CRAN with Hybrid RF/FSO Fronthaul Links:
Joint Optimization of RF Time Allocation and Fronthaul Compression
Abstract
This paper considers the uplink of a cloud radio access network (CRAN) comprised of several multiantenna remote radio units (RUs) which compress the signals that they receive from multiple mobile users (MUs) and forward them to a CU via wireless fronthaul links. To enable reliable high rate fronthaul links, we employ a hybrid radio frequency (RF)/free space optical (FSO) system for fronthauling. To strike a balance between complexity and performance, we consider three different quantization schemes at the RUs, namely perantenna vector quantization (AVQ), perRU vector quantization (RVQ), and distributed source coding (DSC), and two different receivers at the CU, namely the linear minimum mean square error receiver and the optimal successive interference cancellation receiver. For this network architecture, we investigate the joint optimization of the quantization noise covariance matrices at the RUs and the RF time allocation to the multipleaccess and fronthaul links for rate region maximization. To this end, we formulate a unified weighted sum rate maximization problem valid for each possible pair of the considered quantization and detection schemes. To handle the nonconvexity of the unified problem, we transform it into a biconvex problem which facilitates the derivation of an efficient suboptimal solution using alternating convex optimization and golden section search. Our simulation results show that for each pair of the considered quantization and detection schemes, CRAN with hybrid RF/FSO fronthauling can achieve a considerable sum rate gain compared to conventional systems employing pure FSO fronthauling, especially under unfavorable atmospheric conditions. Moreover, employing a more sophisticated quantization scheme can significantly improve the system performance under adverse atmospheric conditions. In contrast, in clear weather conditions, when the FSO link capacity is high, the simple AVQ scheme performs close to the optimal DSC scheme.
I Introduction
Cloud radio access network (CRAN) is a novel cellular architecture whereby the baseband signal processing is moved from the base stations (BSs) to a cloudcomputing based central unit (CU) [2, 3, 4, 5]. In a CRAN, the BSs operate as remote radio units (RUs) that relay the mobile users’ (MUs’) data to the CU via fronthaul links. The CU jointly processes the MUs’ data which enables the exploitation of distributed multipleinput multipleoutput (MIMO) multiplexing gains. However, conveying the signals received at the RUs to the CU via the fronthaul links is a major challenge as this may require a huge fronthaul capacity, e.g. on the order of Gbits/sec [3]. Reviews of recent advances in fronthaulconstrained CRANs are provided in [3, 4, 6].
In most of the existing works on CRANs, the fronthaul links are modelled as abstract capacityconstrained channels based on informationtheoric approaches [7, 8, 9]. Among practical transmission media, optical fiber has been prominently considered as a suitable candidate for fronthaul links, mainly due to the large bandwidths available at optical frequencies [4, 10, 6]. However, the implementation and maintenance of optical fiber systems are costly. Another competitive candidate technology are free space optical (FSO) systems, which provide bandwidths comparable to those of optical fiber systems, are more costefficient in implementation and maintenance, and are easy to upgrade [6, 10, 5]. Unfortunately, the performance of FSO systems significantly deteriorates when the weather conditions are unfavorable, e.g. in snowy or foggy weather [11, 6, 10]. On the other hand, radio frequency (RF) systems can preserve link connectivity in a more reliable manner than FSO systems but offer lower data rates. Therefore, hybrid RF/FSO systems, where RF links are employed to support the FSO links, are appealling candidates for CRAN fronthauling. Hybrid systems benefit from both the huge bandwidth of FSO links and the reliability of RF links [10, 12, 6]. In this paper, we consider the uplink of a CRAN with RF multipleaccess links and hybrid RF/FSO fronthaul links. As the RF spectrum is limited, we assume that the multipleaccess and the fronthaul links share the same RF resources in an orthogonal manner.
To fully exploit the fronthaul capacity, we employ quantization at the RUs. To this end, we consider three different quantization schemes, namely perantenna vector quantization (AVQ), perRU vector quantization (RVQ), and distributed source coding (DSC). In contrast to AVQ, RVQ exploits the correlations between the signals received at the different antennas of a given RU, whereas DSC also takes advantage of the correlations among the signals received at different RUs [13, 14]. In addition, for the CU, we consider two different receivers, namely a linear minimum mean square error (MMSE) receiver and the optimal successive interference cancellation (SIC) receiver. The considered quantization and detection schemes offer different tradeoffs between complexity and performance. Thereby, the performance improvements of DSC over RVQ and of RVQ over AVQ come at the expense of an increased complexity. Similarly, the SIC receiver generally outperforms the linear MMSE receiver at the cost of a higher computational complexity. In practice, one can select a suitable pair of quantization and detection schemes given the affordable complexity at the RU and CU.
The goal of this paper is to jointly optimize the quantization noise covariance matrices at the RUs and the RF time allocated to the RF multipleaccess and fronthaul links for rate region maximization. In order to maximize the achievable rate region, for each pair of the adopted quantization and detection schemes, we first formulate a weighted sum rate maximization problem for optimization of the RF time allocation and the RU quantization noise covariance matrices. Then, we develop a unified representation for the resulting optimization problems valid for all considered quantization and detection schemes. Since the obtained unified optimization problem is nonconvex and difficult to solve, we transform it into a biconvex problem, i.e., a problem that is convex in each optimization variable assuming the other variables are fixed. Exploiting this property, we develop an efficient algorithm based on golden section search (GSS) and alternating convex optimization (ACO) to obtain a suboptimal solution. Moreover, we analyze the asymptotic computational complexities of the proposed algorithm and the considered quantization and detection schemes as functions of the numbers of MUs, RUs, and RU antennas.
In contrast to its conference version [1], which studies the sum rate maximization problem of an uplink CRAN with RVQ and the optimal SIC receiver, this paper focuses on the rate region maximization problem of an uplink CRAN employing AVQ, RVQ, or DSC at the RUs and the linear MMSE or the SIC receiver at the CU. We further note that in recent work [5], a CRAN architecture employing RF multipleaccess links and mixed (i.e., not hybrid) RF/FSO fronthaul links was considered, where some of the RUs utilize RF fronthaul links and some use FSO fronthaul links. Moreover, the RUs apply AVQ for RF fronthauling and radio over FSO (RoFSO) for FSO fronthauling, where the RF signals are converted to the optical domain and forwarded over FSO links. The RoFSO system considered in [5] is simpler than the digital FSO system considered in this paper, however, it suffers from clipping noise, which impairs the original RF signal. In addition, unlike in this paper, the RF resources allocated to the multipleaccess and fronthaul links in [5] are not optimized, which implies a less efficient use of the scarce RF resources. In the following, we briefly summarize the main contributions of this paper.

We propose to employ hybrid RF/FSO systems for wireless fronthauling of CRANs and to adaptively optimize the RF transmission time allocated to the multipleaccess and fronthaul links. To the best of the authors’ knowledge, this system architecture has not been considered in the literature before.

We investigate several quantization schemes for use at the RUs and two different detection schemes for use at the CU in order to strike a balance between complexity and performance. Moreover, we formulate a unified weighted sum rate maximization problem which is valid for any pair of the considered quantization and detection schemes and derive an efficient suboptimal solution.

Our results provide several interesting insights for system design. In particular, we show that the proposed RF time allocation and fronthaul compression policies can achieve a significant performance gain compared to pure FSO fronthauling, especially under adverse weather conditions. Furthermore, under such unfavorable conditions, applying an efficient quantization scheme at the RUs is crucial for the overall performance since the limited fronthaul capacity has to be used as effectively as possible. In contrast, under good atmospheric conditions, applying simple AVQ yields a performance close to that achieved with DSC.
The rest of this paper is organized as follows. In Section II, we provide the system and channel models. In Section III, we present the compression and detection strategies for the considered uplink CRAN. In Section IV, the rate region maximization problem is formulated and an adaptive algorithm for RF time allocation and fronthaul compression is developed. In Section V, the complexities of the proposed algorithms and quantization and detection schemes are analyzed. Simulation results are provided in Section VI, and conclusions are drawn in Section VII.
Notations: Boldface lowercase and uppercase letters denote vectors and matrices, respectively. The superscripts , , and denote the transpose, Hermitian transpose, and matrix inverse operators, respectively; and denote the expectation and matrix trace operators, respectively, and denotes the determinant of a matrix or the cardinality of a set. indicates that matrix is positive semidefinite; represents the dimentional identity matrix. Set denotes the complement of set . Moreover, , , and denote the sets of positive real, real, and complex numbers, respectively. We use to denote a block diagonal matrix formed by matrices . Moreover, is defined as , denotes the natural logarithm, is the bigO notation, and is used to indicate that is a random complex Gaussian vector with mean vector and covariance matrix .
Ii System and Channel Models
In this section, we present the system model and the channel models for the multipleaccess and fronthaul links.
Iia System Model
We consider the uplink of a CRAN where MUs denoted by MU, communicate with a CU via intermediate RUs denoted by RU. Fig. 1 schematically shows the considered communication network. We assume that the RUs and the CU are fixed nodes whereas the MUs can be mobile nodes. There is a lineofsight between the RUs and the CU. Moreover, due to the large distance between the MUs and the CU, the direct link between them cannot support communication and is not considered. Furthermore, we assume that the RUs are halfduplex (HD) nodes with respect to (w.r.t.) the RF links. There are two transmission links in the system: ) the MURU RF multipleaccess links and ) the RUCU hybrid RF/FSO fronthaul links. Each MU is equipped with a single RF antenna whereas each RU has RF antennas and an aperture FSO transmitter pointing towards the CU. The CU is equipped with photodetectors (PDs), each of which is directed to the corresponding RU, and RF antennas. We assume that the PDs are spaced sufficiently far apart such that mutual interference between the FSO links is avoided^{1}^{1}1The minimum spacing between PDs required to avoid cross talk mainly depends on the divergence angle of the FSO beams, the distance between the RUs and the CU, and the relative position of the RUs [11].. Furthermore, we assume block fading, i.e., the fading coefficients are constant during one fading block but may change from one fading block to the next. Throughout this paper, we assume that the CU has the instantaneous channel state information (CSI) of all RF and FSO links and is responsible for determining the transmission strategy and informing it to all nodes. Moreover, we assume that the channel states change slowly enough such that the signaling overhead caused by channel estimation and feedback is negligible compared to the amount of information transmitted in one fading block.
IiB Channel Model
In the following, we provide the channel models for the multipleaccess and fronthaul links.
IiB1 MultipleAccess Links
For the RF multipleaccess links, we assume a standard fading additive white Gaussian noise (AWGN) channel. All MUs transmit simultaneously using the same frequency band. The signal received at RU is denoted by and is given by
(1) 
where is the vector containing the signals transmitted by all MUs and is the signal transmitted by MU. We assume , i.e., the signals transmitted by different MUs are independent; and is the transmit power of MU. In addition, is the noise vector at RU. The elements of , i.e., the noise samples at each antenna, are modelled as mutually independent zeromean complex AWGN with variance . Moreover, denotes the channel matrix corresponding to the RF multipleaccess links from the MUs to RU.
IiB2 Fronthaul Links
The fronthaul links are hybrid RF/FSO. For the FSO links, the aperture transmitter of each RU is directed towards the corresponding PD at the CU. We assume an intensity modulation direct detection (IM/DD) FSO system. Particularly, after removing the ambient background light intensity, the signal intensity detected at the th PD of the CU is denoted by and modelled as [15]
(2) 
where is the symbol transmitted by RU, is the zeromean realvalued additive white Gaussian shot noise with variance caused by ambient light at the CU, and denotes the FSO channel gain from RU to the CU’s th PD. Moreover, we assume an average power constraint . The capacity of the IM/DD FSO link from RU to the CU’s th PD is not known. Nevertheless, in [16, Eq. 26], the following rate has been shown to be achievable
(3) 
where is the bandwidth of the FSO signal.
We assume that the multipleaccess and fronthaul links utilize the same RF resources. However, since the RUs are HD nodes w.r.t. the RF links, the RF multipleaccess and fronthaul links cannot be used at the same time. Therefore, we adopt a time division duplex (TDD) protocol. In particular, we divide each RF transmission time slot into fractions such that holds, where in fraction of each time slot, the RF multipleaccess links are active and in , fraction, the RF fronthaul link from RU to the CU is active. The RF signal of RU received at the CU is denoted by and given by
(4) 
where is the vector of signals transmitted over the antennas of RU. We assume that holds, where is the fixed transmit power of RU over the RF fronthaul link. denotes the noise vector at the CU. The noise at each antenna of the CU, i.e., each element of , is modelled as zeromean complex AWGN with variance . Moreover, denotes the channel matrix of the RF fronthaul link from RU to the CU. The capacity of the RF fronthaul link between RU and the CU is obtained via optimal waterfilling as [15]
(5) 
where is the bandwidth of the RF signal. In (5), is the th singular value of and is the water level which is chosen to satisfy the power constraint as the solution of the following equation
(6) 
Iii Compression, Transmission, and Detection Strategies
In this section, we first present the three considered compression schemes for the RUs, then we introduce the transmission strategy over the fronthaul links, and finally, we explain the two considered detection schemes for the CU. The block diagram of the considered uplink CRAN is depicted in Fig. 2.
Iiia Fronthaul Compression
We assume that the RUs employ compressandforward relaying and quantize their received signals. Let and denote the vectors of received and quantized signals at all RUs where is the quantized version of RU’s signal. In particular, the RUs quantize their received imphase/quadrature (I/Q) samples with a sampling rate of and forward the compressed signals to the CU through hybrid RF/FSO fronthaul links. Assuming the Gaussian quantization test channel, is given by [7]
(7) 
where is the quantization noise and is the joint distortion matrix at the RUs. The mean square distortion between the received I/Q sample vector and the corresponding quantized vector is given by the main diagonal entries of . In this paper, we consider three commonly used quantization schemes, namely AVQ, RVQ, and DSC, which offer different tradeoffs between complexity and performance.
IiiA1 PerAntenna Vector Quantization
One simple way of compressing the received signals at the RUs is to independently quantize the I/Q samples at each antenna of the RUs. In this case, is a diagonal matrix. Let be the distortion at the th antenna of RU, i.e., , where and denote the quantized and the original versions of the signal received at the th antenna of RU, respectively. Therefore, is obtained as . Moreover, based on the lossy source coding theorem [17], the output rate of the quantizers at RU, denoted by , has to meet the following constraint in order to guarantee a maximum distortion of
(8) 
where is given by
(9) 
Here, is a vector containing the channel coefficients between the MUs and the th antenna of RU.
IiiA2 PerRU Vector Quantization
A higher compression rate can be achieved if each RU exploits the correlation between the signals it receives at different antennas and jointly quantizes the I/Q samples of the antennas. In this case, is a block diagonal matrix with matrix entries , where , i.e., the covariance matrix of at RU, is in general nondiagonal. Moreover, according to the lossy source coding theorem [17], the output rate of the quantizer at RU has to satisfy the following constraint if the original signal is to be recovered with maximum distortion
(10) 
where is given by
(11) 
IiiA3 Distributed Source Coding
Since the received signals at different RUs are statistically dependent, the RUs can exploit this dependence via DSC. It has been shown that multiple isolated sources, i.e., the RUs here, can compress data using DSC as efficiently as if they knew each others’ received signals and jointly quantized them [17, 18, 19]. In DSC, each RU quantizes its received signal based on the conditional statistics of (or the correlation model between) the RUs’ signals and not based on the actual signals received at the other RUs [19]. Let denote the joint distortion matrix of the signals received at all RUs. Based on the lossy source coding theorem [17], for DSC, the output rates of the quantizers at the RUs have to meet the following constraints
(12) 
where in vectors and , we collect all and for , respectively, and is a nonempty subset of . Moreover, is given by
(13) 
where matrices and contain all and , respectively, whose indices belong to set .
IiiB Fronthaul Transmission
Due to the limited available RF spectrum, we assume that both the multipleaccess and the fronthaul RF links utilize the same RF resource. Hence, the advantages of employing a hybrid RF/FSO system for fronthauling come at the expense of a bandwidth reduction for the multipleaccess links. In the following, we propose an adaptive protocol which divides the available RF transmission time between the multipleaccess and the fronthaul links in a TDD manner^{2}^{2}2Although TDD based time sharing of RF resources between multipleaccess and fronthaul links is in general suboptimal, we adopt this approach here to avoid interlink interference, which facilitates a simple transceiver design.. Recall that fraction of each RF time slot is allocated to the fronthaul link from RU to the CU. Hence, a fraction of of each RF time slot is available for the multipleaccess links. For future reference, we define , where . In order to recover reliably at the CU, based on the channel coding theorem [17, 18], the rate at the output of the quantizers at RU has to meet the following constraint
(14) 
IiiC Detection at the CU
For detection of the signals transmitted by the MUs at the CU, we consider two detection schemes, namely a suboptimal linear detector and an optimal detector. The linear detector requires lower computational complexity compared to the optimal detector at the expense of a loss in performance. In the following, we explain the detectors employed at the CU in detail. For future reference, let denote the channel vector corresponding to the RF multipleaccess links from MU to all RUs.
IiiC1 Linear Detector
We assume that is a linear filter applied at the CU in order to recover MU’s signal . The corresponding output signal of the filter is given by . Let denote the signaltointerferenceplusnoise ratio (SINR) at the output of the filter for MU which is given by
(15) 
The linear filter that maximizes , denoted by , and the maximum SINR can be obtained using the Rayleigh quotient inequality [20] as
(16)  
(17) 
respectively. As can be observed, (17) is identical to the SINR at the output of an MMSE detector [21]. Hence, the optimal linear detector that maximizes the MUs’ SINRs is the wellknown linear MMSE detector. Therefore, for the Gaussian RF access channel in (1) and the Gaussian quantization test channel in (7), the transmission rate of MU employing linear MMSE detection at the CU is obtained as [7]
(18) 
Using Sylvester’s determinant theorem [22], (18) can be written as follows
(19) 
IiiC2 Optimal Detector
For the multiple access channel (MAC), SIC is capacityachieving. Assume an arbitrary decoding order of , and suppose the MUs are indexed according to the decoding order. The SIC decoder first decodes the signal of MU, subtracts it from the combined signal, , and continues until all MUs’ signals are decoded. Therefore, the signal based on which the message of MU is decoded is given by . For the Gaussian RF MAC in (1) and the Gaussian quantization test channel in (7), the transmission rate of MU employing SIC at the CU is obtained as [7]
(20) 
Remark 1
Employing SIC to achieve (20) is equivalent to applying MMSE decisionfeedback equalization to the signal received at the CU. This is due to the fact that for Gaussian inputs and the AWGN channel, the MMSE filter is information lossless [23, Proposition 5.13]. In other words, holds, where is the MMSE filter after removing MU’s signal .
Iv Rate Region Maximization
In this section, we first formulate the rate region maximization problems for the considered quantization and detection schemes, and then represent them in a unified form. Since the resulting unified problem is nonconvex, we transform the problem to a biconvex problem which enables the derivation of an effective suboptimal solution. Based on this solution, we propose an adaptive protocol for joint optimization of the fronthaul compression and the RF time allocation.
Iva Problem Formulation
Let be the vector of the output rates of the quantizers at the RUs. Moreover, let , where is the weight representing the priority associated with MU. Without loss of generality, we assume that , and . Then, the rate region maximization problem can be formulated as the following weighted sum rate maximization problem
where is the transmission rate of MU given in (19) and (20) for the MMSE and SIC receivers at the CU, respectively. Moreover, constraints (8), (10), and (12) in are the source coding constraints for AVQ, RVQ, and DSC, respectively, and (14) in represents the channel capacity constraint.
To fully characterize the rate region, we have to solve (IVA) for all possible . Note that for a given , the optimal SIC order is obtained by decoding the MUs with smaller weights first [8]. Thereby, without loss of generality, we assume resulting in the decoding order in the following.
Next, we provide a unified representation of (IVA) which enables us to solve this problem for all considered quantization and detection schemes efficiently. To this end, we first define some auxiliary variables. Let denote a subset of , where is the index set of all antennas at the RUs. Moreover, we define as the index set corresponding to all antennas of RU, where is a nonempty subset of . Furthermore, we define for any given set as the pseudo identity matrix of size whose entry in the th row and th column is equal to if the th element of set is equal to , otherwise it is equal to zero. Based on the definition of , we use to select the th entries of and . For instance, for and , holds.
Using the above notations, in the following proposition, we present (IVA) for all adopted quantization and detection schemes in a unified manner.
Proposition 1
The achievable rate region maximization problem (IVA) can be formulated in unified form for the considered quantization and detection schemes as follows
α_0 W^rf∑_k=1^K μ_klog_2—Vk+D——Wk+D—  (22)  
∑_m∈Sr_m ≥α_0 f_s log_2—C(S)+I(TS)D(I(TS))T+σ2I—S—N——I(TS)D(I(TS))T—, ∀S ∈¯S,  
r_m≤C_m^fso+α_mC_m^rf, ∀m ∈M,  
I(T)D(I(T))^T⪰0 and I(T)D(I(T^c))^T=0_—T—×—T^c—, ∀T ∈¯T, 
where is the complement set of w.r.t. . Furthermore, , , and are constant matrices and and are constant sets defined as follows
= {∑j≠kKPjhjhjH+σ2IMN,MMSE∑j¿kKPjhjhjH+σ2IMN,SIC  
= {{∀S⊂M∣—S—=1},AVQ and RVQ{∀S∣S⊆M},DSC 
Proof:
The proof is provided in Appendix A. \qed
Note that optimization problem (22) is jointly nonconvex in . Hence, a bruteforce search may be needed for finding the globally optimal solution which entails a prohibitively high computational complexity. In addition, we note that given , (22) is nonconvex w.r.t. since the objective function of the maximization problem is convex instead of concave. Therefore, in the following, we reformulate the objective function of (22) such that the problem becomes biconvex, i.e., a problem that is convex in each optimization variable assuming the other variables are fixed. This facilitates the application of ACO for finding a suboptimal solution to (22). To further reduce complexity, we transform constraint such that the optimization vector is replaced by scalar optimization variable . The optimal can then be found with a simple one dimensional search.
IvB Problem Transformation
In this section, we present a transformation of the constraints and the objective function which allows us to efficiently tackle nonconvex optimization problem (22).
IvB1 Constraint Transformation
Given , the problem is linear w.r.t. and hence can theoretically be handled by ACO. However, the ACO algorithm fails to converge to an efficient suboptimal solution due to the following reason. For a given in the th iteration, the optimal is found such that constraint holds with equality. Therefore, in the next iteration, given , the optimal returns the same solution as in the previous iteration, i.e., . In other words, the ACO algorithm will be trapped at the initial point . To overcome this challenge, we reformulate the constraints in terms of such that the reformulated problem does not exhibit the aforementioned issue. The following lemma provides an equivalent representation of the channel capacity constraint in (22) which is useful for handling .
Lemma 1
Constraint in (22) can be written in the following equivalent form
(23) 
where , , and denotes a nonempty subset of .
Proof:
The proof is provided in Appendix B. \qed
Remark 2
The equivalence of constraint in (23) and in (22) is analogous to the equivalence of the capacity region of the MAC and the rate region achieved via timesharing and SIC, see [18, Chapter 15]. The advantage of Lemma 1 is that the dimensional optimization variable in in (22) reduces to the onedimensional optimization variable in in (23). This comes at the expense of increasing the number of constraints from to .
IvB2 Golden Section Search (GSS)
Now, since variable is bounded in the interval , the optimal can be obtained via a full search assuming the optimal have been already obtained for any given . Note that since is onedimensional and bounded, a full search is feasible; however, the optimal can be found even more efficiently as described in the following. In particular, assuming that the weighted sum rate is a unimodal function w.r.t. , i.e., a function that has only one optimal point in a given bounded interval [24], the efficient GSS algorithm can be used to find the optimal . Please refer to Appendix C for a detailed discussion of the unimodality of the weighted sum rate and the proof for the special case of . For the general case, a rigorous proof for the unimodality of the weighted sum rate is cumbersome. Nevertheless, our simulation studies in Section VI suggest that the weighted sum rate is in general unimodal w.r.t. . Furthermore, in the following, an intuitive justification for this property is provided. In particular, increasing affects the weighted sum rate in two aspects, namely the RF multipleaccess time interval increases and the distortion caused by the quantization also increases as the RF fronthaul time interval decreases. In other words, by increasing , the weighted sum rate first increases owing to the increasing RF multipleaccess time interval, but ultimately decreases due to the decreasing fronthaul capacity and the resulting larger distortion. We note that the required number of iterations, denoted by , for finding with an accuracy of is for the GSS and for the full search [24]. Hence, we employ the GSS to find the optimal . In the following, we assume is fixed and find the corresponding optimal .
IvB3 Objective Function Transformation
The problem in (22) is nonconvex in since the objective function of the maximization problem is convex in (instead of concave). We use the following lemma to convexify the objective function of (22) w.r.t. .
Lemma 2 ([25])
For any matrix which satisfies , the following equation holds
(24) 
where the optimal solution of the righthand side of (24) is given by .
Defining and , where is a new auxiliary optimization matrix, and applying Lemma 2 to the objective function of (22) and replacing the original constraint with the equivalent constraint from Lemma 1, we reformulate optimization problem (22) as follows
α_0 W^rf∑_k=1^K μ_klog_2—V_k+D—+μ_klog_2—B_k—μkln(2)Tr(B_k(W_k+D))  
∑_m∈Sr_m ≥α_0 f_s log_2—C(S)+I(TS)D(I(TS))T+σ2I—S—N——I(TS)D(I(TS))T—, ∀S ∈¯S,  
∑_∀m∈S r_m G_m(S)≤(1α_0)G(S)+∑_∀m∈SG_m(S)C_m^fso, ∀S ⊆M,  
I(T)D(I(T))^T⪰0 and I(T)D(I(T^c))^T=0_—T—×—T^c—, ∀T ∈¯T. 
Although, for a given , optimization problem (IVB3) is still jointly nonconvex in , the problem is convex w.r.t. the individual variables. This allows the use of ACO to find a suboptimal solution of the problem in terms of for any given . In particular, given , the problem is convex w.r.t. and can be solved using standard semidefinite programming (SDP) solvers [26]. Moreover, given , the problem is convex w.r.t. and has the following optimal closedform solution based on Lemma 2
(26) 
IvC Proposed Algorithm
We propose an algorithm consisting of an outer loop, i.e., Algorithm 1, to find based on GSS, and an inner loop, i.e., Algorithm 2, to find for a given based on ACO, respectively. In the following, we describe the outer and inner loops in detail.
Outer Loop (Algortihm 1): In this loop, we employ the iterative GSS algorithm to maximize the weighted sum rate w.r.t. . Suppose is the search interval in a given iteration. In each iteration, the GSS algorithm requires the evaluation of the weighted sum rate at the intermediate points using the inner loop, i.e., Algorithm 2. Then, it compares the values of the objective function at and denoted by and , respectively, and updates the search interval for the next iteration as follows [24]
(27) 
where are obtained as
(28) 
Here, and with the socalled golden ratio [24]. The GSS algorithm is summarized in Algorithm 1 for the problem considered in this paper. In Algorithm 1, is a small positive number which is used in line 10 to terminate the iteration if the desired accuracy of the GSS algorithm is achieved.
Inner Loop (Algortihm 2): For the inner loop, is provided by the outer loop. Here, we employ ACO to find a stationary point of the problem w.r.t. , , and for the fixed . The proposed ACO method is concisely given in Algorithm 2 where is a small positive number and is the maximum number of iterations that are used in the termination condition in line 7. In the next section, we provide a modified ACO (MACO) method, which is easier to implement in popular numerical solvers such as CVX and also included in Algorithm 2 in blue italic font.
Once is known, the optimal RF time allocation variable for fronthaul link RUCU, , is obtained as
(29) 
IvD Implementation in CVX
Although for given , , and , optimization problem (IVB3) is convex in , its solution using popular numerical solvers, such as CVX [26], can be challenging. More specifically, the implementation of line 5 of Algorithm 2 in CVX is not directly possible since the current version of CVX, i.e., CVX 2.1, does not have a builtin convex function that can directly handle the righthand side of in (IVB3) [26]. In the following, we propose a transformation of denoted by to address this issue. Defining and , where is a new auxiliary optimization matrix, and using (24) in Lemma 2, the following inequality holds for the righthand side of in (IVB3)
(30) 
In (IVD), equality holds if
(31) 
Substituting (IVD) into in (IVB3), we obtain as follows
(32) 
which is convex in if is fixed and vice versa. We note that the feasible set of w.r.t. is identical to that of w.r.t. . In fact, since for any (IVD) holds, the feasible set of cannot be larger than that of . On the other hand, since contains as a special case, the feasible set of cannot be smaller than that of either. Hence, the feasible sets of and are identical. Therefore, we can rewrite problem (IVB3) in the following equivalent form
Based on (IVD), an MACO algorithm w.r.t. can be developed. The corresponding changes are provided in blue italic font in Algorithm 2.
Remark 3
In summary, we note that the global optimal solutions of the three nonconvex optimization problems (22), (IVB3), and (IVD) lead to identical values for the weighted sum rate. However, unlike original problem (22), problem (IVB3) can be solved suboptimally using ACO, and problem (IVD) can be solved suboptimally using ACO via CVX. The obtained suboptimal solutions are local optima of original problem (22) because of the convergence properties of ACO, see e.g. [27, 28]. Therefore, we use (IVD) to generate the simulation results shown in Section VI.
V Complexity Analysis
In this section, we first analyze the computational complexity of Algorithms 1 and 2. Subsequently, we characterize the complexity of the quantization and detection schemes adopted in this paper for online transmission.
Va Algorithms 1 and 2
In the complexity analysis presented below, we only consider the operations that entail the highest computational complexity. In particular, calculating the matrix inversion in Step 4 and solving the SDP problem in Step 5 of Algorithm 2 dominate the overall complexity of both Algorithms 1 and 2 and are discussed in the following.
Step 4: The computational complexity of inverting a matrix of size is [29, 30]. The ACO in Algorithm 2 includes a matrix inversion for finding for MUs which entails a complexity of . For MACO, matrix inversions are needed to find both and . The complexity of computing depends on the type of quantization employed at the RUs, cf. (31), and is for AVQ since is diagonal, for RVQ since is block diagonal, and for DSC since is a general matrix and has to be computed for . Therefore, the overall complexity of computing and in Step 4 of MACO is for AVQ and RVQ and for DSC.
Step 5: The computational complexity required per iteration for solving an SDP with a numerical convex program solver is given by [31, 27], where and denote the number of semidefinite cone constraints and the dimension of the semidefinite cone, respectively. Moreover, the number of iterations needed to solve the SDP problem with an accuracy of is on the order of iterations [31, 27]. This leads to a complexity order of . For AVQ, we do not have semidefinite cones; hence the complexity of Step 4 is higher than the complexity of Step 5. Moreover, for RVQ, we have and . Hence, the complexity of solving one SDP is . For DSC, and , and the corresponding complexity for solving one SDP is .
Overall: The overall complexity of the proposed algorithms is limited by the complexity of Steps 4 and 5 of Algorithm 2. Moreover, Steps 4 and 5 are repeated at most times for Algorithm 2 to converge and times for Algorithm 1 to converge. Therefore, the overall complexity order of the proposed algorithm is given by
(34) 
As can be seen from (34), the complexities of both ACO and MACO grow linearly w.r.t. the number of MUs, , for all considered quantization and detection schemes. However, the growth speed of the complexity of ACO/MACO in terms of the number of RUs, , and their number of antennas, , depends on the adopted quantization scheme. For instance, the complexity of ACO/MACO in terms of the number of antennas, , for AVQ is cubic while it is on the order of for RVQ and DSC.