Group Sparse Precoding for Cloud-RAN with Multiple User Antennas
Cloud radio access network (C-RAN) has become a promising network architecture to support the massive data traffic in the next generation cellular networks. In a C-RAN, a massive number of low-cost remote antenna ports (RAPs) are connected to a single baseband unit (BBU) pool via high-speed low-latency fronthaul links, which enables efficient resource allocation and interference management. As the RAPs are geographically distributed, the group sparse beamforming schemes attracts extensive studies, where a subset of RAPs is assigned to be active and a high spectral efficiency can be achieved. However, most studies assumes that each user is equipped with a single antenna. How to design the group sparse precoder for the multiple antenna users remains little understood, as it requires the joint optimization of the mutual coupling transmit and receive beamformers. This paper formulates an optimal joint RAP selection and precoding design problem in a C-RAN with multiple antennas at each user. Specifically, we assume a fixed transmit power constraint for each RAP, and investigate the optimal tradeoff between the sum rate and the number of active RAPs. Motivated by the compressive sensing theory, this paper formulates the group sparse precoding problem by inducing the -norm as a penalty and then uses the reweighted heuristic to find a solution. By adopting the idea of block diagonalization precoding, the problem can be formulated as a convex optimization, and an efficient algorithm is proposed based on its Lagrangian dual. Simulation results verify that our proposed algorithm can achieve almost the same sum rate as that obtained from exhaustive search.
Keywords:cloud radio access networksparse beamformingblock diagonalizationgroup-sparsityantenna selection
Cloud radio access network (C-RAN)CRAN2011 () is a promising and flexible architecture to accommodate the exponential growth of mobile data traffic in the next-generation cellular network. In a C-RAN, all the base-band signal processing is shifted to a single base-band unit (BBU) poolShi2015 (). In the meantime, the conventional base-stations (BSs) are replaced by geographically distributed remote antenna ports (RAPs) with only antenna elements and power amplifiers, which are connected to the BBU pool via high-speed low-latency fronthaul links by fiberLiu2013a (). The simple structure of RAP enables the probability to deploy ultra-dense RAPs in a C-RAN with low cost.
With highly densed geographically distributed RAPs, significant rate gains can be expected over that with the same amount of co-located antennas in both the single-user and multi-user casesLiu2014 (); Liu2016 (); Wang2015 (). Due to the huge differences among the distance between the user and the geographically distributed RAPs, it has been shown in Liu2014 () that the capacity in the single-user case is crucially determined by the access distance from the user to its closest RAP. This motivates us to investigate whether it is possible to achieve a significant proportion of the sum rate by using a subset of the RAPs. In the single-user case, it is straightforward to avoid distant RAPs transmitting, as they have little contribution to improve the capacity. In the multi-user case, the problem becomes challenging, as the beamformers of all users should be jointly designed.
To tackle this problem, a branch of sparse beamforming technologies are therefore proposed, where the beamforming vectors are designed to be sparse with respect to the total number of transmit antennasZhao2013 (); Mehanna2013 (); Dai2013 (). Motivated by the recent theoretical breakthroughs in compressive sensingZhang2015 (), the sparse beamforming problem is formulated by including the norm of the beamforming vectors as a regularization such that the problem becomes convex. By iteratively updating the weights of the norm, the sparse beamformer that minimizes the total transmit power can be obtained by iteratively solving a second-order-conic-programming (SOCP)Zhao2013 () or a semi-definite-programming (SDP)Mehanna2013 (). The problem can be further simplified to a uplink beamformer design problem via uplink-downlink dualityDai2013 (). Nevertheless, when each RAP includes multiple antennas, one RAP will be switched off only when all the coefficients in its beamformer are set to be zero. In other word, all antennas at a RAP should be selected or ignored simultaneously, which requires group sparsity instead of individual sparsity as in conventional compressive sensing. Recently, group sparse beamforming problem are proposed which can be formulated by inducing a mixed norm regularization Dai2014a (); Shi2014 (); Shi2015a (). For instance, by inducing the -norm as a regularization, the weighted sum rate maximization problem is formulated as a weighted minimum mean square error (WMMSE) minimization and can be solved via a quadratical-constrained-quadratic-programming (QCQP)Dai2014a (). Compared to the individual sparse beamforming, the group sparse beamforming further reduces the network power consumption, and the energy efficiency can be improved as well.
So far, most algorithms focus on the situation where each user has a single antennaZhao2013 (); Mehanna2013 (); Dai2013 (); Dai2014a (); Shi2014 (); Shi2015a (). As suggested by the multiple-input-multiple-output (MIMO) theory, the capacity increases linearly with the minimum number of transmit and receive antennasTelatar1999 (). In a C-RAN, it is desirable to employ multiple antennas at each user to exploit the potential multiplexing gains, which, however, further complicates the sparse precoder design. The difficulty originates from the fact that the problem is typically non-convex, and the transmit and receive beamformers, which are mutual coupling, should be jointly designed. This paper focuses on designing group sparse precoder based on block diagonalization (BD) Spencer2004 (), which has gained widespread popularity thanks to its low complexity and near-capacity performance when the number of transmit antennas is largeShen2006 (); Shen2007 (); Shim2008 (); Ravindran2008 ().
In this paper, we address the joint problem of RAP selection and joint precoder design in a C-RAN with multiple antennas at each user and each RAP. Whereas the problem is typically NP-hard, we show that the problem becomes convex by inducing the reweighted norm of a vector that indicates the transmit power at each RAP as a regularization. Based on its Lagrangian dual problem, we propose an iterative algorithm by iteratively updating the weights of the norm to generate sparse solution. Simulation results verify that the proposed algorithm can achieve almost the same sum rate as that achieved from exhaustive search.
The rest of this paper is organized as follows: Section II introduces the system model and formulates the problem. Section III proposes an iterative algorithm to solve the group sparse precoding problem. Simulation results are presented and discussed in Section IV. Section V concludes this paper.
Throughout this paper, italic letters denote scalars, and boldface upper-case and lower-case letters denote matrices and vectors, respectively. The superscripts and denote transpose and conjugate transpose, respectively. denotes the expectation operator. denote the ceiling operator. denotes the norm of vector . and denote the trace and determinant of matrix , respectively. denotes an diagonal matrix with diagonal entries . denotes an identity matrix. and denote matrices with all entries zero and one, respectively. denotes the cardinality of set .
2 System Model and Problem Formulation
Consider a C-RAN with a set of remote antenna ports (RAPs), denoted as , and a set of users, denoted as , with and , as shown in Fig. 1. Suppose that each RAP is equipped with antennas, and each user is equipped with antennas. The baseband units (BBUs) are moved to a single BBU pool which are connected to the RAPs via high-speed fronthaul links, such that the BBU pool has access to the perfect channel state information (CSI) between the RAPs and the users, and the signals of all RAPs can be jointly processed. With a high density of geographically distributed RAPs, the access distances from each user to the RAPs varies significantly, and the distant RAPs have little contribution to improve the capacity. This motivates us to find a subset of RAPs that can provide near optimal sum rate performance.
In particular, let denotes the set of active RAPs, with . To utilize the multiplexing gains from the use of multiple user antennas, we assume that . The received signal at user can be then modeled as
where and denote the transmit and receive signal vectors, respectively. is the channel gain matrix between the active RAPs and user . denotes the additive noise, which is modeled as a Gaussian random vector with zero mean and covariance . With linear precoding, the transmit signal vector can be expressed as
where is the precoding matrix. is the information bearing symbols. It is assumed that Gaussian codebook is used for each user at the transmitter, and therefore . The transmit covariance matrix for user can be then written as . It is easy to verify that . The sum rate can be written from (1) as
Note that it is difficult to find the optimal linear precoder that maximizes the sum rate due to the non-convexity of (3), and the mutual coupling of the transmit and receive beamformers makes it difficult to jointly optimize the beamformersCai2011 (). In this paper, we assume that block diagonalization (BD) Spencer2004 () is adopted, where an interference-free block channel is obtained by projecting the desired signal to the null space of the channel gain matrices of the other users, such that , or equivalently for all . With BD, the sum rate can be obtained as
As the signals come from more than one RAPs, they need to satisfy a set of per-RAP power constraint, i.e.,
where denotes the per-RAP power constraint. is a diagonal matrix, whose diagonal entries is defined as
This paper focuses on the tradeoff between the sum rate and the group sparsity. In particular, the problem can be formulated as the following optimization problem:
where (7b) is the zero-forcing (ZF) constraint, which ensures that the inter-user interference can be completely eliminated at the optimum. is the tradeoff constant, which controls the sparsity of the solution, and thus the number of active RAPs. With , the problem reduces to a BD precoder optimization problem with per-RAP power constraint. The group sparsity can be improved by assigning a larger .
Note that the optimization problem (7) needs to jointly determine the subset and design the transmit covariance matrices for users, which is a combinatorial optimization problem and is NP-hard. A brute-force solution to a combinatorial optimization problem like (7) is by exhaustive search. Specifically, we must check all possible combinations of the active RAPs. For each combination, we must search for the optimal that satisfies the constraints (7b-7d). In the end, we pick out the combination that maximizes the sum rate. However, the complexity grows exponentially with , which can not be applied to real-world application. Instead, we use the concept of norm to reformulate problem (7). In particular, define as
where , with denoting the transmit signal vector from all RAPs in to user . The -th to the -th entries of are zero if RAP is inactive. It is clear that the -th entry is the transmit power of RAP , which is non-zero if and only if RAP . It is easy to verify that . We then have the following lemma:
Problem (7) is equivalent to the following optimization problem:
where is the channel gain matrix from all RAPs in to user , .
Please refer to Appendix A for detailed proof.
Lemma 2.1 indicates that instead of searching over the possible combinations of and then optimizing according to the corresponding channel gain matrices , (7) can be solved based on the channel between the users and all RAP antennas, i.e., . However, the problem (9) is non-convex due to the existence of the norm, making it difficult to find the global optimal solution.
In compressive sensing theory, the norm is usually replaced by a norm, and sparse solution can be achieved. However, simply substituting by in (9) will not necessarily produce sparse solution in general, as equals the sum power consumption instead of the number of non-zero entries. By replacing by , the transmit power at all RAPs still tend to satisfy the power constraints with equality at the optimum, leading to a non-sparse solution. In this paper, we propose to solve (9) heuristically by iteratively relaxing the norm as a weighted norm. In particular, at the -th iteration, the norm is approximated by
where , with denoting a small positive constant. (9) can be then reformulated as
It is clear that problem (11) is convex, and can be solved by standard convex optimization techniques, e.g. interior point method Boyd2004 (), which, however, is typically slow. In fact, by utilizing the structure of BD precoding, the problem can be efficiently solved by its Lagrangian dual.
3 Reweighted Based Algorithm
In this section, the algorithm to solve the group sparse linear precoding problem will be presented. To solve (11), it is desirable to remove the set of ZF constraints in the first place. It has been proved in Zhang2010 () that the optimal solution for BD precoding with per-RAP constraint is given by
where . is given from the singular value decomposition (SVD) of as
where is the last columns of the right singular matrix of . It is easy to verify that for all .
Note that (15) is a convex problem, its Lagrangian dual can be written as
where , with denoting the Lagrangian dual variables. The Lagrangian dual function can be given as
We can then obtain the Lagrangian dual problem of (15) as
Since the problem (15) is convex and satisfies the Slater’s condition, strong duality holds. The respective primal and dual objective values in (15) and (18) must be equal at the global optimum, and the complementary slackness must hold at the optimumBoyd2004 (), i.e.,
where and are the optimal primal and dual variables, respectively. As BD requires that the total number of transmit antennas should be equal or greater than the total number of user antennas, the minimum number of active RAPs should be . We can then conclude from (19) that the maximum number of positive ’s is
For fixed , the Lagrangian dual function can be obtained by solving
where . By noting that , we have
Let . (20) can be written as
It is clear that for given , the problem (22) can be solved from uncoupled subproblems:
Let us then solve the subproblems (23). By introducing the (reduced) SVD:
where and are unitary matrices. , with denoting the -th singular value of . The optimal can be then obtained from the standard waterfilling algorithmCover2006 ():
where , with
where . The optimal for given can be then obtained as
With the optimal is achieved for given , we can then find the Lagrangian dual variables by the projected subgradient method. Projected subgradient methods following, e.g., the square summable but not summable step size rules, have been proved to converge to the optimal valuesBertsekas2003 (). In particular, a subgradient of with respect to is . With a step size , the dual variables can be updated as
Once the optimal is obtained, the optimal precoding matrices can be achieved by using the fact that as
The optimal set and the corresponding precoding matrices can be obtained from . The algorithm is summarized as Algorithm 1.
Initilization: Set iteration counter , Lagrangian dual variable , , and initial ellipsoid .
Update according to (28).
Update the iteration counter and
Stop if , where is a pre-defined tolerance threshold.
4 Simulation Results
In this section, simulation results are presented to illustrate the results in this paper. We assume that the channel gain matrices from RAP to user are independent over and for all and , and all entries of are independent and identically distributed (iid) complex Gaussian random variables with zero mean and variance . The path-loss model from RAP to the user is
and , where is in the unit of kilometer. The large-scale fading coefficient from RAP to user can be then obtained as
The transmit power constraint at each RAP are assumed to be identical, which is set to be dBm/Hz, , and the noise variance is set to be dBm/Hz.
We consider the case that RAPs with antennas each and users with antennas each. The positions of users and RAPs and the corresponding small-scale fading coefficients are randomly generated. Fig. 2 shows how the number of active RAPs111A RAP is said to be active if its transmit power . varies with iterations, where the tradeoff constant is set to be , and . The number of active RAPs corresponds to . Fig. 2 shows that the number of active RAPs decreases with . Specifically, with , the problem reduces to a BD precoder design with per-RAP constraint, and all RAPs are active. With , the sparsest solution can be achieved, i.e., . We should mention that the value of that corresponds to the sparsest solution varies with the system configuration.
As Fig. 2 shows, the first several iterations lead to biggest improvement. As iterations go on, there is no further improvement after the 15-th iteration. Compared to the full cooperation case, i.e., , as RAPs are switched off, 70% of the power consumption can be saved with, however, limited rate performance loss, which will be illustrated later in this section. Fig. 3 plots the transmit power distribution over all 10 RAPs. From Fig. 3, we can see that as the iterations progress, RAP , and form a serving cluster and transmit with almost the maximum power, while all the other RAPs eventually drop their transmit power to zero after 15 iterations.
Fig. 4 plots how the average sum rate varies with the number of active RAPs. The average sum rate is obtained by averaging over 20 realizations of small-scale fading and 30 realizations of the positions of RAPs and users. The results of exhaustive search is also presented for comparison, which is obtained by searching over all possible combinations of active RAPs, and compute its achievable sum rate by the algorithm given in Zhang2010 (). For the proposed algorithm, we simulate a series of different ’s to get different points along the curve. As we can see from Fig. 4, our proposed algorithm achieves almost the same average sum rate as the exhaustive search, which verifies the optimality of our proposed algorithm. Fig. 4 further plots the average sum rate with a fixed number of instead of uniformly distributed antennas are deployed for comparison, which is denoted as “Fixed ” in Fig. 4. We can clearly see that the proposed algorithm can achieve much better rate performance over that with the same amount of transmit RAP. With 6 selected antennas, for instance, the group sparse precoder reduces the average sum rate for only 3 bit/s/Hz, whereas if only antennas were installed instead of , an additional of 9 bit/s/Hz rate loss can be observed compared to the proposed algorithm. Moreover, the rate gap between the proposed algorithm and that with a fixed number of full cooperative RAPs further increases as the number of selected RAPs decreases. It highlights the importance of group sparse precoder design in C-RAN with a large number of distributed RAPs.
In this paper, we study the group sparse precoder design that maximizes the sum rate in a C-RAN under per-RAP power constraint. We show that the joint antenna selection and precoder design problem can be formulated into an norm problem, which is, however, combinatorial and NP-hard. Inspired by the theory of compressive sensing, we propose an approach that solves the problem via reweighted norm. Simulation results verify the optimality of our proposed algorithm in that it achieves almost the same performance as that obtained form the exhaustive search. Compared to full cooperation, the group sparse precoding can achieve a significant proportion of the maximum sum rate that achieved from full cooperation with, however, much fewer active RAPs, which highlights the importance to employ sparse precoding in C-RAN with ultra-dense RAPs.
Appendix A Proof of Lemma 2.1
Define the channel gain matrix and the precoding matrix between user and the antennas of RAP as and , respectively, and . Let us reorder the channel gain matrices and the precoding matrices as
respectively. We have
for all , where (a) follows from the fact that if RAP is not active, i.e., .
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