Cross-Layer Design for Downlink Multi-Hop Cloud Radio Access Networks with Network Coding

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1 Cross-Layer Design for Downin Muti-Hop Coud Radio Access Networs with Networ Coding Liang Liu, Member, IEEE and Wei Yu, Feow, IEEE Abstract arxiv: v1 [cs.it] 29 Jun 2016 There are two fundamentay different fronthau techniques in the downin communication of coud radio access networ (C-RAN): the data-sharing strategy and the compression-based strategy. Under the former strategy, each user s message is muticast from the centra processor (CP) to a the serving remote radio heads (RRHs) over the fronthau networ, which then cooperativey serve the users through joint beamforming; whie under the atter strategy, the user messages are first beamformed then quantized at the CP, and the compressed signa is unicast to the corresponding RRH, which then decompresses its received signa for wireess transmission. Previous wors show that in genera the compression-based strategy outperforms the data-sharing strategy. This paper, on the other hand, points out that in a C-RAN mode where the RRHs are connected to the CP via muti-hop routers, data-sharing can be superior to compression if the networ coding technique is adopted for muticasting user messages to the cooperating RRHs, and the RRH s beamforming vectors, the user-rrh association, and the networ coding design over the fronthau networ are jointy optimized based on the techniques of sparse optimization and successive convex approximation. This is in comparison to the compression-based strategy, where information is unicast over the fronthau networ by simpe routing, and the RRH s compression noise covariance and beamforming vectors, as we as the routing strategy over the fronthau networ are jointy optimized based on the successive convex approximation technique. The observed gain in overa networ throughput is due to that information muticast is more efficient than information unicast over the muti-hop fronthau of a C-RAN. Index Terms Coud radio access networ (C-RAN), cross-ayer design, data-sharing strategy, compression-based strategy, beamforming, networ coding, routing, fronthau constraints, sparse optimization, successive convex approximation. I. INTRODUCTION As a promising candidate for the 5G ceuar roadmap, coud radio access networ (C-RAN) enabes a centraized processing architecture, using mutipe reay-ie base stations (BSs), named remote radio heads (RRHs), to serve mobie users cooperativey under the coordination of a centra processor (CP). In the downin, the benefit of the C-RAN architecture arises from the abiity to cooperativey transmit signas from RRHs to minimize the effect of interference. It is worth noting that messages intended for different users in the networ originate from the The authors are with The Edward S. Rogers Sr. Department of Eectrica and Computer Engineering, University of Toronto, Toronto, Ontario, Canada, M5S3G4, (e-mai:ianguot.iu@utoronto.ca; weiyu@comm.utoronto.ca). This wor is supported by the Natura Sciences and Engineering Research Counci (NSERC) of Canada through a Coaborative Research and Deveopment (CRD) grant.

2 2 CP. As a resut, a ey question is to decide the most effective way to convey the usefu information about the user messages to the RRHs over the finite-capacity fronthau ins for wireess transmission so as to minimize the unwanted interference seen by the users. In the iterature, a considerabe amount of effort has been dedicated to the efficient utiization of the fronthau capacities in the downin communication in C-RAN (see e.g., [1] and the references therein). Among them, the data-sharing strategy and compression-based strategy have attracted a great dea of attention. Specificay, under the data-sharing strategy, the CP shares user messages with the RRHs over the fronthau networ, which then encode the user messages into wireess signas and cooperativey transmit them to users [2] [4]. Generay speaing, due to the finite-capacity fronthau ins, the message of each user can ony be sent to a subset of RRHs for cooperative transmission. Consequenty, the user-rrh association strategy pays an essentia roe on the downin throughput achieved by the data-sharing strategy. In [3], the reweighted 1 -norm based technique is empoyed to optimize the RRH s beamforming vectors and user-rrh association so as to baance between the cooperation gain over the wireess networ as we as the data traffic over the fronthau networ. Instead of sharing direct user messages, another approach for enabing cooperation is to centray compute the beamformed signas to be transmitted by the RRHs at the CP. Under the compression-based strategy, the CP compresses these beamformed signas and sends the compressed signas to the corresponding RRHs over the fronthau ins for wireess transmission. However, the compression process at the CP introduces quantization noises that imit the system performance. In [5], the transmit covariance for the users and compression noise covariance for the RRHs are jointy optimized to maximize the weighted sum-rate of the users subject to the fronthau capacity constraints. Most previous wors in this area focus on the beamforming and/or compression designs across the RRHs aone. However, besides the transmission strategy in the physica-ayer, the routing strategy in the networ-ayer can significanty affect the throughput of downin C-RAN as we, especiay when the fronthau networ consists of edge routers and networ processors over mutipe hops, as iustrated in Fig. 1. This paper aims to jointy optimize the transmission and routing strategies in the downin muti-hop C-RAN under both the data-sharing strategy and compression-based strategy and investigate which strategy achieves better throughput performance subject to the fronthau capacity constraints. The main contributions of this paper are summarized as foows. This paper proposes a cross-ayer framewor to improve the throughput performance of the downin mutihop C-RAN, where the resources avaiabe in the physica-ayer and networ-ayer are jointy optimized. Under the date-sharing strategy, a ey observation is that such a cross-ayer design provides an opportunity

3 3 CP router user RRH Fig. 1. System mode of downin muti-hop C-RAN. to everage the networ coding technique [6] for muticasting user data to the corresponding RRHs over the muti-hop fronthau networ. A weighted sum-rate maximization probem is thus formuated, where RRH s beamforming vectors, user-rrh association, and networ coding based routing are optimized in an overa design. Under the compression-based strategy, simpe routing is used to unicast the compressed signa to each RRH. Weighted sum-rate maximization is formuated such that the RRH s compression noise covariance and beamforming vectors and the routing strategy are jointy optimized. Efficient agorithms with monotonic convergence are proposed to sove the formuated weighted sum-rate maximization probems under the data-sharing strategy and compression-based strategy, respectivey. Specificay, under the data-sharing strategy, we propose a two-stage agorithm to efficienty sove the studied probem by appying the techniques of sparse optimization and successive convex approximation: first, we approximate each user-rrh s discrete association indicator function by a continuous function and obtain a user-rrh association soution; then we fix this user-rrh association and find the corresponding beamforming and networ coding strategy. Furthermore, under the compression-based strategy, a successive convex approximation based agorithm is proposed to sove the weighted sum-rate maximization probem. Both of the proposed agorithms are proved to yied ocay optima soutions that satisfy the Karush-Kuhn-Tucer (KKT) conditions of the studied probems. By numerica resuts, it is shown that in the downin muti-hop C-RAN, the data-sharing strategy can outperform the compression-based strategy in terms of throughput. This is because in the muti-hop fronthau networ, information muticast under the data-sharing strategy is more efficient than information unicast under the compression-based strategy. This compements the concusions in [7], [8] which show that if the routing

4 4 strategy is not considered, the compression-based strategy in genera outperforms the data-sharing strategy in the downin C-RAN in terms of the spectra efficiency and energy efficiency. It is worth noting that under the data-sharing strategy, the joint beamforming and user-rrh association design in the downin C-RAN has been previousy investigated in [3], but without considering the optimization of the routing strategy. Further, [9] proposes to jointy design the transmission and routing strategies in the downin C-RAN, but in the mode of [9] each user is soey served by one RRH, and the CP unicasts the data of each user to its associated RRH. Our paper differs from [3], [9] in aowing cooperative beamforming among RRHs and in the utiization of networ coding technique over the fronthau networ for information muticast. Finay, under the compression-based strategy, the cross-ayer design of the muti-hop C-RAN has been studied in the upin in [10], where the RRHs utiize a compress-and-forward strategy. However, to the authors best nowedge, the cross-ayer design in the downin muti-hop C-RAN has not been investigated prior to this wor. The rest of this paper is organized as foows. Section II presents the system mode for the downin mutihop C-RAN. Sections III and IV introduce the transmit and routing strategies under the data-sharing scheme and compression-based scheme, respectivey. Section V formuates the weighted sum-rate maximization probems subject to the routing constraints for both schemes. Sections VI and VII present the proposed soutions for the two formuated probems, respectivey. Section VIII provides numerica resuts to verify the effectiveness of the proposed cross-ayer design and compares the performance between the data-sharing and compression-based strategies. Finay, Section IX concudes the paper. II. SYSTEM MODEL Consider the downin communication in C-RAN where N RRHs, denoted by the set N = {1,,N}, cooperativey serve K users, denoted by the set K = {1,,K}, under the coordination of the CP. It is assumed that each RRH is equipped with M 1 antennas, whie each user is equipped with one singe antenna. For the wireess networ, it is assumed that the N RRHs communicate with the K users over quasi-static fat-fading channes over a given bandwidth of B Hz. The channe from RRH n to user is denoted by h,n C M 1, n,. In this paper, it is assumed that the channes to a the K users are perfecty nown at the CP. Moreover, we assume that the CP and RRHs communicate over a muti-hop fronthau networ consisting of J routers, denoted by the set J = {1,,J}, and L digita fronthau ins, denoted by the set L = {1,,L}, as shown in Fig. 1. The capacity of each in L is denoted by C bits per second (bps). This paper considers two fundamentay different fronthau techniques, namey data-sharing strategy and compressionbased strategy, in the downin muti-hop C-RAN. Under the data-sharing strategy, the CP muticasts each user s

5 5 message to a the serving RRHs via the muti-hop fronthau networ using the networ coding technique [6], and each RRH then encodes the user messages into wireess signas and sends them to the users. Under the compression-based strategy, the CP first pre-forms and quantizes the beamformed signa for each RRH in an independent manner, then unicasts each RRH s compressed signa to the corresponding RRH by routing over the fronthau networ. Each RRH then decompresses its received signa and sends it to the users. In the foowing, we introduce in detai the proposed cross-ayer architecture for the downin muti-hop C-RAN under the data-sharing strategy and compression-based strategy, respectivey. III. DATA-SHARING STRATEGY In this section, we derive the throughput achieved by the data-sharing strategy in the downin muti-hop C-RAN. A. Beamforming in the Physica-Layer With the data-sharing strategy, user messages are transmitted to the RRHs by the CP via the fronthau networ (refer to Section III-B for more detai). The equivaent baseband transmit signa of RRH n is x n = K w,n s, n, (1) =1 where s CN(0,1) denotes the message intended for user, which is modeed as a circuary symmetric compex Gaussian (CSCG) random variabe with zero-mean and unit-variance, and w,n C M 1 denotes RRH n s beamforming vector for user. Suppose that RRH n has a transmit sum-power constraint P n ; from (1), we have E[x n x H n ] = The received signa of user can be expressed as y = N h H,n x n +z = n=1 K w,n 2 P n, n. (2) =1 N h H,n w,ns + n=1 N h H,n n=1 i where z CN(0,σ 2 ) denotes the additive white Gaussian noise (AWGN) at user. w i,n s i +z,, (3) The signa-to-interference-pus-noise ratio (SINR) for user is expressed as 2 N h H γ DS,n w,n n=1 h H = 2 = w 2 N h H,n w h H i,n +σ 2 w, (4) i 2 +σ2, i i n=1 whereh = [h T,1,,hT,N ]T denotes the effective channe from a RRHs to user, andw = [w T,1,,wT,N ]T denotes the effective beamforming vector for user across a RRHs. The achievabe rate of user in bps under

6 6 the data-sharing strategy is given by Bog 2 (1+γ DS ),. (5) r DS B. Networ Coding in the Networ-Layer Next, consider the data transmission from the CP to RRHs over the digita muti-hop fronthau networ. It is worth noting that if w,n 0, then user is served by RRH n; otherwise, user is not served by RRH n. As a resut, we can define the user-rrh association indicator function α,n (w,n ) as foows: { 1, if w,n 2 0, α,n (w,n ) =,n. (6) 0, otherwise, If user is served by RRH n, i.e., α,n (w,n ) = 1, the CP needs to send the message s to RRH n over the muti-hop fronthau networ at a rate of r DS bps; otherwise, the CP does not need to send s to RRH n. To summarize, there are K muticast sessions in the muti-hop fronthau networ, i.e., s 1,,s K, and each session s has a set D = {n : α,n (w,n ) = 1,n = 1,,N} of destinations. The traditiona approach for information muticast is to mae each router repicate and forward its received information to the downstream routers. However, the optimization of such muticast routing is equivaent to the Steiner tree pacing probem, which is NP-hard [11], [12]. Moreover, this repicate-and-forward based routing strategy is suboptima since the coding operations at routers are necessary to achieve the muticast capacity [6]. In this paper, we propose to appy the networ coding technique to muticast each session to its destinations independenty, but do not code between different sessions for the foowing reasons. First, this strategy resuts in an easy characterization of the routing region, therefore maing the optima muticast routing probem poynomia time computabe. Second, intersession coding provides margina throughput gains over this approach [11], [12]. Networ coding aows fows for different destinations of a muticast session to share networ capacity by being coded together. The pioneering wor [6] shows that for each singe muticast session, the maximum muticast rate can be achieved for the entire muticast session if and ony if it can be achieved for each muticast receiver independenty. Moreover, with coding the actua physica fow on each in need ony be the maximum of the individua destinations fows. As a resut, the routing constraints for the muti-hop fronthau networ can be

7 7 formuated as [11], [12] α,n (w,n )r DS O(J j) d,n = I(N n) I(J j) d,n,,n, (7) d,n,, n, j, (8) d,n f, n,,, (9) K f C,, (10) =1 f 0, d,n 0,,n,, (11) where d,n denotes the conceptua fow rate on in L for the th muticast session to its potentia destination RRH n, f denotes the actua fow rate on in for muticast session, N n and J j denote RRH n and router j, respectivey, I(N n ) denotes the set of ins that are incoming to RRH n, and I(J j ) and O(J j ) denote the set of ins that are incoming to and outgoing from router j, respectivey. The first constraint guarantees that if n D, then the th session must fow at rate r DS to its destination RRH n. The second constraint represents the aw of fow conservation for conceptua fows. Note that the fow conservation constraint for the CP is not considered because it is automaticay guaranteed by constraints (7) and (8). The third constraint indicates that the actua fow rate of the th muticast session at each in is the maximum rate of the conceptua fows of that in to a the destinations, which is the benefit of networ coding. The fourth constraint guarantees that the overa information fow rate at each in does not exceed the in capacity. The ast constraint guarantees a positive fow rate for a the muticast sessions on a the ins. 1 IV. COMPRESSION-BASED STRATEGY In this section, we derive the throughput achieved by the compression-based strategy in the downin muti-hop C-RAN. A. Joint Beamforming and Quantization in the Physica-Layer Different from the above data-sharing strategy for which the user messages are sent to the RRHs for beamforming, under the compression-based strategy, the CP pre-forms the beamformed signa for each RRH instead. Simiar to (1), the beamformed signa for RRH n can be expressed as x n = K =1 w,ns, n. Then, the CP compresses the beamformed signas and sends the quantization indices to the corresponding RRHs over the fronthau networ 1 Given any fow rate soution satisfying constraints (7) (11), the code design which determines the content of each fow being transmitted across the networ can be found according to [13], [14].

8 8 (pease refer to Section IV-B for more information). The compression noise is modeed as a Gaussian random vector, i.e., x n = x n +e n = K w,n s +e n, n, (12) =1 where e n CN(0,Q n ) C M 1, and Q n 0 denotes the covariance of the compression noise at RRH n. Next, RRH n transmits x n to the users, n. The transmit power constraint for RRH n is then expressed as K E[x n x H n ] = w,n 2 +tr(q n ) P n, n. (13) =1 The baseband received signa at user is N N y = h H,n x n +z = h H,n w,ns + i n=1 n=1 n=1 n=1 N h H,n n=1 i w i,n s i + N h H,n e n +z,. (14) The SINR of user is thus expressed as 2 N h H γ COM,n w,n n=1 h H = 2 = w 2,. (15) N h H,n w i,n + N h H,n Q nh,n +σ 2 h H w i 2 + N h H,n Q nh,n +σ 2 The achievabe rate of user in bps under the compression-based strategy is given by B. Routing in the Networ-Layer r COM i n=1 n=1 Bog 2 (1+γ COM ),. (16) In this paper we assume that the compression process is done independenty across RRHs. According to the rate-distortion theory, the fronthau capacity in bps required to convey the compressed signa x n given in (12) to RRH n is expressed as ( ) K T n = BI(x n ; x n ) = Bog 2 w,n w H,n +Q n / Q n, n. (17) =1 Note that instead of muticasting the information to the RRHs as in the data-sharing strategy, under the compressionbased strategy, the CP merey unicasts each compressed signax n to its destination, i.e., RRHn. As a resut, a simpe routing strategy can be adopted for the information unicast over the fronthau networ. The routing constraints for the mutihop fronthau networ G can then be formuated as ( ) K Bog 2 w,n w H,n +Q n / Q n O(J j) =1 d n = I(J j) d n I(N n) d n, n, (18), n,j, (19) N d n C,, (20) n=1 d n 0, n,, (21)

9 9 where d n denotes the fow rate on in L for the nth unicast session, i.e., x n. The first constraint guarantees that the nth unicast session must fow at rate T n to its destination RRH n. The second constraint represents the aw of fow conservation at each router. Note that the fow conservation constraint for the CP is not considered because it is automaticay guaranteed by constraints (18) and (19). The third constraint guarantees that the overa information fow rate at each in does not exceed the in capacity. The ast constraint guarantees a positive fow rate for a the unicast sessions on a the ins. Remar 1: By comparing Sections III and IV, it can be observed that the ey difference between the data-sharing strategy and compression-based strategy ies in how to utiize the fronthau networ. On one hand, user messages are transmitted over the fronthau networ with the former scheme, whie compressed signas are transmitted with the atter scheme. On the other hand, the data-sharing strategy requires information muticast over the fronthau networ since each user s message is sent to a the RRHs serving this user, whie the compression-based strategy merey requires information unicast since each RRH s compressed signa is sent to this RRH aone. Such different approaches generate different traffic in the fronthau networ, thus eading to different throughput in the considered muti-hop C-RAN, as wi be shown in Section VIII. V. PROBLEM FORMULATIONS In this paper, we aim to maximize the throughput of downin muti-hop C-RAN via a joint optimization of the resources avaiabe in the physica-ayer and networ-ayer under both the data-sharing strategy and the compression-based strategy. A. Data-Sharing Strategy For the data-sharing strategy introduced in Section III, we design the beamforming vectors at a RRHs, i.e., w,n s, and networ coding strategy, i.e., d,n s and f s, to maximize the weighted sum-rate of a the users subject to each RRH s transmit power constraint over the wireess networ as we as the networ coding constraints in the muti-hop fronthau networ, i.e., maximize {w,n,r DS,d,n,f } where µ > 0 denotes the positive rate weight for user. K =1 µ r DS (22a) subject to (2), (5), (7) (11), (22b) It is worth noting that without the routing constraints given in (7) (11), each user shoud be served by a the RRHs, i.e., α,n (w,n ) = 1. However, with the constraints given in (7) (11), in genera each RRH cannot

10 10 support a the users in the downin transmission, and as a resut, from (6), for each RRH n, ony a subset of users are associated with it, for which the corresponding user association function α,n (w,n ) and beamforming vector w,n are non-zero. Moreover, the user association functions α,n (w,n ) s aso affect the networ coding design since they determine the destinations of each muticast session. Therefore, the RRH s beamforming, user-rrh association, and networ coding are couped together and need to be jointy optimized in probem (22), which is a chaenging probem in genera. It is aso worth noting that constraint (7) induces a sparse beamforming soution to probem (22). In the iterature, sparse optimization technique has been previousy used for the downin beamforming design probem [8], [15]. Probem (22) differs from prior wor in two aspects. First, [8], [15] encourage a sparse beamforming soution by penaizing the objective function with a sparsity term. However, probem (22) considered in this paper imposes a set of sparsity constraints which need to be stricty satisfied. Second, in [8], [15] the sparsity penaty is independent of the beamforming soution, but in constraint (7) of our studied probem they are couped. As a resut, the existing sparse optimization techniques, e.g., east-absoute shrinage and seection operator (LASSO), cannot be appied in this paper. B. Compression-based Strategy For the compression-based strategy introduced in Section IV, we design the beamforming vectors at a RRHs, i.e., w,n s, compression noise covariance across the RRHs, i.e., Q n s, and routing strategy, i.e., d n s, to maximize the weighted sum-rate of a the users subject to each RRH s transmit power constraint over the wireess networ as we as the fronthau capacity constraints in the muti-hop fronthau networ, i.e., maximize {w,n,r COM,Q n,d n } K =1 µ r COM (23a) subject to (13), (16), (18) (21). (23b) It is worth noting that both the user rates given in (16) and the fronthau rates given in (17) are non-concave functions over the beamforming vectors w,n s and the compression noise covariance Q n s. As a resut, probem (23) is a non-convex optimization probem, and cannot be soved by the conventiona convex optimization techniques. In the foowing two sections, we propose efficient agorithms to obtain ocay optima soutions to the nonconvex probems (22) and (23), respectivey.

11 11 VI. OPTIMIZATION OF DATA-SHARING STRATEGY In this section, we propose an efficient agorithm to sove probem (22) based on the techniques of sparse optimization as we as successive convex approximation. One main chaenge for soving probem (22) is the discrete indicator function α,n (w,n ) defined in (6). By appying standard sparse optimization technique, in this paper we use the foowing continuous function to approximate α,n (w,n ): g Φ (w,n ) = 1 e Φ w,n 2,,n, (24) where Φ 1. It can be observed that when w,n 2 = 0, then g Φ (w,n ) = α,n (w,n ) = 0. Otherwise, if w,n 2 > 0, we have g Φ (w,n ) α,n (w,n ) = 1 with Φ 1. By using g Φ (w,n ) to approximate α,n (w,n ),,n, probem (22) becomes the foowing continuous probem. maximize {w,n,r DS,d,n subject to,f } K =1 µ r DS g Φ (w,n )r DS I(N n) d,n,,n, (25a) (25b) (2), (5), (8) (11). (25c) However, since g Φ (w,n ) is stricty ess than one when w,n 2 > 0, the soution to probem (25), which satisfies constraint (25b), may not satisfy constraint (7) in probem (22). As a resut, in this paper we propose to sove probem (22) in two steps as foows. First, we sove probem (25) and obtain the beamforming soution, denoted by ŵ,n s. The user-rrh association soution is then obtained as foows: { 1, if gφ (ŵ,n ) ψ, α,n (ŵ,n ) =,n, (26) 0, otherwise, where 0 ψ 1 is a threshod to contro the user association soution. 2 Second, we fix this user association soution in probem (22) and sove the foowing simpified probem to refine the beamforming and networ coding strategy: maximize {w,n,r DS,d,n subject to,f } K =1 µ r DS α,n (ŵ,n )r DS I(N n) d,n,,n, w,n 2 = 0, α,n (ŵ,n ) = 0, (27a) (27b) (27c) In the foowing, we show how to sove probems (25) and (27), respectivey. 2 In our simuation, we set Φ = 50 and ψ = 0.5. (2), (5), (8) (11). (27d)

12 12 A. The First Stage: Soution to Probem (25) Probem (25) is a non-convex probem due to constraints (5) and (25b). As a resut, the conventiona convex optimization technique cannot be directy appied. In this section, we propose an efficient agorithm to sove probem (25) suboptimay based on the technique of successive convex approximation. First, we consider constraint (5), which is equivaent to h H w 2 2rDS h H w i 2 +σ 2 B 1,. (28) i By introducing a set of auxiiary variabesη 0 s, = 1,,K, it can be shown that constraint (28) is equivaent to the foowing two constraints: h H w (2 rds B 1)η,, (29) h H w i 2 +σ 2 η,. (30) i As a resut, η can be interpreted as the interference constraint for user. Constraint (30) can be further transformed into the foowing convex second-order cone (SOC) constraint: [h H w 1,,h H w 1,h H w +1,,h H w K] T η σ 2,. (31) For constraint (29), (2 rds /B 1)η is not a convex function. However, given any β, the foowing convex function is an upper bound for (2 rds /B 1)η : f β (r DS,η )= β η 2 2rDS B + 2 β 1 (2 rds B 1)η,, (32) where the equaity hods if and ony if β = (2 rds /B 1)/η. As a resut, we use the foowing convex constraint to approximate constraint (29): h H w β η 2 DS + 2r B 1,. (33) 2 β After approximating the non-convex constraint (5) by the convex ones (31) and (33), we come to constraint (25b). First, we tae the natura ogarithm of the eft-hand side (LHS) and right-hand side (RHS) of inequaity constraint (25b), which resuts in og(1 e Φ w,n 2 )+og(r DS ) og d,n I(N n),,n. (34) It can be shown that og( I(N n) d,n ) is a concave function over d,n s. However, the LHS of constraint (34) is sti non-convex. Since og(1 e Φx ) is a concave function over x, its first-order approximation serves as its

13 13 upper bound. Specificay, given any x, the first-order approximation of og(1 e Φx ) can be expressed as og(1 e Φx ) Φe Φ x (x x) 1 e Φ x +og(1 e Φ x ), (35) where the equaity hods if and ony if x = x. By substituting x with w,n 2, given any w,n, a convex upper bound for og(1 e Φ w,n 2 ) is expresses as where og(1 e Φ w,n 2 ) Φe Φ w,n 2 w,n 2 1 e Φ w,n 2 +φ( w,n ),,n, (36) φ( w,n ) = Φe Φ w,n 2 w,n 2 1 e Φ w,n 2 +og(1 e Φ w,n 2 ). The equaity hods if and ony if w,n = w,n. Simiary, given any point r DS, the concave function og(rds) can be approximated by its first-order approximation as foows: og(r DS ) rds r DS where the equaity hods if and ony if r DS = r DS. r DS +og( r DS ),, (37) With (36) and (37), the non-convex constraint (34) can be approximated by the foowing convex constraint: Φe Φ w,n 2 w,n 2 + rds r DS 1 e Φ w,n 2 r DS +φ( w,n )+og( r DS ) og d,n,,n. (38) I(N n) To summarize, given r DS s, w,n s, and β s, the non-convex constraints (5) and (25b) in probem (25) are approximated by the convex constraints given in (31), (33), and (38). As a resut, with any given r DS s, w,n s, and β s, probem (25) is approximated by the foowing convex probem. maximize {w,n,r DS,η,d,n,f } K =1 µ r DS (39a) subject to (2),(31),(33),(38),(8) (11). (39b) Since probem (39) is a convex probem, it can be gobay soved by CVX [16]. The successive convex approximation method based agorithm to probem (25) is summarized in Agorithm 1, which iterativey updates r DS s, w,n s, and β s based on the soution to probem (39) as shown in Step 2). The convergence behaviour of Agorithm 1 is guaranteed in the foowing proposition. Proposition 1: Monotonic convergence of Agorithm 1 is guaranteed, i.e., K =1 µ (r DS )(t) K =1 µ (r DS )(t 1). Moreover, the converged soution satisfies a the constraints as we as the KKT conditions of probem (25). Proof: Pease refer to Appendix A.

14 14 Agorithm 1 Proposed Agorithm for Soving Probem (25) Initiaization: Set the initia vaues for w,n s, r DS s, and β s and set t = 1; Repeat: 1) Find the optima soution to probem (39) using CVX as {w (t),n,(rds 2) Update w,n = w (t),n, rds = (r DS)(t), and β = 3) t = t+1. Unti convergence Agorithm 2 Proposed Agorithm for Soving Probem (27) Initiaization: Set the initia vaues for β s and set t = 1; Repeat: (2 (rds )(t) /B 1)/η (t),,n; 1) Find the optima soution to probem (40) using CVX as {w (t),n,(rds 2) Update β = (2 (rds )(t)/b 1)/η (t),,n; 3) t = t+1. Unti convergence )(t),η (t),(d,n )(t),η (t),(d,n ) (t),(f )(t) }; ) (t),(f )(t) }; B. The Second Stage: Soution to Probem (27) Given the user association in probem (27), constraint (27b) becomes convex. By using (31) and (33) to approximate the non-convex constraint (5), given any β s, probem (27) can be approximated by the foowing convex probem. maximize {w,n,r DS,d,n,f } K =1 µ r DS (40a) subject to w,n 2 0, α,n (ŵ,n ) = 0, (40b) (2),(31),(33),(27b),(8) (11). (40c) Since probem (40) is a convex probem, it can be efficienty soved. The successive convex approximation based agorithm to probem (27) is summarized in Agorithm 2. Simiar to Proposition 1, the convergence behaviour of Agorithm 2 is guaranteed in the foowing proposition. Proposition 2: Monotonic convergence of Agorithm 2 is guaranteed, i.e., K =1 µ (r DS )(t) K =1 µ (r DS )(t 1). Moreover, the converged soution satisfies a the constraints as we as the KKT conditions of probem (27). The overa two-stage agorithm to probem (22) is summarized in Agorithm 3. Remar 2: It is worth noting that [3] studies a simiar probem of jointy optimizing the user-rrh association with the beamforming vectors. To dea with the discrete user-rrh association indicator functions (6), in [3] the reweighted 1 -norm technique is empoyed to approximate the fronthau constraint (7) by a set of weighted per-rrh power constraints. Then, an aternating optimization based iterative agorithm is proposed to find a beamforming

15 15 Agorithm 3 Overa Agorithm for Soving Probem (22) 1) Sove probem (25) based on Agorithm 1 and obtain the user-rrh association according to (26); 2) Sove probem (27) based on Agorithm 2 and obtain the beamforming and networ coding soution. and user-rrh association soution. Athough the agorithm in [3] wors we in practice, a rigorous convergence proof is not avaiabe. In contrast, the agorithm proposed in this paper aways converge, but the performance depends on the tuning of the approximation parameters Φ and ψ. VII. OPTIMIZATION OF COMPRESSION-BASED STRATEGY In this section, we propose an efficient agorithm to sove probem (23) based on the technique of successive convex approximation. There are two chaenges to sove probem (23): the non-convex user rate constraint given in (16) and fronthau constraint given in (18). In the foowing, we show how to circumvent the above two chaenges. First, simiar to Section VI, by introducing a set of auxiiary variabes η 0 s, = 1,,K, it can be shown that constraint (16) is equivaent to the foowing two constraints: h H w (2 rcom B 1)η,, (41) N h H w i 2 + h H,n Q nh,n +σ 2 η,. (42) i n=1 Constraint (42) can be further transformed into the foowing convex SOC constraint: [h H w 1,,h H w 1,h H w +1,,h H w K] T η N tr(h,n Q n ) σ 2,, (43) where H,n = h,n h H,n. Moreover, since the non-convex constraint (41) has the same form as constraint (29) in Section VI, we can use the convex constraint given in (33) to approximate it, where r DS is substituted by r COM. As a consequence, the non-convex constraint (16) is approximated by the convex constraints (33) and (43). Next, we dea with the non-convex constraint (18). Since og 2 X n is a concave function over X n 0, its first-order approximation function at any point Xn 0 is an upper bound for it, i.e., og 2 X n og 2 X n tr( X n n2 (X n X n )), (44) where the equaity hods if and ony if X n = X n. By setting X n = K =1 w,nw H,n + Q n, at any point n=1

16 16 Agorithm 4 Proposed Agorithm for Soving Probem (23) Initiaization: Set the initia vaues for β s, w,n s, and Q n s, and set t = 1; Repeat: 1) Find the optima soution to probem (47) using CVX as {w (t) 2) Update β = (2 (rcom 3) t = t+1. Unti convergence,n,(rcom ) (t) /B 1)/η (t), Xn = K =1 w(t),n (w(t),n )H +Q (t) n,,n; ) (t),η (t),(dn )(t),q (t) n }; X n = K =1 w,n w H,n + Q n, we have K w,n w H,n +Q n =1 T n =og 2 Q n ( ( K )) tr X 1 og 2 X n w,n w H,n +Q n X n =1 n + og n2 2 Q n ( K ) 1 X n w 1,n +tr( X n Q n I) =og 2 X n + w H,n =1 n2 og 2 Q n, n. (45) As a resut, in this paper we approximate the non-convex constraint (18) by the foowing convex one: ( og 2 X n + 1 K ) w H 1 1,n X n w,n +tr( X n Q n2 n I) og 2 Q n d n, n. (46) =1 I(N n) To summarize, given β s, w,n s, and Q n s, probem (23) is approximated by the foowing convex probem. maximize {w,n,r COM,η,d n,q n } K =1 µ r COM (47a) subject to (13),(33),(43),(46),(19) (21). (47b) Since probem (47) is a convex probem, it can be gobay soved by CVX. The successive convex approximation method based agorithm to probem (23) is summarized in Agorithm 4, which iterativey updates β s, w,n s, and Q n s based on the soution to probem (47) as shown in Step 2). Simiar to Section VI, the convergence behaviour of Agorithm 4 is guaranteed in the foowing proposition. Proposition 3: Monotonic convergence of Agorithm 4 is guaranteed, i.e., K =1 µ (r COM ) (t) K =1 µ (r COM ) (t 1). Moreover, the converged soution satisfies a the constraints as we as the KKT conditions of probem (23). Remar 3: It is worth noting that a simiar probem to probem (23) is studied in [5], where the RRHs are assumed to be directy connected to the CP via fronthau ins without routers, and the users are assumed to be equipped with mutipe antennas. The successive convex optimization technique is aso used to jointy optimize the transmit covariance for each user and compression noise covariance for each RRH so as to maximize the weighted

17 17 TABLE I SYSTEM PARAMETERS OF THE NUMERICAL EXAMPLE Channe Bandwidth 10 MHz Custer Radius 1 m Number of RRHs 5 Number of Antennas per RRH 2 Number of Users 10 RRH Transmit Power Constraint 43 dbm Antenna Gain 15 dbi Path Loss Mode og 10 (D) db Log-Norma Shadowing 8 db Rayeigh Sma Scae Fading 0 db AWGN Power Spectrum Density 169 dbm/hz sum-rate of a the users subject to the fronthau in capacity constraints. Note that in this paper, each user is assigned with one data stream s since it is equipped with one antenna, and the transmit covariance for each user is thus of ran one. As a resut, if we optimize the transmit covariance as in [5] instead of the beamforming vectors, it is necessary to add the ran-one constraints for the transmit covariance matrices, which are non-convex. On the contrary, in this paper we directy optimize the beamforming vector for each user as shown in Agorithm 4. The obtained soution is shown to satisfy the KKT conditions of probem (23). VIII. NUMERICAL RESULTS In this section, we evauate the performance of the proposed networ coding based data-sharing strategy and routing-based compression-based strategy in the downin muti-hop C-RAN. In this numerica exampe, there are N = 5 RRHs, each equipped with M = 2 antennas, and K = 10 users randomy distributed in a circe area of radius 1000m. The bandwidth of the wireess in is B = 10MHz. The channe vectors are generated from independent Rayeigh fading, whie the path oss mode of the wireess channe is given as og 10 (D) in db, where D (in iometer) denotes the distance between the user and the RRH. The transmit power constraint for each RRH is P n = 43dBm, n. The power spectra density of the AWGN at each user receiver is assumed to be 169dBm/Hz, and the noise figure due to the receiver processing is 7dB. The above simuation parameters are summarized in Tabe I. Moreover, the fronthau networ topoogy together with the capacities of the fronthau ins (denoted by 2C or C/2) are shown in Fig. 2. Last, for convenience, the rate weights are assumed to be one for a the users in both probems (22) and (23), i.e., sum-rate maximization is considered for the data-sharing strategy and compression-based strategy.

18 18 2C 2C 1 C/2 C/2 5 2 C/2 C/2 4 3 C/2 C/2 C/2 C/2 C/2 C/2 C/ C/2 C/2 C/2 C/2 C/ C/2 Router RRH Fig. 2. The muti-hop fronthau topoogy for C-RAN. A. Effectiveness of the Proposed Data-Sharing Strategy First, we verify the effectiveness of our agorithm proposed in Section VI to the weighted sum-rate maximization probem (22) under the data-sharing strategy. Fig. 3 shows the convergence behaviour of the proposed iterative agorithms to probems (25) and (27), i.e., Agorithms 1 and 2, when C = 200Mbps and C = 400Mbps in Fig. 2. Monotonic convergence is observed for both Agorithms 1 and 2 with different vaues of C, which verifies Propositions 1 and 2. Moreover, it is observed that both agorithms converge within 10 iterations. Last, for both vaues of C, the converged sum-rate of Agorithm 2 is very cose to that of Agorithm 1, which verifies that the continuous function g Φ (w,n ) given in (24) is a good approximation to the discrete user-rrh association function α(w,n ) given in (6) such that the soution to the reaxed probem (25) is very cose to the origina probem (22). Next, we verify the effectiveness of our proposed data-sharing strategy. Towards this end, we consider the foowing three benchmar schemes for performance comparison. For the first benchmar scheme, we consider a strategy where each user is ony served by one RRH, as proposed in [9]. Specificay, we first aocate each user to the RRH with the strongest channe power, i.e., 1, if n = arg max α,n = h,n 2, 1 n N 0, otherwise,,n. (48) Given the above user-rrh association soution, the CP unicasts each user s data to its associated RRH via routing over the fronthau networ. Note that in a unicast networ, the networ coding constraints given in (7) (11)

19 Sum-rate (Mbps) Agorithm 1: C=200Mbps Agorithm 2: C=200Mbps Agorithm 1: C=400Mbps Agorithm 2: C=400Mbps Iteration Fig. 3. Convergence behaviour of Agorithms 1 and 2 in the first and second stages for soving the weighted sum-rate maximization probem (22) under the data-sharing strategy. reduce to the unicasting constraints. As a resut, the sum-rate of a the users achieved by this scheme can be obtained by soving probem (27) with the user-rrh association soution given in (48). For the second benchmar scheme, we aow each user to be served by mutipe RRHs. Specificay, we et each user be served by the 3 RRHs with the first three strongest channe power. Given the above user-rrh association soution, the sum-rate of a the users achieved by this scheme can be obtained by soving probem (27) using Agorithm 2. For the third benchmar scheme, we sti et each user be served by the 3 RRHs with the first three strongest channe power. However, instead of encoding the received information, in this scheme we assume that each router simpy repicates and forwards its received information to the other routers in the muti-hop fronthau networ. Note that with the above repicate-and-forward scheme, the routing constraints (27b), (8) (11) in probem (27) need to be modified. Specificay, the muticast of each user s message is buit by Steiner trees. Define T as the set of a the Steiner trees for muticasting user s message, which is determined by the user-rrh association, and L t as the set of a the fronthau ins in a Steiner tree t. According to [17], the routing constraints for the repicate-and-forward scheme can be formuated as r DS t T τ t,,, (49) K,t T, L t τ t, C,, (50) τ t, 0, t,, (51)

20 sum-rate (Mbps) Data-Sharing Strategy: Dynamic Custering with Networ Coding Data-Sharing Strategy: Singe User-RRH Association Data-Sharing Strategy: Stastic Custering with Networ Coding Data-Sharing Strategy: Stastic Custering with Repicate-and-Forward C (Mbps) Fig. 4. Throughput versus fronthau in capacity of the data-sharing strategy. where τ t, denotes the rate for muticasting user s message via Steiner tree t. Via repacing the inear constraints (27b), (8) (11) by the inear constraints (49) (51) in probem (27), we are abe to obtain the sum-rate achieved by the repicate-and-forward based data-sharing strategy. Fig. 4 shows the users sum-rate achieved by different schemes under the data-sharing strategy versus different vaues of C. It is observed that our proposed data-sharing strategy achieves much higher throughput than its counterpart without cooperation between RRHs, especiay when the vaue of C is arge. This is because our proposed scheme provides a joint beamforming design gain. It is aso observed that the proposed networ coding based scheme provides up to 30% throughput gain as compared to the scheme when each user is served by three RRHs with strongest channe power. This shows that the user-rrh association pays a significant roe on the throughput performance and thus shoud be carefuy optimized. Last, it is observed that when each user is served by three RRHs with strongest channe power, the sum-rate achieved by the repicate-and-forward based datasharing strategy is very cose to that achieved by its counterpart based on networ coding. This impies that for the information muticast over the fronthau networ, the gain of the networ coding technique over the optimized repicate-and-forward scheme is not significant. (Note that simiar observations are aso found in the iterature, e.g. [18].) However, as shown in Section III-B, the Steiner tree pacing probem arising from the repicate-and-forward scheme is NP-hard, thus from the agorithm design point of view, the networ coding technique is preferred. B. Effectiveness of the Proposed Compression-based Strategy In this subsection, we evauate the performance of the proposed compression-based strategy in the downin muti-hop C-RAN. Fig. 5 shows the convergence behaviour of Agorithm 4 for probem (23) when C = 200Mps

21 Sum-rate (Mbps) Agorithm 4: C=200Mbps Agorithm 4: C=400Mbps Iteration Fig. 5. strategy. Convergence behaviour of Agorithm 4 for soving the weighted sum-rate maximization probem (23) under the compression-based Sum-rate (Mbps) Data-Sharing Strategy: Muti-Hop Compression-based Strategy: Muti-Hop Data-Sharing Strategy: Singer-Hop Compression-based Strategy: Singe-Hop C (Mbps) Fig. 6. Performance comparison between the data-sharing strategy and compression-based strategy for the muti-hop topoogy of Fig. 2 versus the singe-hop topoogy of Fig. 7. and 400Mbos. Simiar to Fig. 3, monotonic convergence is observed for Agorithm 4, which verifies Proposition 3. Moreover, it is observed that Agorithm 4 converges in ess than 10 iterations for both vaues of C. C. Comparison between Data-Sharing Strategy and Compression-based Strategy It is worth noting that the data-sharing strategy and compression-based strategy are two fundamentay different approaches to utiize the fronthau networ in the downin C-RAN. Under the former strategy, user messages

22 22 C C C C C Fig. 7. The singe-hop C-RAN. are muticast to the RRHs, whie under the atter strategy, each compressed signa is unicast to the corresponding RRH. In this subsection, we aim to answer the foowing question by simuation resuts: in the downin muti-hop C-RAN, which strategy is more efficient for the utiization of the imited capacity in the fronthau networ? Fig. 6 provides a performance comparison between the data-sharing strategy and compression-based strategy in terms of the sum-rate of a the users versus the fronthau in capacity. For the purpose of iustration, we aso provide the throughput performance of the data-sharing strategy and compression-based strategy in the case when each RRH is directy connected to the CP via a fronthau in with capacity C, as shown in Fig. 7. Note that in both the setups in Figs. 2 and 7, the capacity of the information fow to each RRH is C, whie the difference is that the routing strategy aso infuences the throughput performance in the first setup. It is observed from Fig. 6 that in the muti-hop C-RAN, the sum-rate achieved by the data-sharing strategy is higher than that achieved by the compression-based strategy amost for a the vaues of C. Note that this is in sharp contrast to the previous resuts in [7], [8], which shows that if the routing strategy over the fronthau networ is not considered, in genera the compression-based strategy outperforms the data-sharing strategy in terms of both spectra and energy efficiency. Specificay, in this numerica exampe, it is observed that in the singe-hop C-RAN, the compression-based strategy can provide up to 25% performance gain over the data-sharing strategy. By comparing the cases of muti-hop and singe-hop C-RAN, it is concuded that athough sending the compressed signas is a better option than sending the user messages if the routing strategy is not considered, the data-sharing strategy can utiize the information muticast technique over the fronthau networ, which is more efficient than information unicast of the compression-based strategy, to mae up the above disadvantage.

23 23 IX. CONCLUSION This paper investigates two fundamentay different techniques for the downin muti-hop C-RAN, namey data-sharing strategy and compression-based strategy. Different from prior wors, apart from the resources in the wireess in, the routing strategy over the muti-hop fronthau networ is considered as we for maximizing the achievabe throughput of the downin C-RAN under both strategies. Specificay, under the data-sharing strategy, the networ coding technique is utiized to muticast each user s messages to a the RRHs serving this user, whie under the compression-based strategy, a simpe routing technique is used to unicast each RRH s compressed signa to the destination. Efficient agorithms with monotonic convergence are proposed under the above cross-ayer optimization framewor for each strategy, and the obtained soutions are proved to satisfy the KKT conditions of the probems of interests. Prior wors show that if the routing strategy is not considered, the compression-based strategy generay outperforms the data-sharing strategy in terms of spectra efficiency. The main contribution of this paper is that if the routing strategy is jointy optimized with the transmission strategy, the data-sharing strategy can achieve better system throughout than the compression-based strategy in the downin C-RAN, since information muticast is more efficient than information unicast over the muti-hop fronthau networ. This impies that the data-sharing strategy is aso a promising candidate for the downin communication of the emerging C-RAN. A. Proof of Proposition 1 APPENDIX First, it can be shown that in the tth iteration of Agorithm 1, the soution obtained in the (t 1)th iteration is aso feasibe to probem (39) given w,n = w (t 1),n, rds = (r DS)(t 1), and β = (2 r(t 1) 1)/η (t 1),,n. In other words, K =1 µ (r DS )(t 1) is achievabe to probem (39) in the tth iteration. As a resut, the optima weighted sum-rate to probem (39) in the tth iteration, i.e., K =1 µ (r DS )(t), is no smaer than the optima weighted sumrate achieved in the (t 1)th iteration, i.e., K =1 µ (r DS )(t 1). Monotonic convergence of Agorithm 1 is thus proved. Next, since in Agorithm 1 we use upper-bound to approximate the non-convex functions in probem (25), as shown in (32), (35), and (37), any feasibe soution to probem (39) satisfies a the constraints of probem (25). As a resut, the soution from Agorithm 1 must be feasibe to probem (25). Last, according to [19, Theorem 1], if in an optimization probem, each non-convex constraint f(x) 0 is iterativey approximated by a convex constraintf opp (x, x) 0, where x is the optima soution to the approximated

24 24 probem in the previous iteration, and f opp (x, x) is a convex function satisfying f opp (x, x) f(x), (52) f opp ( x, x) = f( x), (53) f opp (x, x) x= x = f(x) x= x, (54) then the successive convex approximation agorithm can aways yied a soution satisfying the KKT conditions of the probem. In the foowing, we show that constraint (33) is an approximation to constraint (29) satisfying the above conditions. First, the inequaity (32) impies that (r DS f β,η ) is an upper bound to (2 rds/b 1)η, where the equaity hods if and ony if β = (2 rds/b 1)/η. Moreover, in Agorithm 1, β is set as (2 (rds )(t)/b 1)/η (t) in each iteration. As a resut, the conditions (52) and (53) are satisfied. Next, it can be shown that (r DS f β,η ) = β (2 η 2 = r DS/B 1)η = (2 rds/b 1)η. (55) 2 η Simiary, it can be shown that (r DS f β,η )/ r DS = (2 rds/b 1)η / r DS. As a resut, constraint (33) is an approximation to constraint (29) satisfying the constraints given in (52) (54). Moreover, it can be shown that constraint (38) is an approximation to constraint (34) satisfying the conditions given in (52) (54). As a resut, the soution obtained by the successive convex approximation based Agorithm 1 must satisfy the KKT conditions of probem (25). REFERENCES [1] O. Simeone, A. Maeder, M. Peng, O. Sahin, and W. Yu, Coud radio access networ: virtuaizing wireess access for dense heterogeneous systems, to appear in J. Commun. and Networs. [Onine]. Avaiabe: [2] R. Zahour and D. Gesbert, Optimized data sharing in mutice MIMO with finite bachau capacity, IEEE Trans. Signa Process., vo. 59, no. 12, pp , Dec [3] B. Dai and W. Yu, Sparse beamforming and user-centric custering for downin coud radio access networ, IEEE Access, vo. 2, pp , [4] L. Liu and R. Zhang, Downin SINR baancing in C-RAN under imited fronthau capacity, in Proc. IEEE Internationa Conference on Acoustics, Speech, and Signa Processing (ICASSP), Shanghai, China, Mar [5] S. H. Par, O. Simeone, O. Sahin and S. Shamai, Joint precoding and mutivariate bachau compression for the downin of coud radio access networs, IEEE Trans. Signa Process., vo. 61, no. 22, pp , Nov [6] R. Ahswede, N. Cai, S. R. Li, and R. W. Yeung, Networ information fow, IEEE Trans. Inf. Theory, vo. 46, no. 4, pp , Ju [7] P. Pati, B. Dai, and W. Yu, Performance comparison of data-sharing and compression strategies for coud radio access networs, in Proc. European Signa Processing Conference (EUSIPCO), Sept [8] B. Dai and W. Yu, Energy efficiency of downin transmission strategies for coud radio access networs, to appear in IEEE J. Se. Areas Commun.. [Onine]. Avaiabe: [9] W.-C. Liao, M. Hong, H. Farmanbar, X. Li, Z.-Q. Luo, and H. Zhang, Min fow rate maximization for software defined radio access networs, IEEE J. Se. Areas Commun., vo. 32, no. 6, pp , Sept [10] S. H. Par, O. Simeone, O. Sahin, and S. Shamai (Shitz), Mutihop Bachau Compression for the Upin of Coud Radio Access Networs, to appear in IEEE Trans. Veh. Techn.. [onine]. Avaiabe: