. In both schemes, BSs apply point-point quantization to receive signals and send the quantization messages to the associated BBUPs over fronthaul links for joint decoding.

Simulation results show that, for small and moderate fronthaul capacities, the achievability gap between lower and cut-set bounds decreases with an inverse of the BBUP-BS ratio. Therefore, the cut-set bound is almost achieved even for small cluster sizes at this range of fronthaul capacities. For higher fronthaul capacity prelogs, the achievability gap is not always tight but decreases with processing capacity prelog.

The finite SNR analysis shows that the proposed schemes outperform the naive schemes at all SNR ranges and, under all scenarios, the interfering clustering cases at all SNR range under strong interference regime when the BBUP processing capacity is scarce and moderate, and the 3-sector decoding case at all SNR range under strong interference regime if the BBUP processing capacity is moderate and high. In other scenarios for interfering clustering and 3-sector decoding cases, the proposed schemes always achieve higher sum-rates except low SNR values.

In general, the results provide valuable insights into appropriate clustering ways for mobile users/sectors, emphasizing the isolation of clusters, particularly if inter-cell interference is highly detrimental.