Joint Power Control and Fronthaul Rate Allocation for Throughput Maximization in OFDMA-based Cloud Radio Access Network
aa r X i v : . [ c s . I T ] A p r Joint Power Control and Fronthaul RateAllocation for Throughput Maximization inOFDMA-based Cloud Radio Access Network
Liang Liu, Suzhi Bi, and Rui Zhang
Abstract
The performance of cloud radio access network (C-RAN) is constrained by the limited fronthaullink capacity under future heavy data traffic. To tackle this problem, extensive efforts have beendevoted to design efficient signal quantization/compression techniques in the fronthaul to maximizethe network throughput. However, most of the previous results are based on information-theoreticalquantization methods, which are hard to implement practically due to the high complexity. In this paper,we propose using practical uniform scalar quantization in the uplink communication of an orthogonalfrequency division multiple access (OFDMA) based C-RAN system, where the mobile users are assignedwith orthogonal sub-carriers for transmission. In particular, we study the joint wireless power controland fronthaul quantization design over the sub-carriers to maximize the system throughput. Efficientalgorithms are proposed to solve the joint optimization problem when either information-theoreticalor practical fronthaul quantization method is applied. We show that the fronthaul capacity constraintshave significant impact to the optimal wireless power control policy. As a result, the joint optimizationshows significant performance gain compared with optimizing only wireless power control or fronthaulquantization. Besides, we also show that the proposed simple uniform quantization scheme performsvery close to the throughput performance upper bound, and in fact overlaps with the upper bound whenthe fronthaul capacity is sufficiently large. Overall, our results reveal practically achievable throughputperformance of C-RAN for its efficient deployment in the next-generation wireless communicationsystems.
Index Terms
Cloud radio access network (C-RAN), fronthaul constraint, quantize-and-forward, orthogonal fre-quency division multiple access (OFDMA), power control, throughput maximization.
I. I
NTRODUCTION
A. Motivation
The dramatic increase of mobile data traffic in the recent years has posed imminent chal-lenges to the current cellular systems, requiring higher throughput, larger coverage, and smallercommunication delay. The 5G cellular system on the roadmap is expected to achieve up to1000 times of throughput improvement over today’s 4G standard. As a promising candidate forthe future 5G standard, cloud radio access network (C-RAN) enables a centralized processingarchitecture, using multiple relay-like base stations (BSs), named remote radio heads (RRHs),
L. Liu and S. Bi are with the Department of Electrical and Computer Engineering, National University of Singapore (e-mail: { liu liang,[email protected] } ).R. Zhang is with the Department of Electrical and Computer Engineering, National University of Singapore (e-mail:[email protected]). He is also with the Institute for Infocomm Research, A*STAR, Singapore. Remote Radio Head (RRH) Mobile UserCentral Unit (CU)
Core Network
Wireless Link Backhaul LinkFronthaul Link C-RAN ClusterC-RAN Cluster
Fig. 1. An illustration of the cluster-based C-RAN in the uplink. to serve mobile users cooperatively under the coordination of a central unit (CU) [1]. For thepractical deployment of C-RAN, a cluster-based C-RAN system is shown in Fig. 1, where thesame frequency bands could be reused over non-adjacent or even adjacent C-RAN clusters toincrease spectral efficiency through coordination among CUs by applying certain interferencemanagement techniques such as dynamic resource allocation [2]. Within each C-RAN cluster, theRRHs are connected to a CU that is further connected to the core network via high-speed fiberfronthaul and backhaul links, respectively. In a C-RAN, a mobile user could be associated withmultiple RRHs. However, unlike the BSs in conventional cellular systems which encode/decodeuser messages locally, the RRHs merely forward the signals to/from the mobile users, whileleaving the joint encoding/decoding complexity to a baseband unit (BBU) in the CU. The use ofinexpensive and densely deployed RRHs, along with the advanced joint processing mechanism,could significantly improve upon the current 4G system with enhanced scalability, increasedthroughput and extended coverage.The distributed antenna system formed by the RRHs enables spectrum efficient spatial divisionmultiple access (SDMA) in C-RAN, which has gained extensive research attentions [3]–[11].In the uplink communication of an SDMA based C-RAN, all mobile users in the same clustertransmit on the same spectrum and at the same time, while the BBU performs multi-user detection(MUD) to separate the user messages. In practice, however, the implementation of MUD ishurdled by the high computational complexity and the difficulty in signal synchronization aswell as perfect channel estimation. Similarly, the downlink communication using SDMA is alsoof high complexity in the encoding design to mitigate the co-channel interference. With thisregard, orthogonal frequency division multiple access (OFDMA) is an alternative candidate forC-RAN because of its efficient spectral usage and yet low encoding/decoding complexity. In
OFDMA-based C-RAN systems, users are allocated with orthogonal subcarriers (SCs) free ofco-channel interference. In this case, simple maximal-ratio combining (MRC) technique could beperformed at the CU over the signals received from different RRHs to decode a user’s messagetransmitted on its designated SC. Moreover, OFDMA is compatible with the current wirelesscommunication systems such as 4G LTE. Considering its potential implementations in futurewireless systems and compatibility with the current 4G standards, we consider OFDMA for thecluster-based C-RAN (see Fig. 1) in this paper.
B. Prior Work
The performance of a C-RAN system is constrained by the fronthaul link capacity. Withdensely deployed RRHs, the fronthaul traffic generated from a single user signal of MHzbandwidth could be easily scaled up to multiple Gbps [1]. In practice, a commercial fiber linkwith tens of Gbps capacity could thus be easily overwhelmed even under moderate mobile traffic.To tackle this problem, many signal compression/quantization methods have been proposed tooptimize the system performance under fronthaul capacity constraints. Specifically, the so-called“quantize-and-forward” scheme is widely adopted for the uplink communication in C-RAN toreduce the communication rates between the BBU and RRHs [4]–[10], where each RRH samples,quantizes and forwards its received signals to the BBU over its fronthaul link. The quantize-and-forward scheme is initially studied in relay channel as an efficient way for the relay todeliver the received signal from the source to the destination [14], [15], [16]. In the uplinkcommunication of C-RAN, which can be viewed as a special case of relay channel model with awireless first-hop link and wired (fiber) second-hop link, quantize-and-forward scheme is studiedunder an information-theoretical Gaussian test channel model with the uncompressed signals asthe input and compressed signals as the output corrupted by an additive Gaussian compressionnoise. Then, the quantization methods are designed through setting the quantization noise levelsat different RRHs to maximize the end-to-end throughput subject to the capacity constraints ofindividual fronthaul links. Specifically, the optimal quantization design needs to consider thesignal correlation across the multiple RRHs, where methods based on distributed source coding,e.g., Wyner-Ziv coding, are widely used to jointly optimize the noise levels at the RRHs (seee.g., [6]–[10]). Besides, quantization method based on distributed source coding is also studiedin the downlink communication of C-RAN in [11].Despite of their respective contributions to the understanding of the theoretical limits of C-RAN, most of the proposed quantization methods are based on information-theoretical models,e.g., Gaussian test channel and distributed source coding, which are practically hard to implement.On one hand, although the quantization noise levels across different RRHs that maximize the end-to-end throughput are found in [4]–[10] under different system setups, it is still unknown how to practically design quantization codebook at each RRH to achieve the required quantization noiselevel for the Gaussian test channel model. On the other hand, the decompression complexity ofdistributed source coding grows exponentially with the number of sources (e.g., RRHs in theuplink communication). In practice, the complexity can be prohibitively high in a C-RAN with alarge number of cooperating RRHs. Therefore, it still remains as a question about the practicallyachievable throughput of C-RAN using practical quantization methods, such as uniform scalaror vector quantization used in common A/D modules [12], which are independently applied overRRHs.Furthermore, most of the existing works (e.g., [4]–[10]) only study signal compression methodsin C-RAN under fixed wireless resource allocation. However, the end-to-end performance of C-RAN is determined by both the wireless and fronthaul links. In an OFDMA system, transmitpower allocation over frequency SCs directly determines the spectral efficiency of wireless link.For an OFDMA-based system without fronthaul constraint, the optimal power allocation problemis extensively studied, e.g., it follows the celebrated water-filling policy for a single user case[13]. However, the behavior of optimal SC power allocation in a fronthaul constrained systemlike C-RAN is still unknown to the authors’ best knowledge.
C. Main Contribution
In this paper, we address the above problems in an OFDMA-based C-RAN. In particular,we consider using simple uniform scalar quantization instead of the information-theoreticalquantization method based on Gaussian test channel, and propose joint wireless power controland fronthaul rate allocation design to maximize the system throughput performance. Our maincontributions are summarized as follows: • In the uplink communication of an OFDMA-based C-RAN, we derive the end-to-end sum-rate of all the users subject to each RRH’s fronthaul capacity constraint achieved by a simpleuniform scalar quantization at each RRH together with independent compression amongRRHs. Different from prior works based on Gaussian test channel model, this providesfor the first time an achievable rate result for C-RAN with a practically implementablequantization method. • With the derived rate under uniform scalar quantization, we formulate the optimizationproblem of joint wireless power control and fronthaul rate allocation to maximize the sum-rate performance in OFDMA based C-RAN. We also formulate the problem based onthe Gaussian test channel model to obtain performance benchmark. Efficient algorithmsare proposed to solve the formulated joint optimization problems based on the alternatingoptimization technique. • By investigating the single-user and single-RRH special case, we obtain important insightson the optimal wireless power control and fronthaul rate allocation over SCs. For example,with a fixed fronthaul rate allocation, we show that the optimal power allocation over SCsis a threshold based policy depending on the channel power of a SC, i.e., no power isallocated to a SC if the channel power is below the threshold. Interestingly, we find that thepower allocation under fronthaul rate constraint in general does not follow a water-fillingpolicy that always allocates more power to SC with higher channel power. The inconsistencyis especially evident in low-fronthaul-rate region, where the SC with the highest channelpower may receive the least transmit power, and vice versa. We also theoretically quantifythe performance gap between the proposed simple uniform quantization scheme from thethroughput upper (cut-set) bound. By simulations we show that the throughput performanceof the simple uniform quantization scheme is very close to the performance upper bound,and in fact overlaps with the upper bound when the fronthaul capacity is sufficiently large.
D. Organization
The rest of this paper is organized as follows. We first introduce in Sections II and III thesystem model of C-RAN and the quantization techniques used in the fronthaul signal processing,respectively. In Section IV, we formulate the end-to-end sum-rate maximization problems forboth the Gaussian test channel and uniform scalar quantization models. Sections V and VI solvethe formulated problems for the special case of single-user and single-RRH and general caseof multi-user and multi-RRH, respectively. Finally, we conclude the paper and point out somedirections for future work in Section VII.II. S
YSTEM M ODEL
We consider the uplink of a clustered C-RAN. As shown in Fig. 1, each cluster consistsof one BBU, M single-antenna RRHs, denoted by the set M = { , · · · , M } , and K single-antenna users, denoted by the set K = { , · · · , K } . It is assumed that each RRH m , ∀ m ∈ M ,is connected to the BBU through a noiseless wired fronthaul link of capacity ¯ T m bps. In theuplink, each RRH receives user signals over the wireless link and forwards to the BBU via itsfronthaul link. Then, the BBU jointly decodes the users’ messages based on the signals from allthe RRHs within the cluster and forwards the decoded information to the core network througha backhaul link. The detailed signal models in the wireless and the fronthaul links are introducedin the following. A. OFDMA-based Wireless Transmission
In this paper, we consider OFDMA-based uplink information transmission between the K users and the M RRHs over a wireless link of a B Hz total bandwidth equally divided into N SQy m,1 ...
P/SP/S: parallel to serial conversion SQ ^^SQ: scalar quantizationy m,1 y m,N y m,N RF Band to Baseband Converter
Wireless Channel Fronthaul Link
FFT ...
S/P
S/P: serial to parallel conversion FFT: fast Fourier transform
Fig. 2. The structure of signal processing at an RRH.
SCs. The SC set is denoted by N = { , · · · , N } . It is assumed that each SC n ∈ N is onlyallocated to one user. Denote Ω k as the set of SCs allocated to user k , ∀ k ∈ K . In practice,dynamic SC allocation could be used to enhance the spectral efficiency by assigning SCs to usersof favorable wireless link conditions, e.g., allocating a SC to the user with the highest signal-to-interference-plus-noise ratio (SINR). However, as an initial attempt to understand the jointdesign of the wireless resource allocation and fronthaul rate allocation in fronthaul constrainedC-RAN, it is assumed for simplicity in this paper that the SC allocations among users, i.e., Ω k ’s,are pre-determined. The interesting case with dynamic SC allocation is left for future study.Specifically, in the uplink each user k , ∀ k ∈ K , first generates an OFDMA modulated signalover its assigned SCs and then transmits to the RRHs in the same cluster. As shown in Fig.2, each RRH m , ∀ m ∈ M , first downconverts the received RF signals to the baseband, thentransforms the serial baseband signals to the parallel ones, and demodulates the parallel signalsinto N streams by performing fast Fourier transform (FFT). Suppose that n ∈ Ω k , then theequivalent baseband complex symbol received by RRH m at SC n can be expressed as y m,n = h m,k,n √ p k,n s k,n + z m,n , (1)where s k,n ∼ CN (0 , denotes the transmit symbol of user k at SC n (which is modelled asa circularly symmetric complex Gaussian random variable with zero-mean and unit-variance), p k,n denotes the transmit power of user k at SC n , h m,k,n denotes the channel from user k toRRH m at SC n , and z m,n ∼ CN (0 , σ m,n ) denotes the aggregation of additive white Gaussiannoise (AWGN) and (possible) out-of-cluster interference at RRH m at SC n . It is assumed that z m,n ’s are independent over m and n . B. Quantize-and-Forward Processing at RRH
To forward the baseband symbols y m,n ’s to the BBU via the fronthaul links, the so-called“quantize-and-forward” scheme is applied, where each RRH first quantizes its baseband receivedsignal and then sends the corresponding digital codewords to the BBU. Specifically, since ateach RRH the received symbols at all the SCs are independent with each other and we assumeindependent signal quantization at different RRHs, a simple scalar quantization on y m,n ’s is optimal as shown in Fig. 2. The baseband quantized symbol of y m,n is then given by ˜ y m,n = y m,n + e m,n = h m,k,n √ p k,n s k,n + z m,n + e m,n , (2)where e m,n denotes the quantization error for the received symbol y m,n with zero mean andvariance q m,n . Note that e m,n ’s are independent over n due to scalar quantization at each SC,and over m due to independent compression among RRHs. Then, each RRH transforms theparallel encoded bits ˆ y m,n ’s into the serial ones and sends them to the BBU via its fronthaullink for joint information decoding.After collecting the digital codewords, the BBU first recovers the baseband quantized symbols ˜ y m,n ’s based on the quantization codebooks used by each RRH. Then, to decode s k,n , the BBUapplies a linear combining on the quantized symbols at SC n collected from all RRHs: ˆ s k,n = w Hn ˜ y n = w Hn h k,n √ p k,n s k,n + w Hn z n + w Hn e n , n ∈ Ω k , k = 1 , · · · , K, (3)where ˜ y n = [˜ y ,n , · · · , ˜ y M,n ] T , h k,n = [ h ,k,n , · · · , h M,k,n ] T , z n = [ z ,n , · · · , z M,n ] T , and e n =[ e ,n , · · · , e M,n ] T . According to (3), the SNR for decoding s k,n is expressed as γ k,n = p k,n | w Hn h k,n | w Hn (cid:0) diag( σ ,n , · · · , σ M,n ) + diag( q ,n , · · · , q M,n ) (cid:1) w n , n ∈ Ω k , k = 1 , · · · , K, (4)where diag( a ) denotes a diagonal matrix with the main diagonal given by vector a . It can beshown that the optimal combining weights that maximize γ k,n ’s are obtained from the well-knownMRC [13]: w ∗ n = (cid:0) diag( σ ,n , · · · , σ M,n ) + diag( q ,n , · · · , q M,n ) (cid:1) − h k,n , n = 1 , · · · , N. (5)With the above MRC receiver, γ k,n given in (4) reduces to γ k,n = M X m =1 | h m,k,n | p k,n σ m,n + q m,n , n ∈ Ω k , k = 1 , · · · , K. (6)III. Q UANTIZATION S CHEMES
The key issue to implement the quantize-and-forward scheme introduced in Section II is howeach RRH should quantize its received signal at each SC in practice. In this section, we firststudy a theoretical quantization model by viewing (2) as a test channel and derive its achievablesum-rate based on the rate-distortion theory, which can serve as a performance upper bound.Then, we investigate the practical uniform scalar quantization scheme in details, which can beeasily applied at each RRH, and derive the corresponding achievable end-to-end sum-rate.
A. Gaussian Test Channel
In this subsection, we assume that the quantization errors given in (2) are Gaussian distributed,i.e., e m,n ∼ CN (0 , q m,n ) , ∀ m, n . With Gaussian quantization errors, (2) can be viewed as aGaussian test channel [17]. As a result, to forward the received data at SC n , the transmissionrate in RRH m ’s fronthaul link is expressed as [17] T ( G ) m,n = BN log (cid:18) | h m,k,n | p k,n + σ m,n q m,n (cid:19) . (7)Since quantization is performed at each RRH independently, { y m, , · · · , y m,N } can be reliablytransmitted to the BBU if and only if [16] T ( G ) m = N X n =1 T ( G ) m,n = BN K X k =1 X n ∈ Ω k log (cid:18) | h m,k,n | p k,n + σ m,n q m,n (cid:19) ≤ ¯ T m , m = 1 , · · · , M. (8)Next, consider the end-to-end performance of the users. With Gaussian noise in (2), theachievable rate of user k at SC n is expressed as R ( G ) k,n = BN log (1 + γ k,n ) = BN log M X m =1 | h m,k,n | p k,n σ m,n + q m,n ! ( a ) = BN log M X m =1 | h m,k,n | p k,n σ m,n + | h m,k,n | p k,n + σ m,n NT ( G ) m,n/B − , (9)where ( a ) is obtained by substituting q m,n by T m,n according to (7). Notice that as the allocatedfronthaul rate T ( G ) m,n → (versus ∞ ), the achievable end-to-end rate in (9) converges to zero (orthat of the wireless link capacity). Then, the achievable throughput of all users is expressed as R ( G )sum = K X k =1 X n ∈ Ω k R ( G ) k,n = BN K X k =1 X n ∈ Ω k log M X m =1 | h m,k,n | p k,n σ m,n + | h m,k,n | p k,n + σ m,n NT ( G ) m,n/B − . (10)From (10), it is clearly seen that the sum-rate performance depends on both the users’ powerallocations, { p k,n } , and the RRHs’ fronthaul rate allocations, { T ( G ) m,n } , over the SCs. B. Uniform Scalar Quantization
In practice, it is very difficult to find the quantization codebooks to achieve the throughputgiven in (10) subject to the fronthaul capacity constraints given in (8). In this subsection, weconsider using practical uniform scalar quantization technique at each RRH and derive theachievable sum-rate.A typical method to implement the uniform quantization is via separate in-phase/quadrature(I/Q) quantization, where the architecture is shown in Fig. 3. Specifically, the received complexsymbol y m,n given in (1) could be presented by its I and Q parts: y m,n = y Im,n + jy Qm,n , ∀ m, n, (11) m,n ¯ y m,n y m,n y m,n I I I m,n ¯ Normalization Uniform Quantization y m,n y m,n y m,n Q Q Q y m,n IQ ^^ Fig. 3. Schematic of uniform scalar quantization. where j = − , and the I-branch symbol y Im,n and Q-branch symbol y Qm,n are both real Gaussianrandom variables with zero mean and variance ( | h m,k,n | p k,n + σ m,n ) / . As a result, each RRH m first normalizes its I-branch and Q-branch symbols at SC n to ¯ y Im,n and ¯ y Qm,n by factors η Im,n and η Qm,n , and then implements uniform scalar quantization to ¯ y Im,n and ¯ y Qm,n with D m,n quantizationbits, separately. For conciseness, we summarize the implementation details of the uniform scalarquantization in Appendix A.In the following, we present the end-to-end achievable throughput of all users subject tothe fronthaul capacity constraints under the uniform scalar quantization technique described inAppendix A. Proposition 3.1:
With the uniform scalar quantization scheme, the transmission rate fromRRH m to the BBU in its fronthaul link is given as T ( U ) m = N X n =1 T ( U ) m,n ≤ ¯ T m , m = 1 , · · · , M, (12)where T ( U ) m,n denotes the transmission rate in RRH m ’s fronthaul link to forward its received dataat SC n , i.e., T ( U ) m,n = 2 BD m,n N , ∀ m, n. (13) Proof:
Please refer to Appendix B.
Proposition 3.2:
With the uniform scalar quantization scheme, an achievable end-to-end through-put of all users is expressed as R ( U )sum = K X k =1 X n ∈ Ω k R ( U ) k,n , (14)where the achievable rate of user k at SC n is expressed as R ( U ) k,n = BN log M X m =1 | h m,k,n | p k,n σ m,n + 3( | h m,k,n | p k,n + σ m,n )2 − NT ( U ) m,nB . (15) Proof:
Please refer to Appendix C. Notice that (15) holds when T ( U ) m,n ≥ (2 B ) /N (i.e., D m,n ≥ ) according to (13). Similar to(10) for the ideal case of Gaussian compression, the sum-rate in (14) with the uniform scalarquantization also jointly depends on both the users’ power allocations, { p k,n } , and the RRHs’fronthaul rate allocations, { T ( U ) m,n } , over the SCs. Furthermore, given the same set of power andfronthaul rate allocations, the achievable rate in (14) is always strictly less than that in (10)provided that T ( G ) m,n = T ( U ) m,n ≥ (2 B ) /N , ∀ m, n .IV. P ROBLEM F ORMULATION
In this paper, given the wireless bandwidth B , each user k ’s SC allocation Ω k ’s as wellas transmit power constraint ¯ P k ’s, and each RRH m ’s fronthaul link capacity ¯ T m ’s, we aimto maximize the end-to-end throughput of all the users subject to each RRH’s fronthaul linkcapacity constraint by jointly optimizing the wireless power control and fronthaul rate allocation.Specifically, for the benchmark scheme, i.e., the theoretical Gaussian test channel based schemein Section III-A, we are interested in solving the following problem. (P1) : Maximize { p k,n ,T ( G ) m,n } R ( G )sum Subject to N X n =1 T ( G ) m,n ≤ ¯ T m , ∀ m ∈ M , X n ∈ Ω k p k,n ≤ ¯ P k , ∀ k ∈ K , where R ( G )sum is given in (10) and T ( G ) m,n is given in (7). Furthermore, for the proposed uniformscalar quantization based scheme in Section III-B, we are interested in solving the followingproblem. (P2) : Maximize { p k,n ,T ( U ) m,n } R ( U )sum Subject to N X n =1 T ( U ) m,n ≤ ¯ T m , ∀ m ∈ M , X n ∈ Ω k p k,n ≤ ¯ P k , ∀ k ∈ K ,T ( U ) m,n = 2 BD ( U ) m,n N , D m,n ∈ { , , · · · } is an integer , ∀ m ∈ M , ∀ n ∈ N , where R ( U )sum is given in (14) and T ( U ) m,n is given in (13).Recall that with the same rate allocations in the fronthaul links for the two schemes, i.e., T ( G ) m,n = T ( U ) m,n ≥ (2 B ) /N , ∀ m, n , R ( G )sum in (10) is always larger than R ( U )sum given in (14). Furthermore,uniform scalar quantization requires that the fronthaul rate allocated at each SC must be aninteger multiplication of (2 B ) /N . Due to the above two reasons, in general the optimal value of problem (P2) is smaller than that of problem (P1), i.e., R ( U )sum < R ( G )sum . It is also worth notingthat user association is also determined from solving problems (P1) and (P2), since if with theobtained solution we have T m,n = 0 , ∀ n ∈ Ω k , RRH m will not quantize and forward user k ’ssignal to the BBU for decoding, or equivalently RRH m does not serve that user at all.It can be also observed that both problems (P1) and (P2) are non-convex since their objectivefunctions are not concave over p k,n ’s and T m,n ’s; thus, it is difficult to obtain their optimalsolutions in general. In the following two sections, we first study the special case of problems(P1) and (P2) with one user and one RRH to shed some light on the mutual influence betweenthe wireless power allocation and fronthaul rate allocation, and then propose efficient algorithmsto solve problems (P1) and (P2) for the general case of multiple users and multiple RRHs.V. S PECIAL C ASE : S
INGLE U SER AND S INGLE
RRHIn this section, we study problems (P1) and (P2) for the special case of K = 1 and M = 1 .For convenience, in the rest of this section we omit the subscripts of k and m in all the notationsin problems (P1) and (P2). A. Gaussian Test Channel
It can be shown that problem (P1) is still a non-convex problem for the case of K = 1 and M = 1 . In this subsection, we propose to apply the alternating optimization technique to solvethis problem. Specifically, first we fix the fronthaul rate allocation T ( G ) n = ˆ T ( G ) n ’s in problem(P1) and optimize the wireless power allocation by solving the following problem. Maximize { p n } N N X n =1 log | h n | p n σ n + | h n | p n + σ n N ˆ T ( G ) n /B − Subject to N X n =1 p n ≤ ¯ P . (16)Let { ˆ p n } denote the optimal solution to problem (16). Next, we fix the wireless power allocation p n = ˆ p n ’s in problem (P1) and optimize the fronthaul rate allocation by solving the followingproblem. Maximize { T ( G ) n } N N X n =1 log | h n | ˆ p n σ n + | h n | ˆ p n + σ n NT ( G ) n /B − Subject to N X n =1 T ( G ) n ≤ ¯ T . (17)Let { ˆ T ( G ) n } denote the optimal solution to problem (17). The above update of { p n } and { T ( G ) n } is iterated until convergence. In the following, we show how to solve problems (16) and (17),respectively. ˆ T ( G ) n (Mbps) f n ( ˆ T ( G ) n ) (a) A plot of f n ( ˆ T ( G ) n ) over ˆ T ( G ) n ˆ T ( G ) n (Mbps) p n ( W ) SC1SC2SC3SC4 (b) A plot of ˆ p n ’s over ˆ T ( G ) n Fig. 4. Threshold-based power allocation.
First, it can be shown that the objective function of problem (16) is concave over p n ’s. As aresult, problem (16) is a convex problem, and thus can be efficiently solved by the Lagrangianduality method [19]. We then have the following proposition. Proposition 5.1:
The optimal solution to problem (16) is expressed as ˆ p n = ( − α n + √ α n − η n , if | h n | σ n > f n ( ˆ T ( G ) n ) , , otherwise . n = 1 , · · · , N, (18)where α n = σ n (2 N ˆ T ( G ) nB + 1) | h n | , (19) η n = σ n N ˆ T ( G ) nB | h n | − σ n (2 N ˆ T ( G ) nB − λN | h n | ln 2 , (20) f n ( ˆ T ( G ) n ) = 2 N ˆ T ( G ) nB λN ln 22 N ˆ T ( G ) nB − , (21)and λ is a constant under which P Nn =1 ˆ p n = ¯ P n . Proof:
Please refer to Appendix D.It can be shown that as ˆ T ( G ) n ’s go to infinity, i.e., the case without fronthaul link constraint inproblem (P1), the optimal power allocation given in (18) reduces to ˆ p n = ( λN ln 2 − σ n | h n | , if | h n | σ n > λN ln 2 , , otherwise , n = 1 , · · · , N, (22)which is consistent with the conventional water-filling based power allocation. In the following,we discuss about the impact of fronthaul rate allocation on the optimal power allocation givenin (18) with finite values of ˆ T ( G ) n ’s.It can be observed from (18) that the optimal wireless power allocation with given ˆ T ( G ) n ’s isthreshold-based. In the following, we give a numerical example to investigate the monotonicity of the threshold f n ( ˆ T ( G ) n ) over ˆ T ( G ) n , ∀ n (note that in (21) λ is also a function of ˆ T ( G ) n ’s). In thisexample, the bandwidth of the wireless link is assumed to be B = 100 MHz, which is equallydivided into SCs. The channel powers are given as | h | = 1 . × − , | h | = 6 . × − , | h | = 2 . × − , | h | = 1 . × − . Moreover, the power spectral density of the backgroundnoise is assumed to be − dBm/Hz, and the noise figure due to receiver processing is dB.The transmit power of the user is dBm. It is further assumed that the fronthaul rates areequally allocated among SCs, i.e., ˆ T ( G ) n = ¯ T / , ∀ n , and thus f n ( ˆ T ( G ) n ) ’s are of the same value.Fig. 4 (a) shows the plot of f n ( ˆ T ( G ) n ) versus ˆ T ( G ) n by increasing the value of ¯ T in problem (16).It is observed in this particular setup (and many others used in our simulations for which theresults are not shown here due to the space limitation) that in general f n ( ˆ T ( G ) n ) is increasing with ˆ T ( G ) n . This implies that as ˆ T ( G ) n increases, more SCs with weaker channel powers tend to be shutdown. The reason is as follows. The dynamic range of the received signal at the SC with strongerchannel power is larger, and thus with equal ˆ T ( G ) n ’s, the corresponding quantization noise level isalso larger. When ˆ T ( G ) n ’s are small, quantization noise dominates the end-to-end rate performanceand thus the relatively small quantization noise level at the SC with weaker channel power mayoffset the loss due to the poor channel condition. However, as ˆ T ( G ) n increases, the quantizationnoise becomes smaller, until the wireless link dominates the end-to-end performance. In thiscase, we should shut down some SCs with poor channel conditions just as water-filling basedpower allocation given in (22).To verify the above analysis, Fig. 4 (b) shows the optimal power allocation among the SCsversus different values of ˆ T ( G ) n = ¯ T / in the above numerical example. It is observed that when ¯ T ( G ) n is small, in general the SCs with poorer channel conditions are allocated higher transmitpower since the quantization noise levels are small at these SCs. As ˆ T ( G ) n increases, the SCs withpoorer channels are allocated less and less transmit power. Specially, when ˆ T ( G ) n ≥ . Mbpsor ¯ T ≥ . Gbps, SC 4 with the poorest channel condition is shut down for transmission. It isalso observed that when ˆ T ( G ) n is sufficiently large such that the quantization noise is negligible,the power allocation converges to the water-filling based solution given in (22).Next, similar to problem (16), it can be shown that problem (17) is a convex problem andthus can be efficiently solved by the Lagrangian duality method. We then have the followingproposition. Proposition 5.2:
The optimal solution to problem (17) can be expressed as ˆ T ( G ) n = (cid:26) BN log − βBβB + BN log ν ( n ) , if ν n > βB − βB , , otherwise , n = 1 , · · · , N, (23)where ν n = | h n | ˆ p n σ n , (24) P o w e r ( W ) Algorithm IEqual Fronthaul Rate AllocationWater−Filling Power Allocation (a) Power Allocation F r on t hau l r a t e ( M bp s ) Algorithm IEqual Fronthaul Rate AllocationWater−Filling Power Allocation (b) Fronthaul Rate AllocationFig. 5. Power and fronthaul rate allocation among SCs. and β < B is a constant under which P Nn =1 ˆ T ( G ) n = ¯ T . Proof:
Please refer to Appendix E.Similar to the optimal power allocation given in (18), it can be inferred from Proposition 5.2that the optimal fronthaul rate allocation with given ˆ p n ’s is also threshold-based. If the receivedsignal SNR, ν n , at SC n is below the threshold βB/ (1 − βB ) , the RRH should not quantize andforward the signal at this SC to the BBU for decoding. On the other hand, if ν n > βB/ (1 − βB ) ,more quantization bits should be allocated to the SCs with higher values of ν n ’s.After problems (16) and (17) are solved by Propositions 5.1 and 5.2, we are ready to proposethe overall algorithm to solve problem (P1), which is summarized in Table I. It can be shownthat a monotonic convergence can be guaranteed for Algorithm I since the objective value ofproblem (P1) is increased after each iteration and it is practically bounded. TABLE I A LGORITHM I : A LGORITHM FOR P ROBLEM (P1)
WHEN K = 1 AND M = 1
1. Initialize: Set T ( G, n = ¯ TN , ∀ n , R (0) = 0 , and i = 0 ;2. Repeata. i = i + 1 ;b. Update { p ( i ) n } by solving problem (16) with ˆ T ( G ) n = T ( G,i − n , ∀ n , according to Proposition 5.1;c. Update { T ( G,i ) n } by solving problem (17) with ˆ p n = p ( i ) n , ∀ n , according to Proposition 5.2;3. Until R ( i ) − R ( i − ≤ ε , where R ( i ) denotes the objective value of problem (P1) achieved by { p ( i ) n } and { T ( G,i ) n } , and ε is a small value to control the accuracy of the algorithm. With the proposed Algorithm I to solve (P1), we provide a numerical example to analyze theproperties of the resulting wireless power and fronthaul rate allocation among SCs. The setupof this example is the same as that for Fig. 4, while the fronthaul link capacity is assumed to be ¯ T = 400 Mbps. Fig. 5 (a) and Fig. 5 (b) show the wireless power allocation and the fronthaulrate allocation at each SC, respectively, obtained via Algorithm I. For comparison, in Fig. 5 (a) we also provide the power allocation at each SC obtained by solving problem (16) withequal fronthaul rate allocation, as well as the water-filling based power allocation at each SC(obtained without considering fronthaul link constraint), and in Fig. 5 (b) the equal fronthaulrate allocation as well as the fronthaul rate allocation obtained by solving problem (17) withwater-filling based power allocation. It is observed in Fig. 5 (a) that Algorithm I results in amore greedy power allocation solution among SCs than the water-filling based method: besidesSC , SC with the second poorest channel condition is also forced to shut down, and the savedpower and quantization bits are allocated to SCs and with better channel conditions. Thisis in sharp contrast to the case of equal fronthaul rate allocation for which SC is allocatedthe highest transmit power and even SC with the poorest channel condition is still used fortransmission. Moreover, in Fig. 5 (b), the fronthaul rate allocations at SCs − obtained byAlgorithm I are . Mbps, . , Mbps, and Mbps, respectively. As a result, differentfrom equal fronthaul rate allocation, Algorithm I tends to allocate more quantization bits to theSCs with strong channel power to explore their good channel conditions, while allocating less(or even no) quantization bits to the SCs with weaker power. A similar fronthaul rate allocationis observed for the water-filling power allocation case.
B. Uniform Scalar Quantization
In this subsection, we study problem (P2) in the case of K = 1 and M = 1 to evaluatethe efficiency of the uniform quantization based scheme. We first solve problem (P2) in thiscase by extending the results in Section V-A. It can be observed that without the last set ofconstraints involving integer D n ’s, problem (P2) is very similar to problem (P1). As a result,in the following we propose a two-stage algorithm to solve problem (P2). First, we ignore theinteger constraints in problem (P2), which is denoted by problem (P2-NoInt), and apply analternating optimization based algorithm similar to Algorithm I to solve it (the details of whichare omitted here for brevity). Let { ˆ p n , ˆ T ( U ) n } denote the converged wireless power and fronthaulrate allocation solution to problem (P2-NoInt). Next, we fix p n = ˆ p n ’s and find a feasible solutionof T ( U ) n ’s based on { ˆ p n , ˆ T ( U ) n } such that D n = N T ( U ) n / B ’s are integers, ∀ n , in problem (P2).This is achieved by rounding each N ˆ T ( U ) n / B to its nearby integer as follows: N T ( U ) n B = ( ⌊ N ˆ T ( U ) n B ⌋ , if N ˆ T ( U ) n B − ⌊ N ˆ T ( U ) n B ⌋ ≤ α, ⌈ N ˆ T ( U ) n B ⌉ , otherwise , n = 1 , · · · , N, (25)where ≤ α ≤ , and ⌊ x ⌋ denotes the maximum integer that is no larger than x . Note thatwe can always find a feasible solution of T n ’s by simply setting α = 1 in (25) since in thiscase we have P Nn =1 T ( U ) n ≤ P Nn =1 ˆ T ( U ) n ≤ ¯ T . In the following, we show how to find a betterfeasible solution by optimizing α . It can be observed from (25) that with decreasing α , the values of T ( U ) n ’s will be non-decreasing, ∀ n . As a result, the objective value of problem (P2)will be non-decreasing, but the fronthaul link constraint in problem (P2) will be more difficultto satisfy. Thereby, we propose to apply a simple bisection method to find the optimal value of α , denoted by α ∗ , which is summarized in Table II. After α ∗ is obtained, the feasible solutionof T ( U ) n ’s can be efficiently obtained by taking α ∗ into (25). Notice that by (25) the numberof quantization bits per SC, D n , is now allowed to be zero, instead of being a strictly positiveinteger as assumed in Sections III and IV. TABLE II A LGORITHM II : A LGORITHM TO F IND F EASIBLE S OLUTION OF T ( U ) n ’ S TO PROBLEM (P2)1. Initialize α min = 0 , α max = 1 ;2. Repeata. Set α = α min + α max ;b. Take α into (25). If T ( U ) n ’s, ∀ n , satisfy the fronthaul link capacity constraint in problem (P2), set α max = α ;otherwise, set α min = α ;3. Until α max − α min < ε , where ε is a small value to control the accuracy of the algorithm;4. Take α into (25) to obtain the feasible solution of T ( U ) n ’s, ∀ n . Next, we evaluate the end-to-end rate performance of the uniform scalar quantization basedscheme in the case of K = 1 and M = 1 . Note that a cut-set based capacity upper bound ofour studied C-RAN is [5] C = min N N X n =1 log (cid:18) | h n | p wf n σ n (cid:19) , ¯ TB ! bps / Hz , (26)where { p wf n } is the water-filling based optimal power solution given in (22). Proposition 5.3:
In the case of K = 1 and M = 1 , let ¯ R ( G )sum denote the optimal value ofproblem (P1) with an additional set of constraints of q n = | h n | p n + σ n NT ( G ) nB − σ n , n = 1 , · · · , N. (27)Then we have ¯ R ( G )sum /B ≥ C − . Proof:
Please refer to Appendix F.Proposition 5.3 implies that with the simple solution { p n = ˇ p n , T ( G ) n = ( B/N ) log (2 + | h n | ˇ p n /σ n ) } with ˇ p n ’s denoting the optimal solution to problem (56) given in Appendix F, theGaussian test channel based scheme can achieve a capacity to within bps/Hz. Next, for theuniform scalar quantization, by setting the quantization noise level given in (43) as q n = 3 σ n , ∀ n , in problem (P2-NoInt), we have the following proposition. In the case of D n = 0 and hence T ( U ) n = 0 , for any SC n , the achievable end-to-end rate for the uniform scalar quantizationgiven in (15) no longer holds, which instead should be set to zero intuitively. Proposition 5.4:
In the case of K = 1 and M = 1 , { p n = ˇ p n , T ( G ) n = ( B/N ) log (1 + | h n | ˇ p n /σ n ) } is a feasible solution to problem (P2-NoInt). Let ¯ R ( U )sum denote the objective valueof problem (P2-NoInt) achieved by the above solution, we then have ¯ R ( U )sum /B > ¯ R ( G )sum /B − . Proof:
Please refer to Appendix G.It can be inferred from Propositions 5.3 and 5.4 that ¯ R ( U )sum /B > ¯ R ( G )sum /B − ≥ C − . As aresult, we have the following corollary. Corollary 5.1:
Without the constraints that the number of quantization bits per SC is aninteger, with the simple solution { p n = ˇ p n , T ( G ) n = ( B/N ) log (1 + | h n | ˇ p n /σ n ) } , the uniformscalar quantization based scheme at least achieves a capacity to within bps/Hz in the case of K = 1 and M = 1 .Corollary 5.1 gives a worst-case performance gap of the proposed uniform quantizationbased scheme to the cut-set upper bound C in (26) if we ignore the constraints that eachquantization level is represented by an integer number of bits. However, it is difficult to analyzethe performance loss due to these integer constraints. In the following subsection, we willprovide a numerical example to show the impact of the integer constraints on the end-to-endrate performance. C. Numerical Example
In this subsection, we provide a numerical example to verify our results for the case of K = 1 and M = 1 . The setup of this example is summarized as follows. The channel bandwidth isassumed to be B = 100 MHz, which is equally divided into N = 32 SCs. The user’s transmitpower is dBm. It is assumed that the distance between the user and the RRH is d = 50 m.The pass loss model is L = 30 . . ( d ) dB. Moreover, it is assumed that the powerspectral density of the AWGN at the RRH is − dBm/Hz, and the noise figure is dB. First,we evaluate the performance of the proposed uniform scalar quantization based scheme againstthat of the Gaussian test channel based scheme as well as the capacity upper bound given in(26). Fig. 6 shows the end-to-end rate achieved by various schemes versus the fronthaul linkcapacity. Note that with the algorithm proposed for problem (P2-NoInt) in Section V-B, we use { p n = ˇ p n , T ( G ) n = ( B/N ) log (1 + | h n | ˇ p n /σ n ) } as the initial point such that the worst-caseperformance gap shown in Corollary 5.1 can be guaranteed. It is observed from Fig. 6 that forvarious values of ¯ T , uniform scalar quantization based scheme without the integer constraintsin problem (P2) does achieve a capacity within bps/Hz to C . Moreover, it is observed thatwith Algorithm II, the performance loss due to the integer constraints is negligible. However,if we simply set α = 1 in (25) to find feasible T ( U ) n ’s, there will be a considerable rate loss.As a result, our proposed Algorithm II is practically useful for setting α such that uniformscalar quantization based scheme can perform very close to the capacity upper bound. Last, it E nd − t o − E nd R a t e ( bp s / H z ) Upper BoundGaussian Test ChannelUniform Quantization: without Integer ConstraintsUniform Quantization: with Integer ConstraintsUniform Quantization: α =1 Fig. 6. Performance of uniform scalar quantization. is observed that the performance gap of all the schemes to the upper bound C vanishes as thefronthaul link capacity increases. This is because if ¯ T is sufficiently large at the RRH, eachsymbol can be quantized by a large number of bits such that the specific quantization methoddoes not affect the quantization noise significantly.To further illustrate the gain from joint optimization of wireless power and fronthaul rateallocation, in the following we introduce some benchmark schemes where either wireless poweror fronthaul rate allocation is optimized, but not both. • Benchmark Scheme 1: Equal Power Allocation.
In this scheme, the user allocates itstransmit power equally to each SC, i.e., p n = ¯ P /N , ∀ n . Then, with the given equal powerallocation, we optimize the fronthaul rate allocation at the RRH to maximize the end-to-endrate. • Benchmark Scheme 2: Water-Filling Power Allocation.
In this scheme, the user ignoresthe fronthaul link constraints and allocates its transmit power based on water-filling solutionas shown in (22). Then, with the given water-filling based power allocation, we optimizethe fronthaul rate allocation at the RRH to maximize the end-to-end rate. • Benchmark Scheme 3: Equal Fronthaul Rate Allocation.
In this scheme, the RRHequally allocates its fronthaul link capacity among SCs, T ( U ) n = ¯ T /N . Then, with the givenequal fronthaul rate allocation, we optimize the transmit power of the user to maximize theend-to-end rate. • Benchmark Scheme 4: Equal Power and Fronthaul Rate Allocation.
In this scheme,the user allocates its transmit power equally to each SC, and the RRH equally allocates itsfronthaul link bandwidth among SCs.Fig. 7 shows the performance comparison among various proposed solutions for the uniformscalar quantization based scheme. It is observed that compared with Benchmark Schemes 1-4where only either wireless power or fronthaul rate allocation is optimized, our joint optimization Fronthaul Capacity (Gbps)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 E nd - t o - E nd R a t e ( bp s / H z ) Joint OptimizationBenchmark Scheme 1Benchmark Scheme 2Benchmark Scheme 3Benchmark Scheme 4
Fig. 7. Performance gain due to joint optimization of wireless power and fronthaul rate allocation. solution proposed in Section V-B achieves a much higher end-to-end rate, especially when thefronthaul link capacity is small, e.g., ¯ T ≤ . Gbps. Furthermore, it is observed from BenchmarkSchemes 1 and 3 that when ¯ T is small, fronthaul rate optimization plays the dominant role inimproving the end-to-end rate performance, while when ¯ T is large, most of the optimizationgain comes from the wireless power allocation. Furthermore, when ¯ T is sufficiently large, theperformance of Benchmark Schemes 2 and 3, for which wireless power allocation is optimized,even converges to the joint optimization solution proposed in Section V-B.VI. G ENERAL C ASE : M
ULTIPLE U SERS AND M ULTIPLE
RRH S In this section, we consider the joint wireless power allocation and fronthaul rate allocationin the general C-RAN with multiple users and multiple RRHs, i.e., K ≥ and M ≥ . A. Gaussian Test Channel
In this subsection, we solve problem (P1). It is worth noting that different from Section V-A,in the case of multiple RRHs, the throughput R ( G )sum given in (10) is not concave over T ( G ) m,n ’s withgiven p k,n ’s due to the summation over m in (6). As a result, the alternating optimization basedsolution proposed in Section V-A cannot be directly extended to the general case of K ≥ and M ≥ .To deal with the above difficulty, we change the design variables in problem (P1). Define ψ m,n = 2 NT ( G ) m,nB − , ∀ m, n. (28)Then, by changing the design variables of problem (P1) from { p k,n , T ( G ) m,n } to { p k,n , ψ m,n } , problem (P1) is transformed into the following problem. Maximize { p k,n ,ψ m,n } BN K X k =1 X n ∈ Ω k log M X m =1 | h m,k,n | p k,n ψ m,n σ m,n ψ m,n + | h m,k,n | p k,n + σ m,n ! Subject to BN N X n =1 log (1 + ψ m,n ) ≤ ¯ T m , ∀ m, N X n =1 p k,n ≤ ¯ P k , ∀ k. (29)Problem (29) is still a non-convex problem. In the following, we propose to apply the techniquesof alternating optimization as well as convex approximation to solve it.First, by fixing ψ m,n = ˆ ψ m,n ’s, we optimize the transmit power allocation p k,n ’s by solvingthe following problem. Maximize { p k,n } BN K X k =1 X n ∈ Ω k log M X m =1 | h m,k,n | p k,n ˆ ψ m,n σ m,n ˆ ψ m,n + | h m,k,n | p k,n + σ m,n ! Subject to N X n =1 p k,n ≤ ¯ P k , ∀ k. (30)Let ˆ p k,n ’s denote the optimal solution to problem (30). Then, by fixing p k,n = ˆ p k,n ’s, we optimizethe fronthaul rate allocation by solving the following problem. Maximize { ψ m,n } BN K X k =1 X n ∈ Ω k log M X m =1 | h m,k,n | ˆ p k,n ψ m,n σ m,n ψ m,n + | h m,k,n | ˆ p k,n + σ m,n ! Subject to BN N X n =1 log (1 + ψ m,n ) ≤ ¯ T m , ∀ m. (31)Let ˆ ψ m,n ’s denote the optimal solution to problem (31). Then, the above update of p k,n ’s and ψ m,n ’s is iterated until convergence. In the following, we provide how to solve problems (30)and (31), respectively.First, we consider problem (30). We have the following lemma. Lemma 6.1:
The objective function of problem (30) is a concave function over { p k,n } . Proof:
Please refer to Appendix H.According to Lemma 6.1, problem (30) is a convex optimization problem. As a result, itsoptimal solution can be efficiently obtained via the interior-point method [19].Next, we consider problem (31). Similar to Lemma 6.1, it can be shown that the objectivefunction of problem (31) is a concave function over ψ m,n ’s. However, the fronthaul link capacityconstraints in problem (31) are not convex. In the following, we apply the convex approximationtechnique to convexify the fronthaul link capacity constraints. Specifically, since according to (28) T ( G ) m is concave over ψ m,n ’s, its first-order approximation serves as an upper bound to it,i.e., T ( G ) m = BN N X n =1 log (1 + ψ m,n ) ≤ BN N X n =1 log (1 + ˜ ψ m,n ) + ψ m,n − ˜ ψ m,n (1 + ˜ ψ m,n ) ln 2 ! , m = 1 , · · · , M. (32)Note that the above inequality holds given any ˜ ψ k,n ’s. As a result, we solve the following problemvia a relaxation of problem (31). Maximize { ψ m,n } BN K X k =1 X n ∈ Ω k log M X m =1 | h m,k,n | ˆ p k,n ψ m,n σ m,n ψ m,n + | h m,k,n | ˆ p k,n + σ m,n ! Subject to BN N X n =1 log (1 + ˜ ψ m,n ) + ψ m,n − ˜ ψ m,n (1 + ˜ ψ m,n ) ln 2 ! ≤ ¯ T m , ∀ m. (33)Problem (33) is a convex problem, and thus its optimal solution, denoted by ˇ ψ m,n ’s, can beefficiently obtained via the interior-point method [19]. Then we have the following lemma. Lemma 6.2:
Suppose that ˜ ψ m,n ’s is a feasible solution to problem (31), i.e., BN P Nn =1 log (1 +˜ ψ m,n ) ≤ ¯ T m , ∀ m . Then, ˇ ψ m,n ’s is a feasible solution to problem (31) and achieves an objectivevalue no smaller than that achieved by the solution ˜ ψ m,n ’s. Proof:
Please refer to Appendix I.Since the optimal solution to problem (31), i.e., ˆ ψ m,n ’s, is difficult to obtain, in the followingwe use ˇ ψ m,n as the solution to (31) according to Lemma 6.2, i.e., ˆ ψ m,n = ˇ ψ m,n , ∀ m, n .After problems (30) and (31) are solved, we are ready to propose the overall iterative algorithmto solve problem (29), which is summarized in Table III. Note that in Step 2.c., we set ˜ ψ m,n = ψ ( i − m,n ’s in problem (33). According to Lemma 6.2, ψ ( i ) m,n ’s will achieve a sum-rate that is nosmaller than that achieved by ψ ( i − m,n ’s. To summarize, a monotonic convergence can be guaranteedfor Algorithm III since the objective value of problem (29) is increased after each iteration andit is upper-bounded by a finite value. TABLE III A LGORITHM
III : A
LGORITHM FOR P ROBLEM (29)1. Initialize: Set ψ (0) m,n = 2 ¯ TmB − , ∀ m, n , R (0) = 0 , and i = 0 ;2. Repeata. i = i + 1 ;b. Update { p ( i ) k,n } by solving problem (30) with ˆ ψ m,n = ψ ( i − m,n , ∀ m, n , via interior-point method;c. Update { ψ ( i ) m,n } by solving problem (33) with ˆ p k,n = p ( i ) k,n and ˜ ψ m,n = ψ ( i − m,n , ∀ m, n , via interior-point method;3. Until R ( i ) − R ( i − ≤ ε , where R ( i ) denotes the objective value of problem (29) achieved by the solution { p ( i ) k,n , ψ ( i ) m,n } ,and ε is a small value to control the accuracy of the algorithm. B. Uniform Scalar Quantization
In this subsection, we propose an efficient algorithm to solve problem (P2) by jointly opti-mizing the wireless power allocation as well as the fronthaul rate allocation. To be consistentwith the solution to problem (P1) proposed in Section VI-A, we define ψ m,n = 2 NT ( U ) m,nB = 2 D m,n , ∀ m, n. (34)Then, by changing the design variables from { p k,n , T ( U ) m,n } into { p k,n , ψ m,n } , problem (P2) istransformed into the following problem. Maximize { p k,n ,ψ m,n } BN K X k =1 X n ∈ Ω k log M X m =1 | h m,k,n | p k,n ψ m,n σ m,n ψ m,n + 3( | h m,k,n | p k,n + σ m,n ) ! Subject to BN N X n =1 log ψ m,n ≤ ¯ T m , ∀ m, X n ∈ Ω k p k,n ≤ ¯ P k , ∀ k,ψ m,n = 2 D m,n , D m,n ∈ { , , · · · } is an integer , ∀ m, n. (35)It can be observed that if we ignore the last set of constraints involving integers D m,n ’s, thenproblem (35) is very similar to problem (29). As a result, we propose a two-stage algorithm tosolve problem (35). First, we ignore the last constraints in problem (35) and apply an alternatingoptimization based algorithm similar to Algorithm III to solve it (the details of which are omittedhere for brevity). Let { ˆ p k,n , ˆ ψ m,n } denote the obtained solution. Then we fix p k,n = ˆ p k,n ’s andfind a feasible solution ψ m,n ’s based on ˆ ψ m,n ’s such that D m,n = log ψ m,n ’s are integers. Forany given m = ¯ m , this is done by rounding log ˆ ψ ¯ m,n ’s, ∀ n , to their nearby integers as follows:
12 log ψ ¯ m,n = ( ⌊ log ˆ ψ ¯ m,n ⌋ , if log ˆ ψ ¯ m,n − ⌊ log ˆ ψ ¯ m,n ⌋ ≤ α ¯ m , ⌈ log ˆ ψ ¯ m,n ⌉ , otherwise , n = 1 , · · · , N, (36)where ≤ α ¯ m ≤ , ∀ ¯ m . Similar to Algorithm II for the special case of K = 1 and M = 1 ,the optimal value of α ¯ m can be efficiently obtained via a simple bisection method, and thus afeasible solution of ψ ¯ m,n ’s, ∀ n , is obtained according to (36). Last, by searching ¯ m from to M , the overall feasible solution { ψ m,n } is obtained. C. Numerical Example
In this subsection, we provide a numerical example to evaluate the sum-rate performance ofthe proposed uniform scalar quantization based scheme in a single C-RAN cluster with M = 7 RRHs and K = 16 users randomly distributed in a circular area of radius m. It is assumedthat the B = 300 MHz bandwidth of the wireless link is equally divided into N = 64 SCs, andeach user is pre-allocated
N/K = 4
SCs. It is further assumed that the capacities of all the Common Fronthaul Capacity (Gbps) E nd - t o - E nd S u m - R a t e ( G bp s ) Gaussian Test ChannelUniform Quantization: Joint OptimizationBenchmark Scheme 1Benchmark Scheme 2Benchmark Scheme 3Benchmark Scheme 4Benchmark Scheme 5
Fig. 8. Uniform scalar quantization: end-to-end sum-rate versus common fronthaul link capacity. fronthaul links are identical, i.e., ¯ T m = T , ∀ m . The other setup parameters are the same as thoseused in Section V-C. Similar to the single-user single-RRH case in Section V-C, we providevarious benchmark schemes. Note that Benchmark Schemes 1-4 introduced in Section V-C canbe simply extended to the general case of K ≥ and M ≥ . Furthermore, to compare the sum-rate performance between our studied OFDMA-based C-RAN and conventional OFDMA-basedcellular networks, we also consider the following benchmark scheme. • Benchmark Scheme 5: Conventional OFDMA.
In this scheme, we assume that each RRHoperates like conventional BS in cellular networks which directly decodes the messages ofits served users, rather than forwarding its received signals to the BBU for a joint decoding.For simplicity, we assume that each user is served by its nearest RRH. Then, the optimalpower solution for each user k among its assigned SCs Ω k is the standard “water-filling”solution given in (22).Fig. 8 shows the end-to-end sum-rate performance versus the common fronthaul link capacity, T , achieved by uniform quantization, Gaussian test channel, as well as Benchmark Schemes1-5 (Note that in Benchmark Scheme 5, since each RRH decodes the messages locally, weassume that the sum-rate is a constant regardless of fronthaul capacities). It is observed thatwith our proposed algorithm in Section VI-B, the sum-rate achieved by the uniform scalarquantization based scheme is very close to that achieved by the Gaussian test channel basedscheme for various fronthaul capacities. Furthermore, this performance gap vanishes as thefronthaul link capacities increase at all RRHs. It is also observed that compared with BenchmarkSchemes 1-4 where only either wireless power or fronthaul rate allocation is optimized, our jointoptimization solution proposed in Section VI-B achieves a much higher sum-rate, especially whenthe fronthaul link capacities are not sufficiently high. By comparing with Fig. 7, it is observedthat the joint optimization gain is more significant over the case of single user and single RRH.Last, it is observed that with joint optimization of wireless and fronthaul resource allocation,the sum-rate achieved by proposed OFDMA-based C-RAN is much higher than that achieved by Benchmark Scheme 5, i.e., conventional OFDMA, under the moderate capacity of currentcommercial fronthaul such as several Gbps.VII. C ONCLUSIONS AND F UTURE W ORK
In this paper, we have proposed joint wireless power control and fronthaul rate allocationoptimization to maximize the throughput performance of an OFDMA-based broadband C-RANsystem. In particular, we have considered using practical uniform scalar quantization instead ofthe information-theoretical quantization method in the system design. Efficient algorithms havebeen proposed to solve the joint optimization problems. Our results showed that the joint designachieves significant performance gain compared to optimizing either wireless power control orfronthaul rate allocation. Besides, we showed that the throughput performance of the proposedsimple uniform scalar quantization is very close to the performance upper (cut-set) bound. Thishas verified that high throughput performance could be practically achieved with C-RAN usingsimple fronthaul signal quantization methods.There are also many interesting topics to be studied in the area of fronthaul-constrainedOFDMA-based C-RAN system. For instance, the impact of imperfect fronthaul link with packetloss of quantized data; dynamic SC allocation among mobile users; multiple users coexistingon one SC to further improve the spectral efficiency; distributed quantization among RRHs toexploit the signal correlations; and joint wireless resource and fronthaul rate allocations in thedownlink, etc. A
PPENDIX
A. Uniform Scalar Quantization
In this appendix, we provide the details on the implementation of uniform scalar quantizationintroduced in Section III-B. First, each RRH normalizes the I-branch and Q-branch symbols ateach SC into the interval [ − , for quantization by the following scaling process: ¯ y χm,n = y χm,n η χm,n , χ ∈ { I, Q } , ∀ m, n. (37)Since y Im,n ’s and y Qm,n ’s are both real Gaussian random variables the instantaneous power ofwhich can go to infinity in some instances, the probability of overflow should be controlled bya proper selection of the scaling factors η Im,n ’s and η Qm,n ’s. In this paper, we apply the so-called“three-sigma rule” [12] to select the scaling factors. Specifically, since the average power of y Im,n and y Qm,n are both ( | h m,k,n | p k,n + σ m,n ) / , we set η Im,n = η Qm,n , η m,n = 3 r | h m,k,n | p k,n + σ m,n , ∀ m, n. (38) As a result, the probability of overflow for both the I-branch and Q-branch symbols is expressedas P ( | ¯ y Im,n | >
1) = P ( | ¯ y Qm,n | >
1) = 2 Q (3) = 0 . , ∀ m, n. (39)Note that in the case of overflow, the quantized value can be set to be if the scaled symbol islarger than or − if it is smaller than − .Next, RRH m implements uniform quantization on the normalized symbols ¯ y Im,n ’s and ¯ y Qm,n ’sat each SC in the interval [ − , . We assume that RRH m uses D m,n ≥ bits to quantize thesymbol received on SC n , resulting D m,n quantization levels, for which the quantization stepsize is given by ∆ m,n = 22 D m,n = 2 − D m,n , ∀ m, n. (40)Furthermore, for each normalized symbol ¯ y Im,n or ¯ y Qm,n , its quantized value is given by ˇ y χm,n = ⌈ D m,n − ¯ y χm,n ⌉ D m,n − − D m,n , χ ∈ { I, Q } , ∀ m, n, (41)where ⌈ x ⌉ denotes the minimum integer that is no smaller than x . Then, ˇ y Im,n ’s and ˇ y Qm,n ’s areencoded into digital codewords ˆ y Im,n ’s and ˆ y Qm,n ’s and transmitted to the BBU.
B. Proof of Proposition 3.1
Note that the I, Q symbols, i.e., y Im,n ’s and y Qm,n ’s, are obtained by sampling of the I, Qwaveforms, the bandwidth of which is B/ N , ∀ m, n . As a result, at each RRH, the Nyquistsampling rate for the I, Q waveforms at each SC is B/N samples per second. Furthermore, sinceat RRH m , each sample at SC n is represented by D m,n bits, the corresponding transmissionrate in the fronthaul link is expressed as T ( U ) m,n = BD m,n N + BD m,n N = 2 BD m,n N . (42)Then, the overall transmission rate from RRH m to the BBU in the fronthaul link is given as T ( U ) m = P Nn =1 T ( U ) m,n , which should not exceed the fronthaul link capacity ¯ T m , ∀ m . Proposition3.1 is thus proved. C. Proof of Proposition 3.2
To derive the end-to-end sum-rate, we need to calculate the power of the quantization errorgiven in (2), i.e., q m,n , ∀ m, n . Note that in (2) we have ˜ y m,n = η m,n (ˇ y Im,n + j ˇ y Qm,n ) , ∀ m, n .According to Widrow Theorem [18], if the number of quantization levels (i.e., D m,n ) is large,and the signal varies by at least some quantization levels from sample to sample, the quantizationnoise can be assumed to be uniformly distributed. As a result, we assume that the quantization errors for both I, Q signals, which are denoted by e Im,n and e Qm,n with e m,n = e Im,n + je Qm,n , areuniformly distributed in [ − η m,n ∆ m,n / , η m,n ∆ m,n / , ∀ m, n . Then we have q m,n = Z ηm,n ∆ m,n − ηm,n ∆ m,n ( e Im,n ) η m,n ∆ m,n de Im,n + Z ηm,n ∆ m,n − ηm,n ∆ m,n ( e Qm,n ) η m,n ∆ m,n de Qm,n = η m,n ∆ m,n | h m,k,n | p k,n + σ m,n )2 − D m,n ( a ) = 3( | h m,k,n | p k,n + σ m,n )2 − NT ( U ) m,nB , (43)where ( a ) is obtained by substituting D m,n by T ( U ) m,n according to (13). Then according to (6),a lower bound for the achievable rate of user k at SC n , by viewing w Hn e n given in (3) as theworst-case Gaussian noise (it is worth noting that the equivalent quantization error given in (3),i.e., w Hn e n , is the summation of N independent uniform distributed random variables e m,n ’s.According to the central limit theory, w Hn e n tends to be Gaussian distributed when N is large),can be expressed as R ( U ) k,n = BN log M X m =1 | h m,k,n | p k,n σ m,n + q m,n ! = BN log M X m =1 | h m,k,n | p k,n σ m,n + 3( | h m,k,n | p k,n + σ m,n )2 − NT ( U ) m,nB . (44)The end-to-end throughput of all users is thus expressed as R ( U )sum = P Kk =1 P n ∈ Ω k R ( U ) k,n . Propo-sition 3.2 is thus proved. D. Proof of Proposition 5.1
The Lagrangian of problem (16) is expressed as L ( { p n } , λ ) = 1 N N X n =1 log | h n | p n σ n + | h n | p n + σ n N ˆ T ( G ) n /B − − λ N X n =1 p n − ¯ P ! , (45)where λ is the dual variable associated with the transmit power constraint in problem (16). Then,the Lagrangian dual function of problem (16) is expressed as g ( λ ) = max p n ≥ , ∀ n L ( { p n } , λ ) . (46)The maximization problem (46) can be decoupled into parallel subproblems all having the samestructure and each for one SC. For one particular SC, the associated subproblem is expressed as max p n ≥ L n ( p n ) , (47) where L n ( p n ) = 1 N log | h n | p n σ n + | h n | p n + σ n N ˆ T ( G ) n /B − − λp n , n = 1 , · · · , N. (48)It can be shown that L n ( p n ) is concave over p n , ∀ n . The derivative of L n ( p n ) over p n is expressedas ∂ L n ( p n ) ∂p n = | h n | σ n (cid:18) N ˆ T ( G ) nB − (cid:19)(cid:18) | h n | p n + σ n N ˆ T ( G ) nB (cid:19) ( | h n | p n + σ n ) N ln 2 − λ, ∀ n. (49)By setting ∂ L n ( p n ) /∂p n = 0 , we have p n + α n p n + η n = 0 , n = 1 , · · · , N, (50)where α n ’s and η n ’s are given in (19) and (20), respectively. If η n < , then there exists a uniquepositive solution to the quadratic equation (50), denoted by ˜ p n = ( − α n + p α n − η n ) / . In thiscase, L n ( p n ) is an increasing function over p n in the interval (0 , ˜ p n ) , and decreasing function inthe interval [˜ p n , ∞ ) . As a result, L n ( p n ) is maximized when p n = ˜ p n . Otherwise, if η ≥ , thereis no positive solution to the quadratic equation (50), and thus L n ( p n ) is a decreasing functionover p n in the interval (0 , ∞ ) . In this case, L n ( p n ) is maximized when p n = 0 .After problem (46) is solved given any λ , in the following we explain how to find the optimaldual solution for λ . It can be shown that the objective function in problem (16) is an increasingfunction over { p n } , and thus the transmit power constraint must be tight in problem (16). Asa result, the optimal λ can be efficiently obtained by a simple bisection method such that thetransmit power constraint is tight in problem (16). Proposition 5.1 is thus proved. E. Proof of Proposition 5.2
Let β denote the dual variable associated with the fronthaul link capacity constraint in problem(17). Similar to Appendix A, it can be shown that problem (17) can be decoupled into the N subproblems with each one formulated as max T ( G ) n ≥ L n ( T ( G ) n ) , (51)where L n ( T ( G ) n ) = 1 N log | h n | ˆ p n σ n + | h n | ˆ p n + σ n NT ( G ) n /B − − βT n , n = 1 , · · · , N. (52)The derivative of L n ( T ( G ) n ) over T ( G ) n is expressed as ∂ L n ( T ( G ) n ) ∂T ( G ) n = 1 B − β − σ n NT ( G ) nB B (cid:18) | h n | ˆ p n + σ n NT ( G ) nB (cid:19) , n = 1 , · · · , N. (53) If β ≥ B , then ∂ L n ( T ( G ) n ) ∂T ( G ) n ≤ , i.e., L n ( T ( G ) n ) is a decreasing function over T ( G ) n , ∀ n . In thiscase, we have ¯ T ( G ) n = 0 , ∀ n , which cannot be the optimal solution to problem (17). As a result,the optimal dual solution must satisfy β < B . In this case, it can be shown that L n ( T ( G ) n ) is an increasing function over T ( G ) n when T ( G ) n < BN log − βB ) | h n | ˆ p n βBσ n , and decreasing functionotherwise. As a result, L n ( T ( G ) n ) is maximized at T ( G ) n = max( BN log − βB ) | h n | ˆ p n βBσ n , . Afterproblem (51) is solved given any β < B , the optimal β that is the dual solution to problem (17)can be efficiently obtained by a simple bisection method over (0 , B ) such that the fronthaul linkcapacity constraint is tight in problem (17). Proposition 5.2 is thus proved. F. Proof of Proposition 5.3
With constraints given in (27), R ( G )sum given in (10) reduces to R ( G )sum B = 1 N N X n =1 log (cid:18) | h n | p n σ n (cid:19) . (54)Moreover, it can be shown from (27) that T ( G ) B = 1 N N X n =1 log (cid:18) | h n | p n σ n (cid:19) = R ( G )sum B + 1 . (55)Thereby, with the additional constraints given in (27), problem (P1) can be simplified as thefollowing power control problem. Maximize { p n } N N X n =1 log (cid:18) | h n | p n σ n (cid:19) Subject to N N X n =1 log (cid:18) | h n | p n σ n (cid:19) + 1 ≤ ¯ TB N X n =1 p n ≤ ¯ P . (56)Let { ˇ p n } and { ˜ p n } denote the optimal power solution to problem (56) and the relaxed versionof problem (56) without the first fronthaul link constraint, respectively. If (1 /N ) P Nn =1 log (1 + | h n | ˜ p n / σ n ) + 1 ≤ ¯ T /B , we have ˇ p n = ˜ p n , ∀ n . Otherwise, it can be shown that any feasiblesolution to the following problem is optimal to problem (56): Find { p n } Subject to N N X n =1 log (cid:18) | h n | p n σ n (cid:19) + 1 = ¯ TB N X n =1 p n ≤ ¯ P . (57) To summarize, the cut-set bound based optimal value of problem (56) is expressed as ¯ R ( G )sum B = min ( N N X n =1 log (cid:18) | h n | ˜ p n σ n (cid:19) , ¯ TB − ) . (58)In the following, we compare this optimal value with the capacity upper bound C given in (26).First, we have N N X n =1 log (cid:18) | h n | p wf n σ n (cid:19) − < N N X n =1 log (cid:18) | h n | p wf n σ n (cid:19) ( a ) ≤ N N X n =1 log (cid:18) | h n | ˜ p n σ n (cid:19) , (59)where ( a ) is because { ˜ p n } is the optimal power solution to problem (56) without the fronthaullink constraint. It then follows that ¯ R ( G )sum B = min ( N N X n =1 log (cid:18) | h n | ˜ p n σ n (cid:19) , ¯ TB − ) ≥ min ( N N X n =1 log (cid:18) | h n | p wf n σ n (cid:19) − , ¯ TB − ) = C − . (60)Proposition 5.3 is thus proved. G. Proof of Proposition 5.4
First, it follows that T ( U ) = BN N X n =1 log (cid:18) | h n | ˇ p n σ n (cid:19) < BN N X n =1 log (cid:18) | h n | ˇ p n σ n (cid:19) ≤ ¯ T . (61)As a result, { p n = ˇ p n , T ( G ) n = ( B/N ) log (1 + | h n | ˇ p n /σ n ) } is a feasible solution to problem(P2-NoInt). Furthermore, with { p n = ˇ p n , T ( G ) n = ( B/N ) log (1 + | h n | ˇ p n /σ n ) } , R ( U )sum given in(14) reduces to R ( U )sum = BN N X n =1 log (cid:18) | h n | ˇ p n σ n (cid:19) . (62)It then follows that ¯ R ( U )sum B > N N X n =1 log (cid:18) | h n | ˇ p n σ n (cid:19) − R ( G )sum B − . (63)Proposition 5.4 is thus proved. H. Proof of Lemma 6.1
Define ϕ m,k,n ( p k,n ) = | h m,k,n | p k,n ˆ ψ m,n σ m,n ˆ ψ m,n + | h m,k,n | p k,n + σ m,n , if n ∈ Ω k , ∀ m, n. (64)Then, it can be shown that ϕ m,k,n ( p k,n ) is concave over p k,n , ∀ m, n . As a result, P Mm =1 ϕ m,k,n ( p k,n ) is concave over p k,n , ∀ k, n . According to the composition rule [19], log (1 + P Mm =1 ϕ m,k,n ( p k,n )) is concave over p k,n , ∀ k, n . It then follows that the objective function of problem (30), i.e., P Kk =1 P n ∈ Ω k log (1 + P Mm =1 ϕ m,k,n ( p k,n )) , is concave over { p k,n } . Lemma 6.1 is thus proved. I. Proof of Lemma 6.2
First, due to the inequality given in (32), any feasible solution to problem (33) must be afeasible solution to problem (31). Thereby, ˇ ψ m,n ’s must be feasible to problem (31). Next, it canbe observed that if ˜ ψ m,n ’s is feasible to problem (31), it must be feasible to problem (33). Since ˇ ψ m,n ’s is the optimal solution to problem (33), the sum-rate achieved by it must be no smallerthan that achieved by ˜ ψ m,n ’s. Lemma 6.2 is thus proved.R EFERENCES [1] “C-RAN: the road towards green RAN,” China Mobile Res. Inst., Beijing, China, Oct. 2011, White Paper, ver. 2.5.[2] R. Zhang, Y. C. Liang, and S. Cui, “Dynamic resource allocation in cognitive radio networks,”
IEEE Signal Process. Mag. ,vol. 27, no. 3, pp. 102-114, May 2010.[3] D. Gesbert, S. Hanly, H. Huang, S. Shamai, O. Simeone, and W. Yu, “Multi-cell MIMO cooperative networks: A new lookat interference,”
IEEE J. Sel. Areas Commun. , vol. 28, no. 9, pp. 1380-1408, Dec. 2010.[4] A. Sanderovich, S. Shamai, and Y. Steinberg, “Distributed MIMO receiver - Achievable rates and upper bounds,”
IEEETrans. Inf. Theory , vol. 55, no. 10, pp. 4419-4438, Oct. 2009.[5] A. Sanderovich, O. Somekh, H. V. Poor, and S. Shamai (Shitz), “Uplink macro diversity of limited backhaul cellularnetwork,”
IEEE Trans. Inf. Theory , vol. 55, no. 8, pp. 3457-3478, Aug. 2009.[6] S. H. Park, O. Simeone, O. Sahin, and S. Shamai, “Joint decompression and decoding for cloud radio access networks,”
IEEE Signal Processing Letters , vol. 20, no. 5, pp. 503-506, May 2013.[7] A. D. Coso and S. Simoens, “Distributed compression for MIMO coordinated networks with a backhaul constraint,”
IEEETrans. Wireless Commun. , vol. 8, no. 9, pp. 4698-4709, Sep. 2009.[8] S. H. Park, O. Simeone, O. Sahin, and S. Shamai, “Robust and efficient distributed compression for cloud radio accessnetworks,”
IEEE Trans. Vehicular Technology , vol. 62, no. 2, pp. 692-703, Feb. 2013.[9] Y. Zhou and W. Yu, “Optimized backhaul compression for uplink cloud radio access network,”
IEEE J. Sel. Areas Commun. ,vol. 32, no. 6, pp. 1295-1307, June 2014.[10] L. Zhou and W. Yu, “Uplink multicell processing with limited backhaul via per-base-station successive interferencecancellation,”
IEEE J. Sel. Areas Commun. , vol. 30, no. 10, pp. 1981-1993, Oct. 2013.[11] S. H. Park, O. Simeone, O. Sahin, and S. Shamai, “Joint precoding and multivariate backhaul compression for the downlinkof cloud radio access networks,”
IEEE Trans. Signal Process. , vol. 61, no. 22, pp. 5646-5658, Nov. 2013.[12] R.M. Gray and D.L. Neuhoff, “Quantization,”
IEEE Trans. Inf. Theory , vol. 44, no. 6, pp. 2325-2383, Oct. 1998.[13] A. Goldsmith,
Wireless Communication , Cambridge Univ. Press, 2005.[14] A. Avestimehr, S. Diggavi, and D. Tse, “Wireless network information flow: A deterministic approach,”
IEEE Trans. Inf.Theory , vol. 57, no. 4, pp. 1872-1905, Apr. 2011.[15] S. H. Lim, Y. H. Kim, A. El Gamal, and S. Y. Chung, “Noisy network coding,”
IEEE Trans. Inf. Theory , vol. 57, no. 5,pp. 3132-3152, May 2011.[16] A. Sanderovich, S. Shamai, Y. Steinberg, and G. Kramer, “Communication via decentralized processing,”
IEEE Trans. Inf.Theory , vol. 54, no. 7, pp. 3008-3023, July 2008.[17] A. El Gamal and Y. H. Kim,
Network information theory , Cambridge Univ. Press, 2011.[18] G. F. Franklin, J. D. Powell, and M. L. Workman,
Digital Control of Dynamic Systems , Addison Wesley, 3rd edition, 1990.[19] S. Boyd and L. Vandenberghe,