Backhaul has emerged as a new challenge of 5G networks due to the ultra-dense deployment of small cells and the explosive growth of data traffic [2, 3]. Traditional fiber-based backhaul can provide high data rates, but the prohibitive cost and the geographical limitations make it impossible to deploy in many practical scenarios. In-band wireless backhauling, also referred to as self-backhauling, is a promising and viable alternative since it utilizes the same spectrum and the same infrastructure with the access link and enables low-cost and plug-and-play installation . Using self-backhauling, a small base station (SBS) can easily receive data from a macro base station (MBS) in the downlink (or a mobile user in the uplink) and then transmit the data to a mobile user (or an MBS) over the same wireless radio spectrum. Combining recent advances in in-band full-duplex (IBFD) technique further facilitates self-backhauled SBSs to transmit and receive at the same time, thereby potentially doubling the spectrum efficiency .
Radio resource management across the access and backhaul links is crucial for performance optimization in self-backhauled wireless networks. It is also an important study issue mentioned in the 3GPP technical report on integrated access and backhaul . The challenge lies in the newly introduced cross-link interference by spectrum sharing between the access and backhaul links. Recently, several research efforts have been made to address the resource management issue for full-duplex self-backhauled wireless networks [6, 7, 8]. The authors in  demonstrated that compared with a conventional time-division duplex (TDD)/frequency-division duplex (FDD) self-backhauled network, the downlink rate in the full-duplex self-backhauled heterogeneous network is nearly doubled, but at the expense of reduced coverage due to higher interference. By employing massive MIMO and transmit beamforming at the MBS, the authors in  studied a joint cell association and power allocation problem for energy efficiency maximization. An advanced block digitalization precoding scheme was proposed to eliminate the cross-link interference and multi-user interference. The authors in  studied a joint scheduling and interference mitigation problem for network utility maximization. A regularized zero-forcing precoding scheme and the SBS operation mode switching (between full-duplex and half-duplex) were jointly considered to mitigate both the co-link interference and the cross-link interference. Note that in [6, 7, 8], it is assumed that each user is associated with only one SBS in the access link and no inter-site cooperation is considered.
Multi-cell cooperation (e.g., CoMP) is a promising technique to mitigate the inter-cell interference by allowing the user data to be jointly processed by several interfering cell sites, thus mimicking a large virtual MIMO system [9, 10]. Exploiting the inter-site cooperation, the authors in  studied the joint access and backhaul resource management for ultra-dense networks. Both joint transmission (CoMP-JT) and coordinated beamforming (CoMP-CB) are considered in the access link. However, the cross-link interference between the access and backhaul links is ignored. The authors in  studied the joint access and backhaul design for the downlink of cloud radio access networks (C-RANs). Therein, a user-centric clustering strategy is assumed in the access link such that each user can be served cooperatively by a cluster of nearby SBSs, and multicast transmission is adopted in the backhaul link by the MBS to deliver the message of each user to the SBSs in its serving cluster simultaneously. The messages of different users are delivered one by one in a time-division manner over the backhaul link. Based on a similar network model in , a different transmission scheme is considered in  to balance the tradeoff between the cooperation benefits and the coordination overhead, where CoMP-CB is adopted in the access link such that each user is served by only one SBS and accordingly, multi-user beamforming is adopted in the backhaul link. Both works, however, are for out-band wireless backhaul and thus did not have the cross-link interference issue.
In this paper, we propose a joint access-backhaul transmission framework for full-duplex self-backhauled wireless networks by considering the inter-site cooperation. As in , an adaptive user-centric clustering strategy is adopted in the access link, where each user is cooperatively served by a cluster of SBSs. To reduce the backhaul traffic load, a user-centric multicast transmission is adopted in the backhaul link, where each user’s data is treated as a multicast message and the SBS cluster receiving the same user’s message forms a multicast group. Different from , we consider IBFD-enabled SBSs which can transmit in the access link while receiving over the backhaul link simultaneously using the same frequency band with potential cross-link interference as well as self-interference. In addition, unlike  where all users’ messages are fetched in a time-division manner over the backhaul link, we adopt a non-orthogonal multicast transmission scheme over the backhaul link so that the MBS delivers all the multicast messages simultaneously, which is more efficient than the orthogonal time-division manner . Under this user-centric joint design framework, we seek the maximum end-to-end achievable rate of the network under the per-BS power constraints through the joint design of multicast beamforming in the backhaul link as well as SBS clustering and beamforming in the access link.
Enabling the inter-site cooperation in general requires acquiring the channel state information (CSI) and exchanging it among cell sites, which is a challenging issue for ultra-dense networks. With a large number of cell sites and mobile users in the network, acquiring full CSI will bring excessive signaling overhead and counteract the performance gains provided by cooperative transmission [15, 16]. Specifically, obtaining the CSI of all cell sites at the user receivers through downlink pilot training may require a long training period that is comparable to the channel coherence time. Moreover, feeding back all the CSI by the user receivers will occupy plenty of the uplink resources . After collecting the CSI, each cell site will forward it to the central controller or share it among the cooperating cell sites for joint signal processing. The overhead will easily overwhelm the backhaul resources, especially in self-backhauled wireless networks, where the backhaul is a scarce resource. Thus, in this work we consider the user-centric joint access-backhaul design with both full CSI and partial CSI. While the former with full CSI can provide the performance upper bound in the ideal case, the latter with partial CSI offers a more practical solution. The contributions of this paper are summarized as follows.
We first consider the joint access-backhaul design with full CSI, where the CSI between each user and each cell site is globally available at a central controller. We formulate an optimization problem for the joint design of multicast beamforming in the backhaul link, SBS clustering and beamforming in the access link to maximize the weighted sum rate of all users under per-BS peak power constraints. This problem is a non-convex mixed-integer non-linear programming (MINLP) problem, which is challenging due to the non-smoothness and the non-convexity of the objective function as well as the combinatorial nature of the SBS clustering. To solve the problem, we first consider the joint access-backhaul beamforming design problem under given SBS clustering. Note that this problem can be approximately transformed into a manifold optimization problem and solved via the Riemannian conjugate gradient (RCG) algorithm, as shown in our prior conference paper 
. However, the RCG algorithm may get stuck in unfavorite local points when the MBS peak power is too large. In this paper, we propose to solve the joint access-backhaul beamforming problem via the successive lower-bound maximization (SLBM) approach. A novel concave lower-bound approximation for the achievable rate expression in the objective function is introduced based on signal-to-interference-plus-noise ratio (SINR) convexification. Simulation results show that the proposed SLBM algorithm with the newly introduced SINR-convexification based lower-bound approximation can avoid the high-power issue in the RCG algorithm. It can also achieve better performance than the well-known weighted minimum-mean-square-error (WMMSE) algorithm. We then develop a heuristic algorithm to determine the SBS clustering based on the iterative link removal technique. The effectiveness of the proposed clustering algorithm is also demonstrated via numerical simulations.
We also consider the joint access-backhaul beamforming design with partial CSI, where only part of the CSI in the access link is available. We formulate a stochastic beamforming design problem to maximize the average weighted sum rate of the network under the per-BS power constraints. We develop a stochastic SLBM algorithm to solve it by adopting the introduced SINR-convexification based lower-bound approximation. Moreover, we derive a deterministic lower-bound approximation for the average achievable rate by using Jensen’s inequality. The original stochastic optimization problem is then approximately solved via solving the resulting deterministic approximation with low complexity. Simulation results demonstrate the performance of the proposed stochastic SLBM algorithm as well as the effectiveness of the proposed deterministic lower-bound approximation. The results also indicate that with a moderate amount of CSI, the proposed algorithms can achieve good performance that is very close to the full CSI case and significantly reduce the channel estimation overhead.
The rest of the paper is organized as follows. Section II introduces the system model. Section III considers the joint access-backhaul design with full CSI and introduces the proposed joint access-backhaul beamforming and SBS clustering algorithms. The joint access-backhaul design with partial CSI is presented in Section IV. Simulation results are provided in Section V. Finally, we conclude the paper in Section VI.
: Boldface lower-case and upper-case letters denote vectors and matrices, respectively. Calligraphy letters denote sets or problems, depending on the context.and denote the real and complex domains, respectively. and denote the absolute value and Euclidean norm, respectively. The operators and correspond to the transpose and Hermitian transpose, respectively.
represents a complex Gaussian distribution with mean
and variance. The real part of a complex number is denoted by . Finally,
denotes the all-zero matrix of dimension.
Ii System Model
Ii-a Network Model
Consider the downlink transmission of a wireless network with full-duplex self-backhauling, where SBSs cooperatively serve users and each of them is connected to the core network through an MBS with in-band wireless backhaul. The MBS is equipped with antennas. Each SBS is enabled by a full-duplex radio with antennas: one for receiving at the wireless backhaul (from the MBS to SBSs) and for transmitting at the access link (from SBSs to users). Each user is equipped with a single antenna. It is assumed that the users that cannot be served by SBSs will be served by the MBS directly on other orthogonal resource blocks, which is not considered in this paper for simplicity.
We consider a user-centric clustering strategy in the access link. Each user is served by a cluster of SBSs cooperatively, denoted as . An example is shown in Fig. 1, where the serving SBS clusters of the three users are , , and
, respectively. Let the binary variableindicate that SBS belongs to the SBS cluster of user and otherwise. Thus, we have . The data intended to user should be fetched at all the SBSs in from the MBS via the backhaul link. Note that the SBS clusters may overlap with each other, which means each SBS may serve multiple users at the same time. We denote as the set of users served by SBS .
In the backhaul link, the MBS adopts a user-centric multicast transmission. Specifically, the MBS treats the message intended to each user as a multicast message and transmits it to the multicast group formed by the SBS cluster associated with this user. Each SBS may receive multiple multicast messages due to the potentially overlapped SBS clusters. All the messages of the users are transmitted simultaneously at the MBS and each SBS decodes the set of messages for its served users using successive interference cancellation (SIC)-based receiver.
|set of all SBSs|
|set of all users|
|binary variable indicating whether or not SBS belongs to the cluster of user|
|set of users served by SBS|
|set of SBSs serving user|
|()||message intended for user in the access (backhaul) link|
|()||additive white Gaussian noise at user (SBS )|
|beamforming vector at SBS for message|
|beamforming vector at the MBS for message|
|()||channel vector between the MBS (SBS ) and user|
|(, )||channel vector between the MBS (SBS ) and SBS|
|SI channel at SBS|
|SI cancellation capability of the SBSs|
|()||peak transmit power of the MBS (SBS )|
Ii-B Signal Model
Let and denote the transmitted signals in the access link and the backhaul link, respectively, that carry the message intended for user . All these signals have normalized power of . Let denote the beamforming vector at SBS for message in the access link and denote the multicast beamforming vector at the MBS for message in the backhaul link. Note that if , which implies that SBS does not participate in the transmission of message . The main notations in the system model are summarized in Table I.
With full-duplex capability, each SBS can deliver messages to its served users in access while receiving messages from the MBS in backhaul simultaneously using the same frequency band.
Ii-B1 Access Link
The received signal at user is given by
where () is the channel vector between the MBS (SBS ) and user , and is the additive white Gaussian noise at user . The channel coefficient is modeled as , where is the large-scale fading coefficient that includes the path loss and shadowing, and is the small-scale fading coefficient modeled as an independent and identically distributed (i.i.d.) random vector. The channel vector is modeled in the same manner. In (1), the first term is the desired signal transmitted cooperatively by all the SBSs in the cluster , the second term is the cross-link interference transmitted by the MBS over the backhaul link, and the third term presents the co-link interference transmitted by all the SBSs in the same access link but intended to other users. Note that the cross-link interference term includes the message for user in the backhaul link, i.e., , but the user does not intend to decode it directly due to the potentially weak channel condition between the MBS and this user.
Based on (1), the achievable data rate of user in the access link can be expressed as
Ii-B2 Backhaul Link
The received signal at SBS is given by
where (, ) is the channel vector between the MBS (SBS ) and SBS and is the additive white Gaussian noise at SBS . The channel coefficient () is modeled as (). In (3), the first term represents the desired signals intended to the set of users served by SBS , the second term is the co-link interference caused by the signals intended to other users, and the third term is the cross-link interference transmitted by all the SBSs over the access link. Note that the cross-link interference term in (3) includes the self-interference (SI) term , where is the SI channel at SBS and models the residual SI due to imperfect SI cancellation. The cross-link interference term also contains the desired signals by users in , i.e., but transmitted by other SBSs. In this work, since the location of SBSs are fixed, we assume that the CSI between SBSs changes very slowly, and hence can be perfectly estimated and made available at all SBSs. Moreover, each SBS already has the knowledge of its transmitted signals, i.e., , the cross-link interference from other SBSs that transmit the same signals can be perfectly canceled at SBS .
Since each SBS has only one receiving antenna for the wireless backhaul, SIC-based multi-user detection is adopted to decode the multiple signals of its served users. In general, the decoding order of these signals at each SBS can be optimized to improve the system performance. In this paper, for simplicity, we assume that the same decoding order is applied for all SBSs, which is given heuristically according to the aggregative channel gain of each user from all SBSs in the access link (i.e., ). This implies that the user with better channel condition has a higher priority to be decoded.
Based on the above discussion, the achievable rate of decoding the message for user at SBS in the backhaul link is given by
where is a term composed of co-link interference, cross-link interference, and residual SI, represents the SI cancellation capability of the SBSs, and is an index set that is jointly determined by the SIC decoding order and the set of users served by SBS .
Since the message for user is multicast to the SBS cluster , to ensure all the SBSs in can decode the message successfully, the overall transmission rate of the message for user in the backhaul link is limited by the SBS with the worst channel condition, given by
Considering the backhaul link from the MBS to SBSs and the access link from SBSs to users, the end-to-end achievable rate of user is given by
Iii Joint Access-Backhaul Design with Full CSI
In this section, we consider the joint access-backhaul design when all the CSI is available. We first provide the problem formulation for the joint design of multicast beamforming in the backhaul link, SBS clustering and beamforming in the access link. We then develop an effective algorithm to solve it via the SLBM approach and the iterative link removal technique.
Iii-a Problem Formulation
Our objective is to maximize the end-to-end weighted sum rate of the network through joint design of the backhaul multicast beamforming , the access beamforming , and the SBS clustering . This problem is mathematically formulated as equationparentequation
where is the peak power of the MBS, is the peak power of SBS , and are the weights accounting for possibly different service grades among all users.
Problem is a non-convex MINLP problem , which is NP-hard in general. Obtaining its optimal solution is challenging due to the non-smoothness and the non-convexity of the rate expression (6) as well as the combinatorial nature of the SBS clustering variable . Even when the SBS clustering is given, problem is still non-convex and computationally difficult.
In the following subsections, we first tackle problem with a given SBS cluster , denoted as . We propose to solve it via the SLBM approach with a newly introduced lower-bound approximation for the achievable rate. We then develop a heuristic algorithm to determine the SBS clustering based on the iterative link removal technique.
Iii-B Successive Lower-Bound Maximization for
For ease of notation, we rewrite problem with a given SBS cluster , i.e., as
Problem is a non-convex problem with non-smooth and non-convex objective function but convex feasible region. We propose to solve it via the SLBM approach . The main idea of the SLBM approach is to successively maximize a sequence of approximate objective functions . Specifically, starting from a feasible point , the algorithm generates a sequence of according to the update rule
where is the point obtained at the -th iteration and is an approximation of at the -th iteration. Typically the approximate function needs to be carefully chosen such that the subproblem (9) is easy to solve. Moreover, to ensure the convergence of the SLBM algorithm, should be a global lower bound for and also be tight at , i.e., and .
Note that the SLBM approach shares the same idea with many important algorithms such as the successive convex approximation (SCA)  and the concave-convex procedure (CCP) , by successively optimizing an approximate version of the original problem. However, they are different mainly in two aspects :
The SCA or CCP approximates both the objective functions and the feasible sets. On the contrary, the SLBM approximates only the objective function.
The SCA or CCP is applicable only to problems with smooth objectives that are differentiable, while the SLBM is able to handle non-smooth objectives.
In this paper, we construct an approximate function by introducing a novel concave lower-bound approximation for the non-convex achievable rate expression in the objective function based on SINR convexification, which allows subproblem (9) to be easily solved with guaranteed convergence. For comparison purpose, we first present an existing lower-bound approximation constructed by the WMMSE method , which has been widely used to deal with varieties of sum rate maximization problems.
Iii-B1 WMMSE-based Concave Lower-Bound Approximation
We take the non-convex achievable rate expression in the access link (2) with given SBS clustering for example. In the WMMSE method, the relationship between the achievable rate and its mean-square-error (MSE) is established. The key transformation is to rewrite it into the following equivalent form :
where , is the receive beamformer, is a scalar variable associated with the -th user, and is the MSE function defined as
Note that in (10), with given SBS clustering, the achievable rate in the access link is a function of the set of beamformers , denoted as .
The unconstrained optimization problem in the right-hand side of (10) can be easily solved by using the first-order optimality condition. Its optimal solution is given by
where is the minimum-mean-square-error (MMSE) receive beamformer and is the resulting MSE.
This function is a global lower bound of , which is derived by the rate-MSE relationship. Then, the WMMSE-based lower-bound approximation for the achievable rate at a feasible point can be constructed as
where is the argument that maximizes with the given , i.e.,
It is easy to check that
Thus, is a lower bound of and it is tight at the point .
Iii-B2 SINR-Convexification based Concave Lower-Bound Approximation
The main idea of SINR convexification is to approximate the SINR expressions in the achievable rates with their concave approximations, thus to convexify the weighted sum rate maximization problem. Specifically, for the achievable rate in (2), we introduce an auxiliary variable such that . Then the SINR expression in can be rewritten into a quadratic-over-linear form , which is jointly convex in and [25, Section 3.1.5]. Taking its first-order Taylor expansion at any feasible point , we have
where the equality holds only when and .
Denote and replace with , then the inequality (18) can be rewritten as
which holds for all . The equality holds only when , where is given by
It is easy to check that the right-hand side of (19) is concave in and , respectively. It is also seen that in (20) is a scaled version of the MMSE receive beamformer in (12). By taking as the receive beamformer, the right-hand side of (19) can be viewed as a lower bound of the SINR expression under the receive beamformer , which is tight at the MMSE receive beamformer .
which holds for all . Then a global lower bound of is given by
Different from the global lower bound in (13), which is derived by the rate-MSE relationship, the above global lower bound is derived via SINR convexification. Since the log function is concave and nondecreasing, according to the composition rules in [25, Section 3.2.4], is concave in and , respectively. There also satisfies that
By checking its first-order optimality condition, the optimal solution is given by the MMSE receive beamformer in (20).
Clearly, there holds
Therefore, is a lower bound of , which is tight at .
Similar to the WMMSE-based lower-bound approximation, our newly introduced lower-bound approximation via SINR convexification can also be used to handle varieties of weighted sum rate maximization problems, e.g., [23, 24]. Its superiority over the WMMSE-based approximation will be demonstrated via numerical simulations in Section V.
Applying the similar method, we can obtain a new concave lower-bound approximation for the non-convex rate expression in (4) as
where the function is given by
It is easy to verify that , where the equality holds when . Moreover, is concave in , since the pointwise minimum of concave functions is also concave [25, Section 3.2.3].
Thus, the subproblem (9) in each iteration of the SLBM algorithm becomes
Since is a lower-bound approximation of the objective function and is also tight at , with a feasible initial point, the iterations of the SLBM algorithm converge to a stationary solution of problem . Note that problem (32) is a convex problem with log functions in the objective. It can be approximated by a sequence of second-order cone programming (SOCP) problems  via the successive approximation method . Each SOCP can then be solved with a worst-case computational complexity of via the interior-point methods  using a general-purpose solver, e.g., SDPT3 in CVX .
The details of the SLBM algorithm using the proposed SINR-convexification based lower-bound approximation for solving problem are summarized in Alg. 1, denoted as SINRC-SLBM.
Iii-C Heuristic Algorithm for
Intuitively, larger SBS cluster size can achieve higher access rate, but result in a lower backhaul rate due to multicast transmission, and vice versa. Thus, by controlling the SBS cluster size of each user, we can make a balance between the access rate and the backhaul rate, such that the end-to-end rate of the two-hop transmission is maximized.
With the above observation, we propose a heuristic algorithm based on the iterative link removal technique. Specifically, starting with full cooperation for each user, i.e., , for all and , we shrink the SBS cluster size by deactivating several (denoted as ) weakest SBS-user links at each iteration and then solve the joint access and backhaul beamforming design problem with the updated SBS cluster. More specifically, at the -th iteration, we solve problem and calculate the transmit power of all the active SBS-user links with as . We then sort them in the ascending order and update the SBS cluster by deactivating SBS-user links with the minimum power. The iterative procedure terminates when all SBS-links are inactive. Comparing the SBS clusters obtained at each iteration, we then choose the one that achieves the maximum objective value to be the final SBS cluster. The details of the algorithm are summarized in Alg. 2.
Complexity: At the -th iteration, we need to solve one problem instance to determine which SBS-user links should be removed. Since the maximum number of iterations is , the overall complexity of the algorithm is .
Iv Joint Access-Backhaul Design with Partial CSI
The joint access-backhaul design presented in the previous section requires full CSI. However, in an ultra-dense network with a large number of cell sites and users, it is challenging to obtain all the CSI due to the excessive signaling overhead but limited training resources. To handle this challenge and reduce the channel estimation overhead, we consider the joint access-backhaul beamforming design only with partial CSI in this section. We first present the detailed assumption on the CSI availability, then provide the problem formulation for the stochastic beamforming design. We develop a stochastic SLBM algorithm by using the proposed SINR-convexification based lower-bound approximation and a low-complexity algorithm based on the deterministic lower-bound approximation to solve this problem, respectively.
Iv-a Assumption on CSI Availability and Problem Formulation
Generally, the location of the cell sites is fixed and high above the ground. The CSI between the cell sites changes very slowly and can be tracked easily. The CSI estimation overhead mainly lies in the channel between the users and the cell sites due to the mobility of the users. Thus, one promising approach to reduce the CSI overhead is to acquire part of the instantaneous CSI between users and cell sites. In this work, we only need to acquire the instantaneous information of those links that have main contribution to the performance gain of network cooperation. Specifically, the instantaneous CSI from each user to its few nearby SBSs with strong large-scale channel gain and the instantaneous CSI from each user to the MBS are needed. The remaining channel links, i.e., the instantaneous channel coefficients of the links between each user and the SBSs that have weak large-scale channel gain can be ignored, since the acquisition of these links will only contribute little to the network performance. By using this way, we can reduce the CSI overhead greatly without losing too much of the performance. Throughout this section, the SBS clustering is assumed to be predetermined based on the large-scale fading, which varies slowly enough. Let denote the predetermined SBS cluster of user .
Recall that we have modeled the channel coefficients using the large-scale fading coefficients and the small-scale fading coefficients as . Then, the following two kinds of CSI are assumed to be available:
Partial instantaneous CSI: The instantaneous CSI between all the cell sites (including the MBS and all SBSs), i.e., , the instantaneous CSI between the MBS and all users, i.e., , and the instantaneous CSI between each user and its serving SBSs, i.e., .
Statistical CSI: The large-scale fading coefficients of the channel links between each user and the SBSs that do not serve it, i.e., , whose instantaneous CSI is unavailable.
Let denote the set of unknown instantaneous CSI. Since the unknown CSI in only involves the channel in the access link, we consider the following average achievable rate for the -th user in the access link: