I Introduction
With the development of the Internet of things and mobile communication networks, the demands for highspeed data applications are growing exponentially recently [Peng2015System]. Meeting such challenging goals should involve new system architectures and advanced signal processing for wireless communications [tony2017cloud]. The paradigm of heterogeneous networks (HetNets) [Gao2018throyghput], composed of a hierarchy of macro cells enhanced by small cells of different sizes, has attracted lots of attention from both industry and academia. Macro cells with highpower base stations (BSs) provide ubiquitous coverage and small cells use various radio access technologies to serve user equipment terminals (UEs) with high datarate demands. Unfortunately, the inter and intratier interferences, resulting from densification of small cells, restrict the improvement of performance gains and commercial applications of HetNets.
At the same time, cloud computing has emerged as a popular computing paradigm for enhancing both spectral and energy efficiencies [Tang2017System]. As an application of cloud computing to radio access networks, cloud radio access networks (CRANs) have been proposed to achieve cooperative gains. In CRANs, radio frequency processing is implemented at remote radio heads (RRHs) whereas baseband processing is centralized in a baseband unit (BBU) pool. However, the performance improvement of CRANs is restricted by capacitylimited fronthaul links. Furthermore, since CRANs are mainly used in hotspots to provide high data rates, control signalling and realtime voice service are not efficiently supported [Peng2015Contract].
To overcome the aforementioned challenges, the concept of heterogeneous CRANs (HCRANs) was proposed in [peng2014heterogeneous]. In HCRANs, HetNets and CRANs complement each other. Specifically, Macro BSs are connected to the BBU pool via backhaul with X2/S1 interfaces and RRHs are connected to the BBU pool through wired/wireless fronthaul. In addition to guaranteeing backward compatibility with existing cellular systems and providing ubiquitous connections, BSs are also responsible for delivering control signals and supporting low datarate services. With the help of macro BSs, unnecessary handover and user reassociation can be avoided. On the other hand, RRHs serve high datarate applications in dedicated zones [tang2015cross]. Besides, the data and control planes are decoupled and the delivery of control signals is shifted from RRHs to macro BSs, thus the signalling overhead is reduced and the burden on fronthaul links is alleviated.
However, the constrained fronthaul capacity is still a bottleneck in HCRANs, due to the large number of UEs and the increasing demands for high datarate service in hotspots. Optical fiber links cannot be used violently because they are expensive. Wireless fronthaul is thought as an economic choice, but the spectrum resource is scarce. Therefore, a sharing strategy where fronthaul links share the spectrum resource with radio access networks has attracted lots of attention recently. Radio access network spectrumbased fronthauling has low sensitivity to propagation conditions, wider coverage, and the reusability of existing equipment. Many works, e.g., references [Yang2016Energy, zhang2017downlink, Nguyen2016Resource, Wang2016Joint, Xia2016Bandwidth], have investigated spectrum resource allocation between radio access networks and backhaul (fronthaul) links. References [Yang2016Energy, zhang2017downlink] studied energy efficiency of HetNets with wireless backhaul, which accounted for the bandwidth and power allocated between macro cells, small cells, and backhaul. Reference [Nguyen2016Resource] considered the joint design of transmit beamforming, power allocation, and spectrum splitting factors that took into account both uplink and downlink transmissions, then formulated a problem of maximizing the achievable sumrate (SR). A joint problem of cell association and bandwidth allocation for wireless backhaul was optimized in [Wang2016Joint]. Bandwidth allocation combined with interference mitigation techniques was optimized to maximize system SR using the largedimensional random matrix theory in [Xia2016Bandwidth].
Besides, compressedandforward schemes are another effective way to deal with the challenge of limited fronthaul capacity [park2014performance, park2014fronthaul]. In the compressedandforward schemes, signals first are compressed at the BBU pool (or the RRHs) using pointtopoint (P2P) compression or WynerZiv (WZ) coding and then transmitted to the RRHs (or the BBU pool) in downlink (or uplink) via fronthaul. Therefore, the communication rates between the BBU pool and RRHs are reduced. Many works, such as references [zhou2014optimized, zhou2016fronthaul, park2013robust, vu2017adaptive], have studied fronthaul compression for the uplink CRAN. In [zhou2014optimized], a weighted SR was maximized via the optimization of compression noise. In [zhou2016fronthaul], the authors jointly designed fronthaul compression and beamforming to maximize the achievable SR in the uplink CRAN with multiantenna RRHs. An optimization problem was formulated in [park2013robust], where the compression and BS selection were performed jointly by introducing a sparsityinducing term into the objective function. For the downlink CRAN, the authors in [Park2013Joint, Park2014Inter] studied the joint design of precoding and backhaul compression to maximize the system SR. Besides, a brief overview of fronthaul compression for both uplink and downlink was presented in [simeone2016cloud, park2014fronthaul]. However, most of these works assumed that the fronthaul was composed of fiber links with high cost and there was no consideration of spectrum resource sharing between radio access networks and fronthaul links. Moreover, these works were performed according to smallscale fading which varies in the order of milliseconds, thus resulting in much communication overhead for collecting channel information and high computational complexity to perform optimization in each coherence time of wireless channels.
Motivated by these facts, we aim to maximize the achievable ergodic SR of uplink HCRANs with less complexity, overhead, and cost. This goal is also in line with the expectations of operators. One possible solution is to design the spectrum sharing strategy and compression noise simultaneously. This paper focuses on the uplink transmission of a HCRAN, where many RRHs are embedded into a macro cell with a multipleantenna BS. For economic reasons, only the BS are connected to the BBU pool via optical fiber links whereas the RRHs are connected to the BBU pool through wireless fronthaul links. Furthermore, the spectrum resource is shared between the radio access network and fronthaul links. Because the capacity of fronthaul links is limited, a twostage compressandforward scheme is adopted. At the BBU side, the quantization bits are decompressed, followed by user message decoding. The contributions of this study are listed as follows.

In this work, the spectrum sharing strategy and the compressandforward scheme are considered together to deal with the challenge of constrained fronthaul capacity. We formulate a joint optimization problem of bandwidth allocation and compression noise to maximize the achievable ergodic SR, which is a mixed timescale issue. Because the bandwidth allocation is executed in a large time span whereas the compression noise design is performed in each small time span due to the dependence on smallscale fading. Besides, we consider not only different compressandforward schemes, i.e., P2P compression and WZ coding, but also different decoding strategies, i.e., the linear receptions with and without successive interference cancellation (SIC). These different compressandforward schemes and decoding strategies have a tradeoff between performance and complexity.

In contrast to existing works [zhou2014optimized, zhou2016fronthaul, Park2013Joint, Park2014Inter] where the SR maximization problems were based on fastchanging smallscale fading, we derive the deterministic approximation for the ergodic SR using largedimensional random matrix theory. Then, an approximate problem of the original problem is introduced, which is a slow timescale issue and only depends on statistical channel information. Therefore, the approximate problem can be solved with less communication overhead for obtaining channel information and lower computational complexity.

On the basis of deterministic expressions, we first give solutions to the two subproblems of the joint optimization problem, then propose an algorithm based on Dinkelbach’s algorithm to find the optimal solution to the approximation problem under different compression schemes and decoding strategies. Furthermore, we also propose a lowcomplexity algorithm to find the near optimal solution to the case of high signaltoquantizationnoise ratio (SQNR).
The remainder of this paper is organized as follows. In Section II, we describe the system model and formulate the optimization problem. Then the deterministic approximation of the ergodic SR is given in Section III. Based on the asymptotic result, we propose the algorithms to find solutions under different decoding schemes in Sections LABEL:analysis1 and LABEL:analysis2, followed by the analysis in special cases in Section LABEL:subanalysis. In Section LABEL:numerical_results, simulation results are presented and discussed. Finally, conclusion is drawn in Section LABEL:conclusion.
Notations
: The notations are given as follows. Matrices and vectors are denoted by bold capital and lowercase symbols.
, , and stand for transpose, conjugate transpose, and trace of , respectively. indicates that is a Hermitian positive semidefinite (definite) matrix. For a matrix , . The notations and are expectation and Euclidean norm operators, respectively. Finally, is a complex Gaussian vector with zeromean and covariance matrix .Ii System Model
As shown in Fig. 1, an uplink twotier HCRAN is considered, consisting of one antenna macro BS, singleantenna RRHs, and one BBU pool. The BS is connected to the BBU pool using control and data interfaces, denoted as X2 and S1, via optical fiber links. The RRHs are connected to the BBU pool through noiseless wireless fronthaul links of capacity in bps, , which meet the constraint [zhou2014optimized]. Such a constraint can model the scenario where the RRHs access to the BBU pool via frequency/time division access scheme, and the number of access slots (frequency or time slots) shared among the RRHs is fixed and limited. The BS, acting as a centre controller, supports seamless coverage and provides service for macro UEs (MUEs). The RRHs cooperate with each other and serve as hotspots for smallcell UEs (SUEs). The MUEs and SUEs each have a single antenna and they simultaneously send messages to BSs and RRHs, respectively. We assume the messages sent from the MUEs can be received by both the BS and RRHs but SUEs’ messages cannot be received by the BS directly (e.g., these SUEs are within far areas from the BS or shadow areas of buildings). SUEs’ messages are transmitted to the BBU pool via the RRHs for processing and MUEs’ messages are processed at the BS and BBU pool simultaneously. Finally, the processed signals are forwarded to the core network through the BS. Furthermore, owing to the limited fronthaul capacity, we assume a compressandforward scheme is implemented at these RRHs. More specifically, the RRHs first compress the received messages from the UEs and then transmit the quantized description toward the BBU pool. The BBU processor first decompresses the compressed version and then decodes the UEs’ messages.
In the frequency domain, the dynamic allocation of bandwidth resource for wireless fronthaul and user communication is adopted. We assume that all the available bandwidth for the whole network is Hz, which can be divided into two orthogonal parts and , such that . The former is employed by the BS and RRHs to serve the MUEs and SUEs and the latter is dedicated to fronthaul links. It’s assumed that where in bps/Hz is constant and depends on the specific propagation environment. Without losing generality, we normalize the fronthaul capacity constraint as where in bps/Hz.
Iia Achievable SR at BS
As discussed above, we assume the messages transmitted from the SUEs are only received at the RRHs while ones from the MUEs can be detected at both the BS and RRHs. Since we are concerned about the achievable SR of the MUEs and SUEs, we simply divide the achievable SR into two parts: one at the BS and the other at the central BBU pool. We first focus on the achievable SR at the BS.
The received signal at the BS from the MUEs through subfrequency band is expressed as
(1) 
where is the channel matrix between the MUEs and BS, , and represent the smallscale and largescale fading coefficients, respectively [chi2017message], describes the independent additive white Gaussian noise (AWGN) [Liu2016Gaussian], is the signal vector of the MUEs with , denotes the transmitted symbols of MUE , and is the transmit power of MUE .
Under the linear reception, the achievable ergodic SR of the MUEs at the BS in bps/Hz is computed as
(2) 
where
(3) 
in which with . Note that the symbol “L” in the superscript suggests the linear reception. However, if the SIC method is applied, without losing generality, the decoding order is assumed as . Then the achievable SR at the BS becomes [simeone2016cloud]
(4)  
(5) 
Obviously, the achievable SR under the linear reception with SIC is greater than that under the linear reception without SIC due to less interference. But this advantage comes at the cost of higher computational complexity introduced by the SIC operation.
IiB Achievable SR at BBU Pool
The received signals at the BBU pool includes those from both the SUEs and MUEs. To the RRHs, the MUEs and SUEs are equivalent. Therefore, for ease of expression, we define as a composite channel matrix, where and represent the channel matrix between the SUEs and RRHs and that between the MUEs and RRHs, respectively. , where , represents the channel coefficient between the th SUE and the th RRH, and and describe the corresponding largescale and smallscale fading coefficients, respectively. is similar to the definition of . We further define as a composite signal vector, where and are the signal vectors from the SUEs and MUEs, respectively. , where , is the signal transmitted by SUE , and denotes the transmit power of SUE . Thus the received signal at RRH is expressed as
(6) 
where , represents the AWGN at AP , denotes the th element of , and is the th element of .
We assume the compressandforward scheme is applied at the RRHs, e.g., pointtopoint (P2P) compression [zhou2013approximate] or WynerZiv (WZ) coding [zhou2014optimized]. The RRHs first quantizes the received signal into (the superscript “” can be P2P or WZ indicating different quantization schemes adopted), and then transmits the compressed version to the BBU pool for central processing through wireless fronthaul links. At the BBU side, the quantization codewords are decompressed and then the messages are decoded [zhou2016fronthaul]. In this work, we obtain the achievable rate region in the case where each RRH only compresses its own received signal. Similar to reference [park2014fronthaul], we model the relationship between the received signal and its compressed description as Gaussian test channel,
(7) 
where is independent quantization noise and
describes the variance of the quantization noise at RRH
.In the case of the linear reception at the BBU pool, the achievable ergodic SR is
(8) 
where
(9) 
in which is an diagonal matrix with diagonal elements ’s, with , and .
When the SIC method is employed, without loss of generality, the decoding order is assumed as [zhang2014the, zhang2017energy, lin2017anew]. Then the achievable ergodic SR of the SUEs and MUEs at the BBU is expressed as
(10)  
(11) 
In what follows we give the fronthaul constraints that must be satisfied to achieve the SRs in (11) and (8) under two different compression strategies, i.e., P2P compression and WZ coding, respectively.
IiB1 P2P Compression Scheme
As a consequence of P2P compression, each RRH produces a binary string that allows the decompressor unit at the BS to identify the quantized signal from a certain codebook in parallel [park2014fronthaul]. According to ratedistortion theory, the signal can be recovered if the fronthaul rate satisfies the condition
(12) 
This constraint suggests that, from an intuitive level, larger ’s cause a smaller and thus less bandwidth is required by fronthaul. However, such separate and independent processing does not take into account the statistical correlation across the signals , , received at different RRHs [simeone2016cloud], which can be used as side information when the BBU decompresses the quantized signals from the RRHs. WZ coding provides an efficient way to leverage the side information available at the RRHs.
IiB2 WZ Coding Scheme
Taking advantage of the correlation of the received signals at all the RRHs which results from the mutual interference between UEs, WZ coding can achieve a higher performance and make better use of limited fronthaul capacity than P2P compression [park2014fronthaul]. WZ coding enables the compressor to use a finer quantizer and associate the same binary string to a subset of codewords, whereas P2P compression associates a distinct binary string with each codeword in the quantization codebook. According to Proposition 1 in [zhou2014optimized], the SRs in (11) and (8) can be achieved if the condition
(13) 
is satisfied.
IiC Problem Formulation
By jointly designing the compression noise variance matrix and the bandwidth allocation factor , the achievable ergodic SR maximization problem can be formulated as:
(14a)  
s.t.  (14b)  
(14c)  
(14d)  
(14e) 
where the symbol “” in the superscript can be “L” or “SIC” representing different decoding strategies (i.e., the linear receptions with and without SIC), the fronthaul constraint (14b) can be either (12) or (13), depending on the specific compression strategy. Constraints (14d) and (14e) indicate that is a diagonal matrix. The objective function is a differenceofconvex problem and some algorithms are proposed in [beck2009gradient, zhou2014optimized, Park2013Joint] to solve related problems. However, since the optimization of (, ) is based on the ergodic SR, Monte Carlo averaging over channels needs a lot of samples to capture the variations of both largescale fading and smallscale fading. Because smallscale fading varies at the level of milliseconds, collecting channel information leads to too much communication overhead and calculating average results over channels also brings more prohibitively computational complexity. Besides, the bandwidth allocation problem is a slow timescale issue because it is usually executed in a large time span whereas the compression noise design problem is a fast timescale issue since it is performed in each small time span (for example, a coherence time interval of wireless channels) due to its dependence on fastchanging smallscale fading. Therefore, the joint problem is a mixed timescale issue. To address these challenges, we first introduce the asymptotic expression of the ergodic SR in the largesystem regime and then find solutions based on the deterministic approximation in the following sections.
Iii Deterministic Equivalents
In the following, we understand the ergodic SR in the largesystem regime [zhang2013large, xia2017large] where , , , and grow infinitely while keeping fixed ratios and such that and [Xia2016Bandwidth]. For notational convenience, we use and to refer to and , respectively. We will derive deterministic approximations , , and of the ergodic SRs , , and , respectively.
Iiia Deterministic Equivalents for and
Since ’s are independently and identically distributed (i.i.d.) complex Gaussian variables whose real and imaginary parts are independent. According to references [wen2013adetermini, zhang2013oncap], we have the following lemma.
Lemma 1.
Given that ’s are i.i.d. complex Gaussian variables with independent real and imaginary parts, as , we have , where and
(15) 
where
(16) 
in which and
(17) 
In the case of the linear reception without SIC, as , the deterministic equivalent of is given as , where and in which
(18) 
with and
(19) 
Proof:
Refer to Appendix LABEL:appendixA.
IiiB Deterministic Equivalents for and
Based on the analysis of in Section IIIA, we use largedimensional random matrix theory again to derive the deterministic equivalents of and in the following lemma.
Lemma 2.
Given that ’s are i.i.d. complex Gaussian variables with independent real and imaginary parts, as , we have , where and
(20) 
where
(21) 
in which and
(22) 
with and .
In the case of the linear reception without SIC, we have , where and in which
(23) 
with and
(24) 