On the Capacity of Cloud Radio Access Networks with Oblivious Relaying
Abstract
We study the transmission over a network in which users send information to a remote destination through relay nodes that are connected to the destination via finitecapacity errorfree links, i.e., a cloud radio access network. The relays are constrained to operate without knowledge of the users’ codebooks, i.e., they perform oblivious processing. The destination, or central processor, however, is informed about the users’ codebooks. We establish a singleletter characterization of the capacity region of this model for a class of discrete memoryless channels in which the outputs at the relay nodes are independent given the users’ inputs. We show that both relaying àla CoverEl Gamal, i.e., compressandforward with joint decompression and decoding, and “noisy network coding”, are optimal. The proof of the converse part establishes, and utilizes, connections with the Chief Executive Officer (CEO) source coding problem under logarithmic loss distortion measure. Extensions to general discrete memoryless channels are also investigated. In this case, we establish inner and outer bounds on the capacity region. For memoryless Gaussian channels within the studied class of channels, we characterize the capacity region when the users are constrained to timeshare among Gaussian codebooks. We also discuss the suboptimality of separate decompressiondecoding and the role of timesharing. Furthermore, we study the related distributed information bottleneck problem and characterize optimal tradeoffs between rates (i.e., complexity) and information (i.e., accuracy) in the vector Gaussian case.
I Introduction
Cloud radio access networks (CRAN) provide a new architecture for nextgeneration wireless cellular systems in which base stations (BSs) are connected to a cloudcomputing central processor (CP) via errorfree finiterate fronthaul links. This architecture is generally seen as an efficient means to increase spectral efficiency in cellular networks by enabling joint processing of the signals received by multiple BSs at the CP and, so, possibly alleviating the effect of interference. Other advantages include low cost deployment and flexible network utilization [2].
In a CRAN network, each BS acts essentially as a relay node; and so it can in principle implement any relaying strategy, e.g., decodeandforward [3, Theorem 1], compressandforward [3, Theorem 6] or combinations of them. Relaying strategies in CRANs can be divided roughly into two classes: i) strategies that require the relay nodes to know the users’ codebooks (i.e., modulation, coding), such as decodeandforward, computeandforward [4, 5, 6] or variants thereof, and ii) strategies in which the relay nodes operate without knowledge of the users’ codebooks, often referred to as oblivious relay processing (or nomadic transmission) [7, 8, 9]. This second class is composed essentially of strategies in which the relays implement forms of compressandforward [3], such as successive WynerZiv compression [10, 11, 12] and quantizemapandforward [13] or noisynetwork coding [14]. Schemes that combine the two apporaches have been shown to possibly outperform the best of the two [15], especially in scenarios in which there are more users than relay nodes.
In essence, however, a CRAN architecture is usually envisioned as one in which BSs operate as simple radio units (RUs) that are constrained to implement only radio functionalities such as analogtodigital conversion and filtering while the baseband functionalities are migrated to the CP. For this reason, while relaying schemes that involve partial or full decoding of the users’ codewords can sometimes offer rate gains, they do not seem to be suitable in practice. In fact, such schemes assume that all or a subset of the relay nodes are fully aware (at all times!) of the codebooks and encoding operations used by the users. For this reason, the signaling required to enable such awareness is generally prohibitive, particularly as the network size gets large. Instead, schemes in which relay nodes perform oblivious processing are preferred in practice. Oblivious processing was first introduced in [7]. The basic idea is that of using randomized encoding to model lack of information about codebooks. For related works, the reader may refer to [8, 16] and [17]. In particular, [8] extends the original definition of oblivious processing of [7], which rules out timesharing, to include settings in which transmitters are allowed to switch among different codebooks, constrained relay nodes are unaware of the codebooks but are given, or can acquire, time or frequencyschedule information^{1}^{1}1Typically, this information is small, e.g., 1 bit that captures on/off activity; and, so, obtaining it is generally much less demanding that obtaining full information about the users’ codebooks.. The framework is referred to therein as “oblivious processing with enabled timesharing”.
In this work, we consider transmission over a CRAN in which the relay nodes are constrained to operate without knowledge of the users’ codebooks, i.e., are oblivious, and only know time or frequencysharing information. The model is shown in Figure 1. Focusing on a class of discrete memoryless channels in which the relay outputs are independent conditionally on the users’ inputs, we establish a singleletter characterization of the capacity region of this class of channels. We show that both relaying àla CoverEl Gamal, i.e., compressandforward with joint decompression and decoding, and noisy network coding are optimal. For the proof of the converse part, we utilize useful connections with the Chief Executive Officer (CEO) source coding problem under logarithmic loss distortion measure [18]. Extensions to general discrete memoryless channels are also investigated. In this case, we establish inner and outer bounds on the capacity region. For memoryless Gaussian channels within the studied class, we provide a full characterization of the capacity region under Gaussian signaling, i.e., when the users’ channel inputs are restricted to be Gaussian. In doing so, we also investigate the role of timesharing. Finally, leveraging the connection with the information bottleneck method (IB) [19] (see [20] for an earlier equivalent formulation of the IB problem), we study the problem of distributed information bottleneck (DIB) in which multiple sensors compress separately their observations in a manner that, collectively, the compressed signals provide as much information as possible about a remote (or hidden) source. For this model, we characterize optimal tradeoffs among the minimum description lengths at which the observed signals are described (i.e., complexity) and the information that the produced descriptions collectively preserve about the remote source (i.e., accuracy or relevant information). This is captured through the model’s informationrate region which we establish here for both discrete memoryless and memoryless vector Gaussian cases. The result in the Gaussian case generalizes that developed by Tishby et al. [21] for the singleencoder Gaussian information bottleneck method to the case of multiple encoders. Since the singleencoder IB method has found application in various contexts of learning and prediction [22], such as word clustering for text classification [23], community detection [24], neural code analysis [25], speech recognition [26] and others, distributed IB methods clearly finds usefulness in the extensions of those applications to the distributed case. BlahutArimoto type algorithms that allow to compute optimal tradeoffs between rate and information have recently been developed in [27]. The reader may refer to [28, 29, 30, 31, 32] for other related works.
Outline and Notation
The rest of this paper is organized as follows. Section II provides a formal description of the model, as well as some definitions that are related to it. Section III contains the main result of this paper, which is a singleletter characterization of the capacity region of a class of discrete memoryless CRANs with oblivious processing at relays and enabled timesharing in which the channel outputs at the relay nodes are independent conditionally on the users’ channel inputs. This section also provides inner and outer bounds on the capacity region of general discrete memoryless CRANs with constrained relays, as well as some discussions on the suboptimality of successive decompression and decoding and the role of timesharing. In Section IV, we study a memoryless vector Gaussian CRAN model with oblivious processing at relays and enabled timesharing, for which we characterize the capacity region under Gaussian signaling. Finally, in Section V, we characterize the rateinformation region of the vector Gaussian distributed information bottleneck problem.
Throughout this paper, we use the following notation. Upper case letters are used to denote random variables, e.g., ; lower case letters are used to denote realizations of random variables ; and calligraphic letters denote sets, e.g., . The cardinality of a set is denoted by . The length sequence is denoted as ; and, for integers and such that , the subsequence is denoted as . Probability mass functions (pmfs), are denoted by ; or for short, as . Boldface upper case letters denote vectors or matrices, e.g., , where context should make the distinction clear. For an integer , we denote the set of integers smaller or equal as . Sometimes, this set will also be denoted as . For a set of integers , the notation designates the set of random variables with indices in the set , i.e., . We denote the covariance of a zero mean vector by ; is the crosscorrelation , and the conditional correlation matrix of given as .
Ii System Model
Consider the discrete memoryless (DM) CRAN model shown in Figure 1. In this model, users communicate with a common destination or central processor (CP) through relay nodes, where and . Relay node , , is connected to the CP via an errorfree finiterate fronthaul link of capacity . In what follows, we let and indicate the set of users and relays, respectively.
Similar to [8], the relay nodes are constrained to operate without knowledge of the users’ codebooks and only know a timesharing sequence , i.e., a set of time instants at which users switch among different codebooks. The obliviousness of the relay nodes to the actual codebooks of the users is modeled via the notion of randomized encoding [7] (see also [33] for an earlier introduction of this notion in the context of coding for channels with unknown states). That is, users or transmitters select their codebooks at random and the relay nodes are not informed about the currently selected codebooks, while the CP is given such information. Specifically, in this setup, user , , sends codewords that depend not only on the message of rate that is to be transmitted to the CP by the user and the timesharing sequence , but also on the index of the codebook selected by this user. This codebook index runs over all possible codebooks of the given rate , i.e., , and is unknown to the relay nodes. The CP, however, knows all indices of the currently selected codebooks by the users. Also, it is assumed that all terminals know the timesharing sequence.
Iia Formal Definitions
The discrete memoryless CRAN model with oblivious relay processing and enabled timesharing that we study in this paper is defined as follows.

Messages and Codebooks: Transmitter , , sends message to the CP using a codebook from a set of codebooks that is indexed by . The index is picked at random and shared with the CP, but not the relays.

Timesharing sequence: All terminals, including the relay nodes, are aware of a timesharing sequence , distributed as for a pmf .

Encoding functions: The encoding function at user , , is defined by a pair where is a singleletter pmf and is a mapping that assigns the given codebook index , message and timesharing variable to a channel input . Conditioned on a timesharing sequence , the probability of selecting a codebook is given by
(1) where for some given conditional pmf .

Relaying functions: The relay nodes receive the outputs of a memoryless interference channel defined by
(2) Relay node , , is unaware of the codebook indices , and maps its received channel output into an index as . The index is then sent the to the CP over the errorfree link of capacity .

Decoding function: Upon receiving the indices , the CP estimates the users’ messages as
(3) where
(4) is the decoding function at the CP.
Definition 1.
A code for the studied DM CRAN model with oblivious relay processing and enabled timesharing consists of encoding functions , relaying functions , and a decoding function .
Definition 2.
A rate tuple is said to be achievable if, for any , there exists a sequence of codes such that
(5) 
where the probability is taken with respect to a uniform distribution of messages , , and with respect to independent indices , , whose joint distribution, conditioned on the timesharing sequence, is given by the product of (1).
For given individual fronthaul constraints , the capacity region is the closure of all achievable rate tuples .
In this work, we are interested in characterizing the capacity region .
IiB Some Useful Implications
As shown in [8], the above constraint of oblivious relay processing with enabled timesharing means that, in the absence of information regarding the indices and the messages , a codeword taken from a codebook has independent but nonidentically distributed entries.
Lemma 1.
Without the knowledge of the selected codebooks indices , the distribution of the transmitted codewords conditioned on the timesharing sequence are given by
(6) 
Thus, the channel output at relay is distributed as
Proof.
Remark 1.
Equation (6) states that, when averaged over the probability of selecting a codebook and over the uniform distribution of the message set, but conditioned on the timesharing variable , the transmitted codeword has a pmf according to a product distribution of independent but nonidentically distributed entries. That is, in the absence of codebook information, the codewords lack structure. When a node is informed of the codebook index , the codebook structure is provided by the selected codebook.
Iii Discrete Memoryless Model
Iiia Capacity Region of a Class of CRANs
In this section, we establish a singleletter characterization of the capacity region of a class of discrete memoryless CRANs with oblivious relay processing and enabled timesharing in which the channel outputs at the relay nodes are independent conditionally on the users’ inputs. Specifically, consider the following class of DM CRANs in which equation (2) factorizes as
(7) 
Equation (7) is equivalent to that, for all and all ,
(8) 
forms a Markov chain in this order. The following theorem provides the capacity region of this class of channels.
Theorem 1.
For the class of DM CRANs with oblivious relay processing and enabled timesharing for which (8) holds, the capacity region is given by the union of all rate tuples which satisfy
(9) 
for all nonempty subsets and all , for some joint measure of the form
(10) 
Remark 2.
Our main contribution in Theorem 1 is the proof of the converse part. As mentioned in Appendix A, the direct part of Theorem 1 can be obtained by a coding scheme in which each relay node compresses its channel output by using WynerZiv binning [34] to exploit the correlation with the channel outputs at the other relays, and forwards the bin index to the CP over its ratelimited link. The CP jointly decodes the compression indices (within the corresponding bins) and the transmitted messages, i.e., CoverEl Gamal compressandforward [3, Theorem 3] with joint decompression and decoding (CFJD)^{2}^{2}2The rate region achievable by this scheme for a general DM CRAN, i.e., without the Markov chain (8), is given by Theorem 2.. Alternatively, the rate region of Theorem 1 can also be obtained by a direct application of the noisy network coding (NNC) scheme of [14, Theorem 1]. The reader may find it useful to observe that the fact that the two operations of decompression and decoding are performed jointly in the scheme CFJD is critical to achieve the full rateregion of Theorem 1, in the sense that if the CP first jointly decodes the compression indices and then jointly decodes the users’ messages, i.e., the two operations are performed successively, this results in a a region that is generally strictly suboptimal. A similar observation can be found in [12].
Remark 3.
Key element to the proof of the converse part of Theorem 1 is the connection with the Chief Executive Officer (CEO) source coding problem^{3}^{3}3Because the relay nodes are connected to the CP through errorfree finiterate links, the scenario, as seen by the relay nodes, is similar to one in which a remote vector source needs to be compressed distributively and conveyed to a single decoder. There are important differences, however, as the vector source is not i.i.d. here but given by a codebook that is subject to design.. For the case of encoders, while the characterization of the optimal ratedistortion region of this problem for general distortion measures has eluded the information theory for now more than four decades, a characterization of the optimal region in the case of logarithmic loss distortion measure has been provided recently in [18]. A key step in [18] is that the logloss distortion measure admits a lower bound in the form of the entropy of the source conditioned on the decoders input. Leveraging on this result, in our converse proof of Theorem 1 we derive a single letter upperbound on the entropy of the channel inputs conditioned on the indices that are sent by the relays, in the absence of knowledge of the codebooks indices . (Cf. the step (56) in Appendix A). The connection with the CEO problem is discussed further in Section V.
Remark 4.
In the special case in which and the memoryless channel (7) is such that for , the source coding counterpart of the problem treated in this section reduces to a distributed source coding setting with independent sources (recall that the users input symbols are independent here) under logarithmic loss distortion measure. Note that, for and general, i.e., arbitrarily correlated, sources, the problem appears to be of remarkable complexity, and is still to be solved. In fact, the BergerTung coding scheme [35] can be suboptimal in this case, as is known to be so for KornerMarton’s modulotwo adder problem [36].
IiiB Inner and Outer Bounds for the General DM CRAN Model
In this section, we study the general DM CRAN model (2). That is, the Markov chains given by (8) are not necessarily assumed to hold. In this case, we establish inner and outer bounds on the capacity region that do not coincide in general. The bounds extend those of [7], which are established therein for a setup with a single transmitter and no timesharing, to the case of multiple transmitters and enabled timesharing.
The following theorem provides an inner bound on the capacity region of the general DM CRAN model (2) with oblivious relay processing and timesharing.
Theorem 2.
For the general DM CRAN model (2) with oblivious relay processing and enabled timesharing, the achievable rate region of the scheme CFJD is given by the union of all rate tuples that satisfy, for all nonempty subsets and all ,
(11) 
for some joint measure of the form
(12) 
Remark 5.
The coding scheme that we employ for the proof of Theorem 2, which we denote by compressandforward with joint decompression and decoding (CFJD), is one in which every relay node compresses its output àla CoverEl Gamal compressandforward [3, Theorem 3]. The CP jointly decodes the compression indices and users’ messages. The scheme, as detailed in Appendix B, generalizes [7, Theorem 3] to the case of multiple users and enabled timesharing.
We now provide an outer bound on the capacity region of the general DM CRAN model with oblivious relay processing and timesharing. The following theorem states the result.
Theorem 3.
For the general DM CRAN model (2) with oblivious relay processing and enabled timesharing, if a rate tuple is achievable then for all nonempty subsets and it holds that
(13) 
for some distributed according to
(14) 
where for ; for some random variable and deterministic functions , for .
Remark 6.
The inner bound of Theorem 2 and the outer bound of Theorem 3 do not coincide in general. This is because in Theorem 2, the auxiliary random variables satisfy the Markov chains , while in Theorem 3 each is a function of but also of a “common” random variable . In particular, the Markov chains do not necessarily hold for the auxiliary random variables of the outer bound.
Remark 7.
As we already mentioned, the class of DM CRAN models satisfying (8) connects with the CEO problem under logarithmic loss distortion measure. The ratedistortion region of this problem is characterized in the excellent contribution [18] for an arbitrary number of (source) encoders (see [18, Theorem 3] therein). For general DM CRAN channels, i.e., without the Markov chain (8) the model connects with the distributed source coding problem under logarithmic loss distortion measure. While a solution of the latter problem for the case of two encoders has been found in [18, Theorem 6], generalizing the result to the case of arbitrary number of encoders poses a significant challenge. In fact, as also mentioned in [18], the BergerTung inner bound is known to be generally suboptimal (e.g., see the KornerMarton lossless modulosum problem [36]). Characterizing the capacity region of the general DM CRAN model under the constraint of oblivious relay processing and enabled timesharing poses a similar challenge, even for the case of two relays. Finally, we mention that in the context of multiterminal distributed source coding with general distortion measure, an outer bound has been derived in [37]; and is shown to be tight in certain cases. The proof technique therein is based on introducing a random source such that the observations at the encoders are conditionally independent on , i.e., a Markov chain similar to that in (8) holds. Note however that the connection of the outer bound that we develop here for the uplink CRAN model with oblivious relay processing with that of [37] is only of high level nature as the proof techniques are different.
IiiC On the Suboptimality of Separate DecompressionDecoding and Role of TimeSharing
For the general DM CRAN model (2), the scheme CFJD of Theorem 2 is based on a joint decoding of the compression indices and users’ messages. That is, the CP performs the operations of the decoding of the quantization codewords and the decoding of the users’ messages simultaneously. A more practical strategy, considered also in [7] and [12], consists in having the CP first decode the quantization codewords (jointly), and then decode the users’ messages (jointly). That is, compressandforward with separate decompression and decoding operations. In what follows, we refer to such a scheme as CFSD. The following proposition provides the rateregion allowed by this scheme for the DM CRAN model (2).
Proposition 1.
As a special instance of the scheme CFSD, we consider compressandforward with successive separate decompressiondecoding performs sequential decoding of the quantization codewords first, followed by sequential decoding of the users’ messages. More specifically, let and be two permutations that are defined on the set of quantization codewords and the set of user message codewords, respectively. An outline of this scheme, which we denote as CFSSD, is as follows. The relay nodes compress their outputs sequentially, starting by relay node . In doing so, they utilize WynerZiv binning [34], i.e., relay node , , quantizes its channel output into a description taking into account as decoder side information. The CP first recovers the quantization codewords in the same order, and then decodes the users’ messages sequentially, in the order indicated by , starting by user . That is, the codeword of user , , is estimated using all compression codewords as well as the previously decoded users codewords . The rateregion obtained with a given decoding order as well as that of the scheme CFSSD, obtained by considering all possible permutations, are given in the following proposition.
Proposition 2.
For the general DM CRAN model (2) with oblivious relay processing and enabled timesharing, the achievable rate region of the scheme CFSSD with decoding order is the union of all rate tuples that satisfy, for all and ,
(16a)  
(16b) 
for some pmf . The rate region achievable by the scheme CFSSD is defined as the union of the regions over all possible permutations and , i.e.,
(17) 
While successive separate decompression and decoding results in a rate region that is generally strictly smaller than that of joint decoding, i.e., with CFJD, in what follows we show that the maximum sumrate that is achievable by this specific separate decompressiondecoding is the same as that achieved by joint decoding. That is, the schemes CFSSD and CFJD achieve the same sumrate (and, so, so does also the scheme CFSD). Specifically, let the maximum sumrate achieved by the scheme CFJD be defined as
Similarly, let the maximum sum rate for the scheme CFSD be defined as
and that of the scheme CFSSD defined as
Theorem 4.
Remark 8.
The proof of Theorem 4 uses properties of submodular optimization; and is similar to that of [12, Theorem 2] which shows that CFJD and CFSD achieve the same sumrate for the class of CRANs that satisfy (8). Thus, in a sense, Theorem 4 can be thought of as a generalization of [12, Theorem 2] to the case of general channels (2).
Remark 9.
Theorem 4 shows that the three schemes CFJD, CFSD and CFSSD achieve the same sumrate and that, in general, the use of timesharing is required for the three schemes to achieve the maximum sumrate. Note that the uplink CRAN is a multiplesource, multiplerelay, singledestination network. If all fronthaul capacities were infinite, then the model would reduce to a standard multiple access channel (MAC) and it follows from standard results that timesharing is not needed to achieve the optimal sumrate in this case [38]. The reader may wonder whether it is also so in the case of finiterate fronthaul links, i.e., whether one can optimally set in the region for sumrate maximization. The answer to this question is negative for finite fronthaul capacities , as shown in Section IV. This is reminiscent of the fact that timesharing generally increase rates in relay channels, e.g., [39, 40]. In addition, when the three schemes CFJD, CFSD and CFSSD are restricted to operate without timesharing, i.e., , CFSSD might perform strictly worse than CFJD and CFSD. To see this, the reader may find it useful to observe that while timesharing is not required for sumrate maximization in a regular MAC, as successive decoding (in any order) is sumrate optimal in this case, it is beneficial when the sumrate maximization is subjected to constraints on the users’ message rates such as when the users’ rates need to be symmetric [41], i.e., the operation point is not in a corner point of the MAC region. Similarly, standard successive WynerZiv (in any order, without timesharing) is known to achieve any corner point of the BergerTung region [42, 43], but timesharing (or ratesplitting àla [42]) is beneficial if the compression rates are subjected to constraints such as when the compression rates are symmetric. An example which illustrates these aspects for memoryless Gaussian CRAN is provided in Section IV.
Iv Memoryless MIMO Gaussian CRAN
In this section, we consider a memoryless Gaussian MIMO CRAN with oblivious relay processing and enabled timesharing. Relay node , , is equipped with receive antennas and has channel output
(19) 
where , is the channel input vector of user , is the number of antennas at user , is the matrix obtained by concatenating the , , horizontally, with being the channel matrix connecting user to relay node , and is the noise vector at relay node , assumed to be memoryless Gaussian with covariance matrix and independent from other noises and from the channel inputs . The transmission from user is subjected to the following covariance constraint,
(20) 
where is a given positive semidefinite matrix, and the notation indicates that the matrix is positive semidefinite.
Iva Capacity Region under TimeSharing of Gaussian Inputs
The memoryless MIMO Gaussian model with oblivious relay processing decribed by (19) and (20) clearly falls into the class of CRANs studied in Section IIIA, since forms a Markov chain in this order for all . Thus, Theorem 1, which can be extended to continuous channels using standard techniques, characterizes the capacity region of this model. The computation of the region of Theorem 1, i.e., , for the model described by (19) and (20), however, is not easy as it requires finding the optimal choices of channel inputs and the involved auxiliary random variables . In this section, we find an explicit characterization of the capacity region of the model described by (19) and (20) in the case in which the users are constrained to timeshare only among Gaussian codebooks. That is, for all and all , the distribution of the input conditionally on is Gaussian (with covariance matrix that can be optimized over so as to satisfy (20)). We denote that region by . Although Gaussian input may generally be suboptimal for uplink CRAN [7], i.e., in general , restricting to Gaussian input for every is appreciable because it leads to rate regions that are less difficult to evaluate. In doing so, we also show that timesharing Gaussian compression at the relay nodes is optimal if the users’ channel inputs are restricted to be Gaussian for all .
Let, for all , the input be restricted to be distributed such that for all ,
(21) 
where the matrices are chosen to satisfy
(22) 
The following theorem characterizes the capacity region of the model with oblivious relay processing described by (19) and (20) under the constraint of fixed Gaussian input and given fronthaul capacities .
Theorem 5.
The capacity region of the memoryless Gaussian MIMO model with oblivious relay processing described by (19) and (20) under timesharing of Gaussian inputs is given by the set of all rate tuples that satisfy
(23) 
for all nonempty and all , for some pmf and matrices and such that and ; and where, for and , the matrix is defined as .
Remark 10.
Theorem 5 extends the result with oblivious relay processing of [7, Theorem 5] to the MIMO setup with users and enabled timesharing, and shows that under the constraint of Gaussian signaling, the quantization codewords can be chosen optimally to be Gaussian. Recall that, as shown through an example in [7], restricting to Gaussian input signaling can be a severe constraint and is generally suboptimal.
IvB On the Role of TimeSharing
In Remark 9 in Section IIIC we commented on the utility of timesharing for sumrate maximization in the uplink of DM CRAN with oblivious relay processing. In this section we investigate further the role of timesharing. Specifically, we first provide an example in which timesharing increases capacity; and then discuss some scenarios in which timesharing does not enlarge the capacity region of the memoryless MIMO Gaussian CRAN model with oblivious relay processing described by (19) and (20).
For convenience, let us denote by the rate region obtained by setting , i.e, without enabled timesharing, in the region of Theorem 5. That is, is given by the set of all rate tuples that for all nonempty and all
(24) 
for some , .
The following example shows that may be contained strictly in .
Example 1.
Consider an instance of the memoryless MIMO Gaussian CRAN described by (19) and (20) in which , , (all devices are equipped with singleantennas), the relay nodes have equal fronthaul capacities, i.e., , and
(25) 
where and , for .
The capacity of this oneuser Gaussian CRAN example can be obtained from Theorem 5 as the following optimization problem
(26) 
where the maximization is over , and , such that and . Due to Theorem 4, is achievable with CFJD, CFSD and CDSSD by using timesharing. Without timesharing, i.e., , the capacity of this oneuser Gaussian CRAN example is achievable with the CFJD scheme and can be obtained easily from (24), as
(27)  
(28) 
With timesharing with, say , the user can communicate at larger rates with CFJD, as follows. The transmission time is divided into two periods or phases, of duration and respectively, where . The user transmits symbols only during the first phase, with power ; and it remains silent during the second phase. The two relay nodes operate as follows. During the first phase, relay node , , compresses its output to the fronthaul constraint ; and it remains silent during the second phase. Observe that with such transmission scheme the input constraint (22) and fronthaul contraints are satisfied. Evaluating the rateregion of Theorem 5 with the choice , , and , yields in this case
(29) 
Figure 2 depicts the evolution of the the capacity enabled with timesharing , the capacity without timesharing , as well as the cutset upper bound, for and , as function of the user transmit power . Also shown for comparison is the achievable rate as given by (29), which is a lower bound on . Obseve that while restricting to CFJD with twophases might be suboptimal, is very close to . As it can be seen from the figure, the utility of timesharing (to increase rate) is visible mainly at small average transmit power. The intuition for this gain is that, for small , the observations at the relay nodes become too noisy and the relay mostly forwards noise. It is therefore more advantageous to increase the power at for a fraction of the transmission. Accordingly, the effective compression rate is increased to , therefore reducing the compression noise. This observation is reminiscent of similar ones in [39] in the context of relay channels with orthogonal components and in [40] in the context of primitive relay channels.
When the three schemes CFJD, CFSD and CFSSD are restricted to operate without timesharing, i.e., , and Gaussian signaling, CFSD and CFSSD might perform strictly worse than CFJD. The rate achievable by the CFSD scheme without timesharing follows by Proposition 1, and it is easy to show that it coincides with in (28), i.e., in this example, CFJD and CFSD achieve the capacity without timesharing. The rate achievable by CFSSD without timesharing and Gaussian test channels , , can be obtained from Proposition 2, as
(30) 
where and .
Figure 3 shows the capacities , and the achievable rates and for and , as function of the transmit power . Note that CFSSD, when restricted not to use timesharing performs strictly worse than CFJD and CFSD without timesharing, i.e., . Observe that in this scenario, the gains due to timesharing are limited. This observation is in line with the fact that for large fronthaul values, the CRAN model reduces to a MAC, for which timesharing is not required to achieve the optimal sumrate. ∎
The above shows that in general timesharing increases rates for the memoryless MIMO Gaussian CRAN model described by (19) and (20), i.e., . In what follows, we discuss two scenarios in which timesharing does not enlarge the capacity region of the model given by (19) and (20), i.e., .
IvB1 Case of Fixed Gaussian Codebook at User Side
Consider the scenario in which the users are not allowed to timeshare among several Gaussian codebooks, but they are constrained to use each a single, possibly different, Gaussian codebook. This may be relevant, e.g., for contexts in which signaling overhead reduction among the users and relays is of prime interest. Conceptually, this corresponds to equalizing all the covariance matrices for given and all . Let
(31) 
The reader may wonder whether allowing the relay nodes to timeshare among compression codebooks can be beneficial in this case. Note that the answer to this question is not clear apriori, because timesharing in general increases the BergerTung rate region if constraints on the rates are imposed. (See Remark 9). The following proposition shows that for the model described by (19) and (20) this does not hold under the constraint (31).
Proposition 3.
IvB2 High SNR Regime
Consider again the model described by (19) and (20). Assume that for all the vector Gaussian noise at relay node has covariance matrix
(32) 
for some and that is independent from .
The following proposition shows that, in this case, the benefit of timesharing in terms of increasing rates vanishes for arbitrarily small .
Proposition 4.
IvC Price of NonAwareness: Bounded Rate Loss
In this section, we show that for the memoryless MIMO Gaussian model that is given by (19) and (20) allowing the relay nodes to be fully aware of the users’ codebooks (i.e., the nonconstrained or nonoblivious setting) increases rates by at most a bounded constant (only !). In other terms, restricting the relay nodes not to know/utilize the users’ codebooks causes only a bounded rate loss in comparison with maximum rate that would be achievable in the nonoblivious setting. The constant depends on the network size, but is independent of the channel gain matrix, powers and noise levels. The result is an easy combination of a recent improved constantgap result of Ganguly and Lim in [44] (which tightens further that of Zhou et al. [12], see Remark 11 below) with our Theorem 5.
For simplicity, we focus on the case in which for all and for all . For the unconstrained case (i.e., with none of the constraints of obliviousness and Gaussian signaling assumed), the capacity region of the model described by (19) and (20), which we denote hereafter as , is still to be found in general; and an easy outer bound on it is given by the maximumflow mincut bound, i.e., the set of all rate tuples for which for all and
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The following theorem shows that the rateregion of Theorem 5 is within a constant gap from , and so from the capacity region of the unconstrained setting .
Theorem 6.
If , then there exists a constant such that , with