I. Introduction
To cope with the continuously increasing number of wireless devices, to satisfy the demand of high data rates, and to meet the stringent qualityofservice (QoS) requirement of the emerging wireless services, various advanced communication technologies have been proposed in recent years [1]. For example, cooperative communication [2], massive multipleinput multipleoutput (MIMO) [3], and network densification [4] are key technologies to achieve high capacity, high data rate, and guarantee network coverage in the fifth generation (5G) mobile cellular systems. Especially for network densification, cloud radio access network (CRAN) is considered as an emerging network architecture that shows significant promises in suppressing interference and coordinately allocating resource for wireless networks [6, 7, 8]. In CRAN, the remote radio heads (RRHs) are connected to a central processor, i.e., baseband signal processing unit (BBU), through fronthaul links [9, 10, 11]. However, performing completely joint processing requires not only signaling overhead but also payload data sharing among all coordinated RRHs, resulting in tremendous burden on fronthaul links [12, 13, 14, 15, 16]. Thus, the design of transmission scheme needs to take the limitation of the capacity of fronthaul links into account. In [12, 13, 14], the maximization problem of weighted sum rate was investigated under the constraint of fronthaul capacity. In [15], the datasharing strategy and compression strategy were studied regarding the energy efficiency of the CRANs. In [16], joint design of fronthaul and radio access links for CRANs was investigated. Note that a common assumption in aforementioned literature is that the coordinated RRHs do not have the ability to cache data files.
One of the most challenging requirements in 5G is the provision of connections with low endtoend latency. In most cases, the latency should be smaller than 100 ms, while sometimes it is smaller than 10 ms [17]. The low latency is demanded by some applications such as augmented reality and vehicular communications [18]. One solution to reduce the endtoend latency as well as the fronthaul bottleneck in CRANs is using distributed caches at the edge of the mobile cellular network, referred to as fog radio access networks (FRANs) [19, 20, 21]. The resulting RRHs are referred to as enhanced RRHs (eRRHs). The eRRHs can prefetch the most frequently requested files to the eRRHs’ local caches during offpeak traffic periods so that the fronthaul overhead can be reduced during the peak traffic periods. In this way, lower latency and higher spectral efficiency can be achieved [21]. In [22], the optimization of hybrid cache placement of data files for coordinated relaying networks was studied. Then the cache placement in FRANs was investigated by fully considering the flexibility of physicallayer transmission and diverse content preferences of different users [23]. In [24], joint design of cloud and edge processing for the downlink of FRANs was investigated for maximizing the minimum user delivery rate. In [25], the problem of minimizing the system cost that accounts for fronthaul overhead while satisfying the target signaltointerferenceplusnoise ratio (SINR) was considered. More recently, the authors of [26] proposed to use wireless caching to enhance the physical layer security of video streaming in cellular networks with limited backhaul capacity.
In general, the sum rate maximization or power minimization problem subject to QoS and transmit power constraints needs to be centrally solved for interference channels [24, 25, 26]. Meanwhile, some decentralized optimization algorithms were also developed to reduce the burden of information collection [27, 28, 30, 29, 31]. In [27]
, a distributed weighted sum rate maximization (WSRMax) method was proposed via exploiting the relation between the user rate and the minimum mean square error (MMSE) for interfering broadcast channel. At each iteration all mobile terminals need to first estimate the covariance matrices of their received signals, and then compute and feed back some parameters to the eRRHs. To avoid the estimation of the covariance matrices at each mobile terminal, the authors of
[28] proposed a distributed approach to maximize the WSRMax problem with limited signaling exchange among eRRHs. In [29], both centralized and distributed beamforming algorithms were proposed to solve power minimization and SINR balancing problem. In [30], the maxmin fairness problem under the total power constraint was tackled from the perspective of transmit power minimization, where a distributed hierarchical iterative algorithm was proposed based on the uplinkdownlink duality. In [31], a decentralized optimization method was proposed by exploiting interference temperature [32] to maximize the system energy efficiency. However, the decentralized optimization algorithms in the aforementioned literature need signaling exchange in each iteration during the optimization process, which not only requires strict synchronization but also generates some backhaul or fronthaul burden among coordinated eRRHs. The signaling exchange per iteration between coordinated eRRHs is nevertheless not desirable for low latency communications.The joint transmission of cached and uncached requested contents in [24, 25, 26] only used the caching to reduce the burden on fronthaul links without making full use of the caching to reduce the delivery latency. However, one of the main motivation of introducing edge caching is to reduce the delivery latency for satisfying the demands of the latencysensitive traffic for future wireless communication system [33]. To the best of the authors’ knowledge, how to fully exploit the capabilities, i.e., caching and baseband signal processing, of eRRHs to reduce the delivery latency and the overhead of fronthaul links is still an enormous challenge for wireless transmission in FRANs. In this paper, we investigate the downlink transmission mechanism for cacheenabled FRANs. We divide the transmission of the requested files into twolevel, i.e., cachelevel transmission and networklevel transmission. In cachelevel transmission, only requested files cached at the local cache are transmitted to the corresponding users. The duration of cachelevel transmission is the delay caused by the transfer between the BBU and eRRHs as well as by the signal processing at the BBU. The remaining requested files are jointly transmitted to the corresponding users at networklevel transmission. Our objective is to maximize the delivery rate under the constraints of fronthaul capacity, maximum transmit power of each eRRH, and size of cached data files. The corresponding optimization algorithms are presented to address the problems of interest. Specifically, the contribution of this work is summarized as follows:

A twolevel transmission scheme including cachelevel and networklevel transmission is proposed to reduce the delivery latency and the burden on fronthaul links by making full use of the local cache and the signal processing capabilities of eRRHs.

For cachelevel transmission, coordinated beamforming is adopted to improve the spectral efficiency. A centralized optimization algorithm is firstly designed to maximize the sum of network delivery rates. To avoid the demand of centralized processing and the exchange of signaling among eRRHs, a decentralized algorithm is then presented to maximize the sum of network delivery rates by introducing a new concept of signaltointerferenceleakagenoise ratio (SILNR) for cachelevel transmission.

For networklevel transmission, joint transmission is adopted to maximize the system performance such that the interuser interference can be effectively controlled. At the same time, an optimization algorithm is developed using convex approximation methods.

All presented algorithms are proved to converge to the KarushKuhnTucker (KKT) solutions of the problems. The computational complexity of all presented algorithms is also analyzed. Numerical results are further provided to corroborate the effectiveness of proposed algorithms and unveil that how to cache files and how many files to cache are key problems for cacheenabled FRANs.
The remainder of this paper is organized as follows. The system model is described in Section II. Section III formulates the optimization problem of the proposed twolevel transmission scheme. The design of optimization algorithm for cachelevel transmission is investigated in Section IV. In Section V, an optimization algorithm for networklevel transmission is developed. Numerical results are provided in Section VI. Finally, Section VII concludes the paper.
Notations
: Bold lowercase and uppercase symbols represent column vectors and matrices, respectively. The superscripts
and represent the matrix transpose and conjugate transpose, respectively. For a set , denotes the cardinality of . For a complexvalued number and a matrix , and denote the absolute value of and the determinant of , respectively. and denote the trace and the Euclidean norm, respectively. and denote the zero matrix and identity matrix, respectively. The circularly symmetric complex Gaussian distribution with mean and covariance matrix is denotes by . denotes a diagonal matrix with the main diagonal given by .II. System Model
As illustrated in Fig. 1, the considered cacheenabled FRANs include a BBU connected to eRRHs serving totally singleantenna users. The index sets of the eRRHs and the users are denoted as and , respectively. The eRRH equipped with antennas is connected to the BBU through an errorfree fronthaul link with the capacity of bits per symbol. The eRRH is also equipped with a cache, which can store bits, , where is the number of symbols transmitted in each downlink transmission interval. The BBU in the “cloud” can perform signal processing globally, while the eRRHs at the edge of the “cloud” can only perform signal processing locally.
Assume that each user requests contents or files from library stored in the BBU. Without loss of generality, we assume that all files in library are of the same size of bits. For simplicity, in this paper, we further assume that the files cannot be partitioned and different eRRH caches different files. The eRRH prestores files in its cache based on the user behavior, the delaysensitive requirement, the longterm information of the popularity distribution, the fronthaul capacity, and etc [24, 25, 22, 23]. The state of cached file ,
, can be modeled by defining binary variables
, , , as(1) 
At the same time, satisfies the conditions or , indicating that different eRRH caches different files. For simplicity, in this work, we assume that the state information , , , is predetermined. Thus, if file requested by user is cached in the local cache of its serving eRRH, it can be retrieved directly from the eRRH without downloading from the BBU^{1}^{1}1Each eRRH can retrieve the information of requested files by learningbased methods and is out of the scope of this work. The developed method in this work can be easily extended to the case where each eRRH caches multiple different files and the case where the file can be partitioned into multiple subfiles.. Otherwise, the information of file needs to be transferred to its serving eRRH from the BBU via the fronthaul link.
(5) 
III. Problem Formulations
In order to make full use of the functionalities of cache as well as the signal processing capability of each eRRH, in this paper, the data transmission is divided into twolevel transmission. The first level transmission is cachelevel transmission where all eRRHs coordinately transmit cached files to scheduled users via coordinated beamforming. The duration of cachelevel transmission is the delay caused by the signal processing at the BBU and the information exchange between the eRRH and the BBU. The second level transmission is networklevel transmission where a quantized version of the precoded signals of the untransmitted requested files is sent to the eRRHs via fronthaul links and then all eRRHs jointly transmit it to users via joint transmission.
A. Cachelevel Transmission
To reduce the latency incurred by the propagation of the fronthaul link and the signal processing at the BBU, eRRH firstly transmits data to users, denoted as , whose requested files are stored in the local cache in cachelevel transmission. Furthermore, , , and . As a numerical example, user set can be given by
(2) 
where is a scalable parameter with and is the requested file index of user . Here we assume that each user requests a single file. For cachelevel transmission, the signal transmitted by the eRRH can be expressed as
(3) 
where is the beamforming vector and for . The signal received at user , , can be expressed as:
(4) 
where denotes the channel coefficients between user and eRRH and is the additive white Gaussian noise (AWGN). Note that in (4), we assume that baseband signal and noise are independent, . Thus, the achievable rate of user served by eRRH is calculated as , where is the signaltointerferenceplusnoise ratio (SINR) that is calculated as (5), at the top of the next page.
The goal is to maximize the sum of delivery rate by optimizing the transmit beamforming vectors at each level transmission. Thus, for cachelevel transmission, the optimization of , , , can be realized by solving the following problem^{2}^{2}2The idea of this work can be extended to the scenario where multiple users request the same file via multicast transmission as well as the scenario where the user equipped with multiple antennas requests multiple files via spatial multiplexing. What we need to do is to adjust the formulation of problems (6) and (10) according to the parameters of the corresponding scenario. However, it is left as our future work.:
(6a)  
(6b)  
(6c)  
(6d) 
where the variable is . denotes the delivery rate of user served by eRRH , , , for cachelevel transmission. In problem (6), constraint (6b) shows that the delivery rate is bounded by the achievable data rate . Constraint (6c) denotes that the number of delivery data cannot exceed the cached file size . Constraint (6d) means the maximum transmit power constraint on the beamforming vectors. Delay is a summation of the signal processing time at the BBU and the information exchange time between eRRH and the BBU^{3}^{3}3For simplicity, we assume that the delay is the same for all eRRHs. For full caching case, i.e., all requested files are stored the local cache, the value of the delay is set be unit in optimization problem (6) and networklevel transmission is not need.. From the description of problem (6), one can see that the duration of cachelevel transmission is also the delay .
B. Networklevel Transmission
For networklevel transmission, all eRRHs coordinately transmit data to all scheduled users by using joint transmission [2]. Although joint transmission can effectively control the intra and intereRRH interference and improve the quality of service for each user, the BBU needs to collect all channel state information (CSI) and allocates the processed data files to each eRRH resulting in a considerable large amount of overhead on the fronthaul links. For networklevel transmission, the signal transmitted by eRRH is expressed as^{4}^{4}4Note that pointtopoint compression algorithm is adopted to quantize the precoded signals for each eRRH during the networklevel transmission. As pointed out in [8], we can adopt some advanced compression algorithm to quantize the precoded signals such that the system performance can be further improved. Therefore, we would like to study the performance of the proposed twolevel transmission scheme with advanced compression algorithm in our future work.:
(7) 
where is the beamforming vector for file used at eRRH . The quantization noise is assumed to be independent of the transmitting signal and distributed as . The signals and for different eRRHs are quantified independently so that the quantization noise signals and are independent [24]. For ease of presentation, let be the covariance matrix of quantization noise, i.e., . For networklevel transmission, the signal received at user , , can be expressed as:
(8) 
where and . For the joint transmission, we implicitly assume that the stringent synchronization of timing and frequency among eRRHs has been finished. Similarly, the achievable rate of user is calculated as , where SINR is given by:
(9) 
where .
Similarly, for networklevel transmission, the transmit beamforming vectors , , are optimized centrally at the BBU by addressing the following problem:
(10a)  
(10b)  
(10c)  
(10d)  
(10e) 
where the variables are . Also note that denotes the delivery rate of user during networklevel transmission, , and with matrix being defined as
with and , where . denotes the maximum transmission rate of user at cachelevel transmission^{5}^{5}5We assume that the BBU can obtain the values of , , , according to the cache state information and the transmission scheme adopted by each eRRH, i.e., (6).. In problem (10), (10d) constrains the size of networklevel transmission file not to exceed the remaining file size ^{6}^{6}6Note that different from [12, 13, 14, 15],[24], and [25], the size of transmitted files in cachelevel transmission is included in the constraints (10d). This provides a chance to increase transmit rate of untransmitted files, i.e., improve the system delivery rate.. Constraint (10e) is due to the fronthaul capacity constraints ensuring the signal can be reliably recovered by eRRH [34, Ch 3]. Note that when the delay is zero, problem (10) becomes the optimization problem of precoding matrices for traditional cloudbased coordinated joint transmission system under the constraints of the fronthaul capacity, the size of data files, and the maximum transmit power of eRRHs.
From the aforementioned analysis, one can see that the minimum delivery content of our proposed twolevel transmission scheme is the sum of the minimum delivery rate achieved by cachelevel transmission and that of networklevel transmission, i.e., . The total minimum delivery content is for the traditional cloudbased coordinated joint transmission system.
Remark 1.
Note that in networklevel transmission, all eRRHs transmit quantized precoding signals received from the BBU. However, in [21], all eRRHs transmit a superposition of precoded signals received from the BBU and locally precoded signals based on the caching files. The coupling of the precoding matrices used at all eRRHs make the optimization problem more difficult such that a central controller is used to optimize the precoding matrices [21]. Thus, the precoding matrices used at each eRRH need to be transmitted to the corresponding eRRH. In our proposed scheme, we use the caching files to simultaneously reduce the delivery latency and the burden on fronthaul links. Namely, in cachelevel transmission, the required files which are cached at eRRHs are first transmitted to the corresponding users during the duration of the delay . Meanwhile, the burden on fronthaul links is also reduced under the constraints of (10d) and (10e).
IV. Design of Optimization Algorithm for Cachelevel Transmission
In this section, we first focus on addressing problem (6) by designing a centralized optimization method. Then, to avoid the exchange of signaling between coordinated eRRHs, a novel concept of signaltointerferenceleakagenoise ratio (SILNR) is introduced to maximize the intraeRRH delivery rate while suppressing intereRRH interference. Finally, we design a decentralized optimization algorithm to obtain the coordinated beamforming vectors for cachelevel transmission.
A. Centralized Optimization for Cachelevel Transmission
For coordinated interference networks, the existing of intraand intereRRH interference leads the user rate to be a nonconcave function. In other words, it is generally difficult to find the global optimal solution of (6). Therefore, this requires the relaxations of the optimization conditions in order to provide reasonable design for practical implementations. The first thing of addressing problem (6) is to transfer it into a solvable form by using some mathematical methods. In the sequel, we introduce some auxiliary variables , , , which denote the interference leakage limitation [35]. Thus, problem (6) is equivalently reformulated as:
(11a)  
(11b)  
(11c)  
(11d)  
(11e) 
where the variables are and . The set is given by
and with redefined as
(12) 
where . Constraint (11c) is activated at the optimal point of problem (11). Unfortunately, problem (11) is still a nonconvex due to the existing of interuser interference in the denominator of defined in (12). To overcome this difficulty, we firstly reformulate problem (11) into the following equivalent form:
(13a)  
(13b)  
(13c)  
(13d)  
(13e)  
(13f)  
(13g) 
where the variables are and . Note that in problem (13), constraints (13c), (13d), and (13e) are activated at the optimal point of problem (13). Although the function in (13c) is convex function, constraint (13c) is nonconvex. Therefore, problem (13) is still nonconvex. If , the corresponding constraint given in (13d) can be removed, implying that the interference suppression among eRRH , is not needed.
In what follows, we try to iteratively convexify the nonconvex constraint (13c) such that an iterative optimization method is designed to address problem (13). In light of [36, 37, 38, 39], we replace the right side of the inequality in (13c) by its lower bound, which now can be obtained by the first order approximation due to the convexity of . In particular, we have
(14) 
where is defined as
(15) 
In (15), and denote the values of variables and obtained at the th iteration, respectively. Thus, we resort to solving iteratively problem (16) to obtain the solution to problem (11) instead of solving problem (13) directly.
(16a)  
(16b)  
(16c) 
where the variables are and . Problem (16) is a convex problem that can be easily solved with classical convex optimization solver, e.g., CVX [40]. The detailed steps used to solve problem (11) are summarized as Algorithm 1 where is a predefined stop threshold.
(17) 
Let denote the objective value of problem (16). If we replace , , , , and with , , , , and , , , , respectively, all constraints are still satisfied, which means that the optimal solution of the th iteration is feasible point of problem (16) in the th iteration. This is because of the approximation in (16c) [37]. Thus, the objective obtained in the th iteration is no smaller than that in the th iteration, i.e., . In other words, Algorithm 1 generates a nondecreasing sequence of objective values . Moreover, the problem has an upper bound due to the power constraints. Therefore, the convergence of Algorithm 1 can be guaranteed by the monotonic boundary theorem [38]. Furthermore, following the same arguments as those in [37, Theorem 1], we can prove that Algorithm 1 converges to a KarushKuhnTucker (KKT) solution of problem (11).
In Algorithm 1, Step 2 solves a convex problem, which can be efficiently implemented by primaldual interior point method with approximate complexity of , where , , and stands for the bigO notation [41]. The overall computational complexity is , where denotes the number of the iteration of Algorithm 1.
B. Decentralized Optimization for Cachelevel Transmission
Though problem (6) can be solved by Algorithm 1, it needs to be implemented centrally at BBU, which results in large burden on fronthaul links. Note that the existing decentralized optimization methods that are designed to address the optimization problem for coordinated multiple point network need signaling exchange among BSs, which results in burden on backhaul link and incurs a certain delivery latency [27, 28, 30, 29, 31]. From the perspective of reducing the delivery latency caused by information exchange and making the best use of the available signal processing capability of each eRRH, Algorithm 1 and the existing decentralized optimization algorithms [27, 28, 30, 29, 31] are not suitable for cacheenabled FRANs.
For FRANs, the objective is to reduce delivery latency and the overhead of fronthaul links. With the help of the cached contents, each eRRH can design its transmitted signal based only on its local cached content. In addition, the signalling exchange between eRRHs results in a large potential latency and requires stringent synchronization between eRRHs. Therefore, the optimization of transmit beamformers is expected to be independently implemented at each eRRH for cachelevel transmission. The denominator of the right item in (3) implies that eRRH cannot independently optimize , , due to the existing of intereRRH interference. In order to release the coupling among eRRHs and to achieve autonomous optimization of the beamformers, we introduce a new concept that is defined as the signaltointerferenceleakageplusnoise ratio (SILNR), i.e., (17), at the top of the next page. In the denominator of (17), the first term denotes the intraeRRH interference and the second term denotes the total power leaked from user to all other user . It is not difficult to see that the SILNR only depends on the local CSI, i.e., the channel coefficient from eRRH to all users, which can be easily obtained, . To design an autonomous optimization, we replace the achievable rate in constraints (6b) with an approximated achievable rate , , . Thus, instead of directly addressing problem (6), the problem of beamformer optimization for the cachelevel transmission is formulated as a series of parallel decentralized optimization problems given by (18) at eRRH .
(18a)  
(18b)  
(18c)  
(18d) 
where the variable is . The goal of problem (18) is to maximize the local delivery rate, while taking the intereRRH interference suppression into account, i.e., maximizing the total delivery rate of cachelevel transmission as much as possible. Similarly, problem (18) can be equivalently transformed into the following form:
(19a)  
(19b)  
(19c)  
(19d)  
(19e)  
(19f)  
(19g) 
where the variables are and . . Furthermore, problem (19) can be independently solved at each eRRH with similar procedure as Algorithm 1 without any signaling exchange among eRRHs. The overall computational complexity is , where denotes the number of the iterations of decentralized optimization algorithm of problem (19).
Remark 2.
Note that to obtain the performance gain, the centralized optimization algorithm needs to collect all channel state information (CSI) and data file to the central processing unit. However, the decentralized optimization algorithm needs only partial CSI that can be obtained by the training sequence of the uplink and optimize the beamforming vector without sharing data files. In time division duplex (TDD) system, each BS can directly estimate the CSI of its own users without any information exchange among eRRHs, by exploiting uplink and downlink reciprocity. In addition, each eRRH can estimate the crosstalk channels to other users served by other eRRHs from the reverse link.
V. Design of Optimization Algorithm for Networklevel Transmission
In this section, we focus on addressing problem (10) with centralized way at the BBU. Different from the signal processing at each eRRHs, due to the fact that the BBU has the whole CSI and data files of all users, the BBU can centrally design the precoders for all users and jointly transmit data files to each user. Thus, the intra and intereRRH interference can be efficiently controlled to maximize the sum of delivery data rate. Following a similar procedure used to solve problem (11), we firstly introduce some auxiliary variables , , , and then reformulate equivalently problem (10) into the following form:
(20a)  
(20b)  
(20c)  
(20d)  
(20e)  
(20f)  
(20g) 
where the variable is . Note that the constraints (20c) and (20d) are activated at the optimal point of problem (20). The difficulties of solving problem (20) lie in (20c) and (20g), as those contraints are nonconvex. To overcome the nonconvexity of the right hands of the inequalities in (20c), we resort to the following inequality [36]
(21) 
where is defined as
(22) 
To convexify the left side of constraint (20g), we need to transform the first function into a convex form. According to the concavity property of function , we have
(23) 
where and . Now, instead of directly solving problem (20), we iteratively solve the following problem to obtain its solution by replacing the right hands of the inequalities in (20c) with , and with the right side of (23) .
(24a)  
(24b)  
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