Robust and Efficient Multi-Cell Cooperation under Imperfect CSI and Limited Backhaul Peter Rost, Member, IEEE Abstract—Future cellular networks need to harvest existing spectral resources more efficiently. Hence, networks will be deployed at a higher density in order to increase the spatial reuse. This will require advanced interference mitigation techniques that allow to cope with the increased interference level. In this paper, the two-way interference channel is analyzed as a model for a typical inter-cell interference scenario. Based on this model, a new inter-cell interference mitigation approach is derived. This new approach reshapes interference by asymmetrically assigning uplink and downlink to communication pairs, i. e., one communication pair operates in uplink while an adjacent communication pair is in downlink. In addition, backhaul resources are taken into account, which are used to exchange support information between radio access points and to support the interference mitigation process. The introduced approach is compared to cooperative multi-point techniques which employ joint transmission and reception algorithms. The evaluation is done under consideration of limited backhaul resources and imperfect channel state information. It shows that assigning uplink and downlink asymmetrically is able to outperform cooperative multi-point techniques for terminals close to the cell border with gains of up to about 20 % compared to noncooperative transmission and 10 % compared to CoMP. Index Terms—Two-way communication, interference channel, uplink-downlink-interference, cooperative communication

I. I NTRODUCTION Next generation mobile networks (NGMN) will be characterized by heterogeneous and very dense deployments which allow for an efficient reuse of the existing spectrum. These deployments will make use of smaller Radio Access Points (RAPs) which are more cost- and energy-efficient than macrobase-stations. However, the dense deployment of networks will imply that terminals receive strong interference from adjacent cells. Hence, communication will become more and more interference-limited, in particular as frequency reuse larger than 1 is not an option due to spectrum scarcity. In addition, heterogeneous deployments imply additional interference between macro-cells and pico-cells. Multi-cell coordination and cooperation techniques are seen as one option to tackle the aforementioned problems. Cooperation techniques across multiple cells are defined by IEEE 802.16m [1] and are currently investigated and partly standardized in 3GPP LTE [2]. We can differentiate four different ways of multi-cell cooperation: inter-cell interference coordination, P. Rost ([email protected]) is with NEC Europe Labs, Network Research Department, 69115 Heidelberg, Germany. Part of this work has been performed in the framework of the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement No 257263 (FLAVIA). The authors would like to acknowledge the contributions of their colleagues in FLAVIA, although the views expressed are those of the authors and do not necessarily represent the project.

joint beamforming requiring joint Channel State Information (CSI), non-coherent joint transmission requiring shared user data but no joint CSI, and joint signal processing requiring joint knowledge of CSI and user data. In this paper, we use the term Cooperative Multi-Point (CoMP) to refer to the last strategy (joint signal processing) and we use the term InterCell Interference Coordination (ICIC) to refer to techniques where multiple access points coordinate their transmissions but do not share CSI. CoMP promises to provide significant gains due to its ability to cancel interference. However, it also requires significant backhaul resources and it is susceptible to imperfect CSI, latency on backhaul, and CSI quantization errors. By contrast to CoMP, ICIC coordinates transmissions instead of transmitting and receiving jointly. This may be done by coordinated scheduling, power control, or coordinated beamforming. ICIC requires less backhaul resources and allows for more flexibility with respect to the CSI quality. However, ICIC may not provide the same performance benefits as CoMP. Hence, there is need either to improve CoMP with respect to its backhaul-efficiency and robustness regarding imperfect CSI, or to improve ICIC to provide similar gains as CoMP through efficient, robust, and flexible approaches. A. Multi-cell cooperation This paper investigates approaches for the two-way Interference Channel (IC) [3], which consists of two communication pairs interfering each other. Hence, it represents a simple model for multi-cell scenarios with inter-cell interference. In addition, this model considers bi-directional communication representing the uplink and downlink of a mobile network. The scope of this paper is limited to Time Division Duplex (TDD) systems, where, usually, uplink and downlink of both communication pairs are aligned. By contrast, this paper considers a setup where uplink and downlink of communication pairs overlap. Due to the complexity and CSI requirements of CoMP, we consider only ICIC for the case of an asymmetric uplink and downlink assignment to both communication pairs. One reason for overlapping uplink (UL) and downlink (DL) transmissions is a dynamic traffic adaptation of cells, e. g. as considered for 3GPP LTE [4]. By dynamically changing the UL-DL-pattern, base stations (BSs) are able to adapt the offered traffic to the demand of user terminals (UTs). However, this may also be used to adaptively change the interference channel properties in the case of strong intercell interference and may result in preferable interference properties. In this paper, we present an ICIC approach which

exploits the interference re-alignment, is robust to imperfect CSI and requires less backhaul resources than CoMP.

multi-cell interference coordination based on joint scheduling, beamforming and power control.

B. Related work

The two-way interference channel was first analyzed in [22], which investigated a deterministic model of a forward datachannel and backward feedback-channel. The authors derived different feedback approaches for full-duplex and half-duplex scenarios. It was shown that under certain conditions, feedback may not increase the sum-capacity of forward and backward channel. More recently, [23] analyzed different two-way channels and derived outer bounds on the capacity with particular focus on the conditions under which feedback provides benefits. Finally, [24] analyzed the two-way interference channel based on a linear deterministic model and derived a specific feedback approach which improves the achievable rates. The derived strategy exploits interference channel properties such that superimposed signals are fed back.

The two-way channel was first investigated by Shannon [5] for the case of point-to-point communication. More recently, [6] derived outer bounds for the two-way three-node channel where one relay node supports a communication pair. While [6] derived more general outer bound regions, [7] investigated a more specific setup with one relay node that amplifies and forwards its channel output. Furthermore, [8] discussed general cut set bounds for networks of two-way communication nodes. The interference channel was introduced and analyzed by Carleial in [9]. It can be divided into three regimes, the very strong [10], the strong [11], and the weak interference channel [12]. Until now, [12] derived the best known inner bound for the interference channel based on superposition coding and joint decoding. More recently, [13] investigated different interference channels and their relation to each other. However, the derived approaches were only analyzed in the context of one-way communication. Further attention was drawn to interference alignment by [14] who analyzed interference channels with many terminals and described an approach which aligns all interference along one signal-dimension at a terminal and all useful information along an orthogonal dimension. One of the earliest work on inter-cell interference coordination is [15]. It analyzed distributed power control to balance the signal-to-interference power ratio of users assigned to different cells. In addition, it considered fading channels and imperfect channel measurements. In [16], the authors give an overview of heterogeneous networks and coordination techniques in time- and frequency-domain as well as using power control. The authors focus on 3GPP LTE and techniques that are applicable to a practical system. You et al. [17] presented a practical scheme in the context of 3GPP LTE, which allows to divide the bandwidth into different regions and to assign UTs dynamically to these different bandwidth regions. The evaluation of this scheme was done using a simulation framework as used for 3GPP LTE. Rangan and Madan tackle in [18] a general class of inter-cell interference coordination and resource allocation problems using a belief propagation framework which allows to optimize different relevant parameters such as transmit power and beamforming vectors. Their solution reduces both the communication overhead and computation complexity. In a recent work [19], Akoum and Heath investigate the problem of base station clustering for inter-cell interference coordination. Based on a Poisson point process model for the BS deployment, they derive how the cluster size depends on the number of antennas at the BS and other parameters. Furthermore, they derive a feedback scheme which takes the signal quality into account in order to reduce the signaling overhead. A similar scheme was presented in [20], where the distance between access point and UT was taken into account to compress the feedback information. Furthermore, [21] investigated an approach for

In order to assess the performance of the introduced approaches, we use the CoMP protocols presented in [25] as benchmark for the uplink evaluation and the CoMP protocols presented in [26] as benchmark for the downlink evaluation. These protocols rely on the lossless source-coding approach introduced by Slepian and Wolf [27], which is applied to support the inter-cell cooperation by exchanging data over an errorless backhaul connection. We further pay particular attention to limited backhaul-capacity as well as imperfect CSI [28], [29], which both affect the performance of multi-cell cooperation. CoMP has been mentioned first for downlink joint processing in [30]. Since then, an extensive literature on CoMP has been published. Among others, uplink multi-cell signal processing was discussed first in [31], [32]. More recently, [33] and [34] analyzed CoMP under limited backhaul-capacity but not taking into account imperfect CSI. Furthermore, [35], [36] provide an overview of recent developments as well as results from testbed trials and [37] provides a thorough overview of relevant literature, theoretical concepts and practically applicable technologies. Based on the seminal work in [38], Ozg¨ur et al. show in [39] that large wireless networks may operator both in a bandwidth and power-limited regime. They show that in such a regime, it is optimal to cooperate locally (CoMP) and to apply multi-hop communication for long-range communication. Cross-slot interference has already been observed in [40] which proposed a scheduling algorithm to avoid cross-slot interference. Similarly, [41] presented a cell-sizing algorithm for CDMA/FDD which avoids cross-slot interference by adapting the cell-size and therefore the traffic demand in UL and DL. In [42], a hybrid scheme of static and dynamic assignment of UL/DL slots was proposed which, in combination with appropriate scheduling and power control, allows to reduce the interference. Finally and more recently, [43] analyzed synchronization issues in systems where UL and DL of adjacent cells overlap.

Backhaul constraint β

C. Contributions and outline This paper applies a cooperation-technique based on Slepian-Wolf (SW)-coding, which is applied to the two-way IC and is compared to CoMP. Our analysis takes into account the constraints of limited backhaul-capacity and imperfect CSI. Available backhaul-capacity highly depends on the deployed backhaul technology, e. g., optical fiber, microwave, copperbased (DSL), or non-line-of-sight backhaul. While optical fiber offers enough bandwidth for extensive cooperation-techniques, other technologies may be capacity-limited and require lowoverhead cooperation techniques. One example is the deployment of small cells which may be connected through capacitylimited backhaul with high latency. In addition, imperfect CSI constitutes a serious challenge as many cooperation techniques require high-quality CSI. However, a practical system has only access to imperfect CSI due to estimation and quantization errors. Analytical results show that particularly towards the celledge an asymmetric assignment of UL and DL slots may outperform CoMP. Furthermore, the two-way IC allows to consider jointly achievable rates in UL and DL, which reflects the bi-directional nature of cellular communication. In the jointly achievable rate region, asymmetric assignments provide more flexibility compared to traditional CoMP. At first, this paper introduces the considered system model in Section II. Based on this system model, an approach for the asymmetric assignment of UL and DL slots is derived in Section III. In Section IV, CoMP approaches are discussed which serve as benchmark for the analysis of the introduced approach. Finally, we evaluate these approaches in Section V, discuss their applicability to mobile networks in Section VI, and conclude the paper in Section VII. II. S YSTEM SETUP AND TRANSMISSION MODEL A. Notation Throughout this paper, underlined symbols x indicate column vectors, boldface uppercase letters H indicate matrices, italic letters n, N indicate scalars, and calligraphic letters S indicate ordered sets. The i-th row and j-th column of matrix H is denoted by [H]i,j and the dimension of a matrix H with N rows and K columns is denoted by H[N ×K] . The trace of a matrix is denoted by tr(·), a transposed matrix is denoted by HT , the Hermitian of a matrix is denoted by HH , ∆(H) sets all off-diagonal elements of√H to 0, and ∆(x) denotes a matrix with x on its diagonal. H denotes an element-wise n square-root of matrix H. x ∼ CN (0, Σx ) denotes a circularly symmetric i.i.d. Gaussian random process with zero mean 0 and co-variance matrix Σx . We further use the matrices I and 0 to denote the identity matrix and zero matrix, and we use 0 to denote a vector of zeros. B. System setup Fig. 1(a) illustrates a conventional, symmetric operation of two BSs (1 and 2) and two UTs (3 and 4). The system is divided into two phases k ∈ {1, 2} of relative length 0 < τ1 < 1 and τ2 = 1 − τ1 which represent the relative share of DL

1

2

Direct link

Inter-cell if.

Direct link

3

4

(a) Symmetric UL/DL assignment

Backhaul constraint β

1

Inter-BS if.

Direct link

3

2

Direct link

Inter-UE if.

4

(b) Asymmetric UL/DL assignment Fig. 1. Symmetric and asymmetric assignment of uplink and downlink in the two-way interference channel with BSs 1 and 2, and UTs 3 and 4.

and UL, respectively. During the DL-phase, both BSs transmit and both UTs receive. In the UL-phase, both BSs receive the transmission of both UTs. Both transmissions in UL and DL are subject to inter-cell interference. This interference may be mitigated by coordinating both BSs over the backhaullink of capacity β. Throughout this paper, backhaul refers to the additional amount of signaling spent for multi-cell cooperation. Hence, β = 0 refers to the conventional noncooperative case. In the following, we use Ri , i ∈ {1, ...4} to denote the rate with which node i is communicating with its destination node, i. e. R1 , R2 denote the downlink rates of both BSs and R3 , R4 denote the uplink rates of both UTs. In Fig. 1(b), we illustrate the asymmetric operation of a TDD-system. By contrast to the symmetric operation, during the first phase of length τ1 BS 1 and UT 4 transmit while BS 2 and UT 3 are receiving the transmissions. In the second phase, this assignment switches such that BS 2 and UT 3 transmit while BS 1 and UT 4 are listening. Hence, UL and DL of both transmission pairs overlap and cause inter-BS as well as inter-UT interference. This interference may be very different from inter-cell interference in a symmetrically operating system, which allows for exploiting the possibly preferable properties of a different interference channel. In addition, this implies that UL and DL rates form a joint rate region in which the point of operation may be chosen depending on the DL-UL-rate demand. Therefore, we use the parameter η = τ1 R1 /τ2 R3 = τ2 R2 /τ1 R4 to identify the actual DL-UL-rate. In [44], a field study has been presented which analyzed typical values for smartphone services. It

shows that a majority of services requires a downlink-to-uplink rate ratio ranging from η = 2/1 to η = 5/1. Similarly, the 3GPP work item eIMTA [45] considers traffic adaptation using different TD-LTE configurations. Simulation setups for this work item consider a set of scenarios ranging from η = 1/2 to 4/1. In the following, we assume that each BS is equipped with NBS antennas and each UT is equipped with NUT antennas. Furthermore, let the transmission power at node i ∈ [1; 4] be constrained by pi . Then, we define the vector pi whose elements determine the per-antenna power assignments of node i and tr(∆(pi )) ≤ pi . For notational convenience, we will use the diagonal matricesPi of size  (NBS +NUT  )×(NBS +NUT ) defined as P1 = ∆( pT1 0T ), P2 = ∆( pT2 0T ), P3 =     ∆( 0T pT3 ), and P4 = ∆( 0T pT4 ). C. Transmission model

Our transmission model uses variables xi and yi to denote channel input and output, respectively, at node i. Variables xk , yk , and nk denote the channel input, channel output, and additional white Gaussian noise in phase k. More specifically, n n let x1 ∼ CN (0, P1 + P4 ), x2 ∼ CN (0, P3 + P2 ), and n nk ∼ CN (0, I). Vector y1 is the stacked receive vector of UT 3 and BS 2, and y2 is the stacked receive vector of BS 1 and UT 4. Both receive vectors are of size (NBS + NUT ) × 1. Correspondingly, xk , k ∈ {1, 2}, are the stacked transmit vectors of BS 1 and UT 4 as well as UT 3 and BS 2. Finally, we use Hi,j = HTj,i to denote the channel matrix between nodes i and j. The transmission is divided into two phases. Hence, we can state two transmission equations: y1 = Hx1 + n1 , 2

T 2

2

y =H x +n , with the channel matrix " [N ×N ] H1,3UT BS H= [N ×N ] H1,2BS BS

# [N ×N ] H4,3UT UT [N ×N ] , H4,2BS UT

(1) (2)

(3)

which is assumed to be constant during a transmission interval. We apply the channel estimation model as described in [25], [26], [46], i. e., the estimated channel at the receiver in phase k is given by ˆk = H+E ˆ k. H (4) This model is an abstraction of a practical channel measurement process. In a practical system, measurements and measurement errors are correlated in time. Due to the complexity of such a model, we use a model where the channel ˆ k is i.i.d. Gaussian and independent of the estimation error E channel event H. This model provides a lower bound on the achievable rate as shown in [25], [46]. A similar model was ˆk + E ˆ k , which leads to the discussed in [47] where H = H same capacity expressions. Both models further assume that the receiver knows the measurement error variance. Due to the fact that the estimated channel depends on the transmission power, it differs for uplink and downlink. Hence,

the transmission needs to be treated separately for uplink and downlink. An equivalent transmission model can be expressed by [25], [26] y1 = He,1 x1 + v1 + n1 , 2

e,2 T 2

y = (H

2

(5) 2

) x +v +n ,

(6)

where vk is an additional Gaussian random variable (r.v.) modeling imperfect CSI. The individual variables are derived as follows [H]i,j (7) [He,k ]i,j = q  k 1 + ΣEˆ i,j /E {|[H]i,j |2 }  1   N P +P  , k = 1  k 1 4 p i,j ΣEˆ i,j = (8) 1    N PT +PT  , k = 2 p 3 2 i,j    e,k ∆ E (P1 + P4 )(Ee,k )H , k = 1   (9) E{vk (vk )H } = ∆ Ee,k (PT3 + PT2 )(Ee,k )H , k = 2 v   u u E {|[H]i,j |2 } · ΣkEˆ i,j  e,k    . (10) E = t i,j E {|[H]i,j |2 } + ΣkEˆ i,j

The channel estimation error variance ΣkEˆ follows from the Cramer-Rao lower bound [48] which gives a lower bound on the error variance based on the number of pilot signals Np and the effective Signal-to-Noise Ratio (SNR) on the used pilot signals. In [49], it has been shown that this bound is achievable with MMSE estimation if the measurement error is i.i.d. Gaussian and pilot signals are uniformly distributed. In the following, we use the abbreviations i i h h e,1 e,1 e,1 e H = He3 = He,1 H H H 2 1,2 4,2 1,3 4,3 h i h i e,2 T e,2 T e,2 T e e T H1 = (H1,3 ) H4 = (H4,3 ) (H1,2 ) (He,2 4,2 ) where Hei denotes the effective channel from all transmitting e nodes to receiving node i. Similarly, the sub-matrices Ei e,k belonging to E are defined. III. I NTER - CELL COOPERATION IN THE TWO - WAY IC In this paper, we derive a cooperation approach which uses superposition coding and inter-cell cooperation based on Slepian-Wolf (SW) coding [27]. SW-coding refers to lossless source-coding where the decoder has access to sideinformation. Consider the following example. The encoder intents to communicate a message W1 to the decoder, which has access to side-information W2 . Then, the necessary rate with which the encoder needs to encode the source-message is given by the conditional entropy R1 ≥ H(W1 |W2 ) even though the encoder does not have access to the side-information but only knows the joint probability density function pW1 ,W2 (·, ·). A similar technique is applied to Decode-and-Forward relaying [50] where the destination-node exploits additional sideinformation provided by the relay node.

ˆ 1 , W2 V U1 , V1 , W1

•

g2

f1

V4 , U4 , V1

  p y1 |u1 , u4

U4 , V4

•

g3

f4

U1 , V1 V4 , W1 •

(a) Phase 1 ˆ 2 , W1 V U2 , V2 , W2

g1

f2

V3 , U3 , V2

  p y2 |u2 , u3

U3 , V3

g4

f3

U2 , V2 V3 , W2

(b) Phase 2 Fig. 2.

Illustration of message exchange in phases 1 and 2

We apply SW-coding to the two-way IC with asymmetric uplink-downlink assignment to support the mitigation of interBS interference. Specifically, SW-coding is used to support the detection, decoding, and subtraction of interference that the transmitting BS causes to the receiving BS. After the interference has been subtracted, the receiving BS detects and decodes the transmission from its assigned UT. Similarly, SWcoding is applied to the inter-UT interference such that each BS supports the interference-mitigation of its assigned UT. We further apply superposition coding which refers to an encoding strategy where two or more messages are superimposed. This strategy has been first applied to the degraded broadcast channel [51], [52]. The individual messages are split up such that the channel input X depends upon another sequence U, i .e., p(x, u) = p(x|u)p(u) and with channel pdf p(y1 , y2 |x, u) = p(y1 , y2 |x). This allows to overlay information for different receivers with different channel conditions to the transmitter. The same method was also used by Han and Kobayashi [12] for the interference channel. In their encoding approach, the channel input is an overlay of a private message and common message. The private message is only decoded by the corresponding communication partner while the common message is decoded by both receivers. Consider Fig. 2, which shows the individual message exchange in the considered scenario and for phases 1 and 2. In phase 1, nodes 1 (BS) and 4 (UT) transmit while nodes 2 (BS) and 3 (UT) receive. We consider the following messages:

Node 4 divides its message into two parts U4 and V4 with sum-rate R4 . Both messages are decoded by node 2. Node 3 only decodes the common message V4 in order to reduce the interference level for all other messages. Node 1 divides its message into three superimposed parts U1 , V1 , and W1 . Message U1 is a private message only intended for node 3, message V1 is a common message decoded by both nodes 2 and 3 in order to reduce the interference level at node 2. Message W1 in block b depends on message V4 in block b − 1 and supports node 3 during the decoding process of V4 . ˆ 1 and Node 1 further forwards the support messages V ˆ 1 is W2 over a errorless backhaul to node 2. Message V used to support the decoding process of message V1 at node 2. W2 is sent in the next block from node 2 to node 4 in order to support the decoding of message V3 .

Consider Table I which details the decoding and encoding procedure in both phases. In the first phase, node 1 transmits its messages U1 , V1 , W1 while node 4 transmits the messages U4 , V4 . Node 2 decodes both messages of node 4. In addition, it decodes message V1 in order to reduce the interference level for the messages sent by node 4. Node 1 supports the decoding of V1 at node 2 by providing additional redundant information, ˆ 1 , over the backhaul link. SW-coding is applied by first V selecting a set of possible candidates for V1 using the channel output Y2 . The correct message is identified by determining ˆ 1 . i. e., V ˆ1 the intersection with all messages identified by V can be seen as a-priori information for the receiver. Similarly, node 3 first decodes message W1 which supports the decoding of V4 sent in the previous block. V4 is decoded in order to reduce the interference for the actual messages U1 , V1 decoded at node 3 and sent in the previous block. Node 1 provides the additional information through W1 . W1 was forwarded by node 2 in the previous block and then forwarded to node 1 over the backhaul link. The exploitation of the additional information is done again through SW-coding. Based on this description and the system model defined in Section II, we formulate the following theorem. Theorem 1: The rate tuple (R1 , R2 , R3 , R4 ) must fulfill the constraint Ri ≤ RUi + RVi , i = 1 . . . 4

(11)

with the following side conditions for the first phase
−1 RU1 = log2 I + Rnn + R3vv + He3 αU4 P4 (He3 )H (12) · He3 αU1 P1 (He3 )H  RV1 = min log2 I + Rnn + R3vv
−1 e H3 αV1 P1 (He3 )H , +He3 (αU1 P1 + αU4 P4 )(He3 )H β − β2′ + log2 I + Rnn + R2vv 
−1 e H2 αV1 P1 (He2 )H + He2 (αU1 P1 + P4 )(He2 )H (13)

Enc

Phase 1 2

Node 1 U1 (b), V1 (b), W1 (b)

Backhaul

2

ˆ 2 (b) 7→ Y1 (b), V U3 (b), V3 (b), V2 (b)

1

ˆ 1 (b) V1 (b) 7→ V V3 (b) 7→ W2 (b + 1)

2

Node 3

U2 (b), V2 (b), W2 (b)

U3 (b), V3 (b) 1. Y3 (b) 7→ W1 (b) 2. W1 (b − 1, b), Y3 (b − 1) 7→ V4 (b − 1) 3. Y3 (b − 1) 7→ U1 (b − 1), V1 (b − 1)

ˆ 1 (b) 7→ Y2 (b), V U4 (b), V4 (b), V1 (b)

1

Dec

Node 2

Node 4 U4 (b), V4 (b)

1. Y4 (b) 7→ W2 (b) 2. W2 (b − 1, b), Y4 (b − 1) 7→ V3 (b − 1) 3. Y4 (b − 1) 7→ U2 (b − 1), V2 (b − 1)

ˆ 2 (b) V2 (b) 7→ V V4 (b) 7→ W1 (b + 1)

TABLE I E NCODING AND DECODING PROCEDURE IN BLOCK b. ROW Enc REFERS TO THE ENCODED MESSAGES AT THE INDIVIDUAL NODES , ROW Dec REFERS TO THE DECODED MESSAGES WHERE Y 7→ X IMPLIES THAT MESSAGE X WAS DECODED USING MESSAGE Y, AND ROW Backhaul REFERS TO THE MESSAGES WHICH ARE EXCHANGED THROUGH THE BACKHAUL CONNECTION BETWEEN NODES 1 AND 2.


−1 RU4 = log2 I + Rnn + R2vv + He2 αU1 P1 (He2 )H (14) · He2 αU4 P4 (He2 )H  RV4 = min RV4′ , log2 I + Rnn + R3vv
−1 +He3 (αU4 P4 + (αU1 + αV1 )P1 )(He3 )H  e H e (15) · H3 (αW1 P1 + αV4 P4 )(H3 ) RV4′ = log2 I + Rnn + R2vv
−1 e H2 αV4 P4 (He2 )H + He2 (αU4 P4 + αU1 P1 )(He2 )H (16)

 RV3 = min RV3′ , log2 I + Rnn + R4vv

+He4 (αU3 PT3 + αU2 + αV2 PT2 )(He4 )H  · He4 (αW PT2 + αV PT3 )(He4 )H 3

2

RV3′

= log2 I + Rnn + R1vv

+ He1 (αU3 PT3 + αU2 PT2 )(He1 )H


−1


−1 RU2 = log2 I + Rnn + R4vv + He4 αU3 PT3 (He4 )H · He4 αU2 PT2 (He4 )H (18)  RV2 = min log2 I + Rnn + R4vv
−1 e H4 αV2 PT2 (He4 )H , +He4 (αU2 PT2 + αU3 PT3 )(He4 )H β − β1′ + log2 I + Rnn + R1vv 
e H T e H −1 e T T e H1 αV2 P2 (H1 ) + H1 (αU2 P2 + P3 )(H1 ) (19)

RU3


−1 = log2 I + Rnn + R1vv + He1 αU2 PT2 (He1 )H · He1 αU PT3 (He1 )H 3

(20)

(21)

He1 αV3 PT3 (He1 )H (22)

min(β2′ , RV3′ ) ≥ log2 I + Rnn + R4vv +He4 (PT3
−1 e H4 αW2 PT2 (He4 )H + (αU2 + αV2 )PT2 )(He4 )H (23)

and with    min(β1′ , RV4′ ) ≥ log2 I + Rnn + R3vv +He3 (P4 ∆ Eej (P2 + P3 )(Eej )H ,
−1 e   H3 αW1 P1 (He3 )H Rjvv = + (αU1 + αV1 )P1 )(He3 )H ∆ Eej (P1 + P4 )(Eej )H , (17) and the following side conditions for the second phase


−1

j ∈ {1, 4} j ∈ {2, 3}

(24)

denoting the interference power caused by imperfect CSI. The set of all achievable rate tuples Rasym is then given by the β convex hull [ Rasym = (τ1 R1 , τ2 R2 , τ2 R3 , τ1 R4 ) (25) β α,τ

over all α· ≥ 0 and τ· > 0 satisfying αU1 + αV1 + αW1 ≤ 1, αU2 + αV2 + αW2 ≤ 1, αU3 + αV3 ≤ 1, αU4 + αV4 ≤ 1, and τ1 + τ2 = 1. The parameters τ1 and τ2 = 1 − τ1 represent the relative share of DL and UL, respectively, and the parameters α· represent the relative share of power for the individual messages at each terminal, which is explained in further detail in Appendix B. Proof: The proof of this theorem is based on an equivalent formulation for the discrete memoryless channel (DMC) in Appendix A, Theorem 2. The proof for Theorem 1 is given in Appendix B.

Equation (11) defines the actual rate constraint as the sum of the achievable rates for the private and common message Ui and Vi . In (12), we see that the achievable rate for the private message of node 1 is limited by the sum of the interference power caused by imperfect CSI, noise power, and the interference caused by the private message of node 4. Message V4 does not appear in (12) because it is decoded beforehand by node 3 (which is expressed by the second part of the min-expression in (15)). By symmetry, we have (18) and the second part of the min-expression in (21). Eq. (13) gives the rate-constraint for node 1’s common message V1 . The first part of the min-expression is the achievable rate for the link between node 1 and node 3, interfered by the private messages of nodes 1 and 4. The second part of the min-expression is the achievable rate for the link between ˆ 1 which node 1 and 2. The decoding process is supported by V ˆ 1, is accounted for with the backhaul capacity available for V ′ i. e., β − β2 . Since both node 3 and 2 need to decode V1 , we use the minimum in (13). By symmetry, we obtain (19). Similar to (12), the achievable rate of U4 in (14) is limited by the sum of the interference power caused by imperfect CSI, noise power, and the interference power caused by the private message of node 1. Since node 2 decodes first the common message V1 , it is not causing interference to message U4 . By symmetry, we obtain (20). As mentioned before, the right part of the min-expression in (15) follows from the constraint that node 3 is first decoding V4 . The left part of the min-expression is detailed in (16) and shows the achievable rate on the link between node 4 and 2. The private messages V1 and V4 are interfering with V4 which limits the achievable rates. By symmetry, we obtain (22). Finally, (17) reflects the constraint that the achievable rate of W1 (right hand side of (17)) must not be larger than the available backhaul resources β1′ and the achievable rate on the link between node 4 and 2. The latter follows because the amount of support information in W1 cannot exceed the information carried by V4 . By symmetry, we obtain (23). IV. T HE B ENCHMARK : C O MP In order to assess the performance of the introduced asymmetric UL-DL-assignment, we compare its performance to CoMP. However, this comparison only accounts for the backhaul overhead, which is necessary to exchange user data between BSs. It does not consider the overhead which is required to exchange CSI as well as the latency on the backhaul. By contrast to CoMP, protocols for an asymmetric UL-DL-assignment can directly obtain the required CSI through pilot signals and need not to exchange CSI over their backhaul link. All protocols and their achievable rate regions were described in [25], [26]. Therefore, we summarize the considered protocols briefly and refer to [25], [26] for the exact rate expressions. A. Uplink CoMP The applied uplink approach has been introduced in [25, Theorem 4] where it is referred to as Distributed Interference

X3

f1

W1

g2

X3 , X4

g1 Y2 Y1 1

2

3

4

Fig. 3. Uplink CoMP protocol DIS which is based on SW-coding and used as benchmark −θBS A1,2 = ABS dBS

1

3 −θUT A1,3 = dUT

4

2 −θUT A2,4 = dUT

Fig. 4. Setup for the analysis of the two-way interference channel. Labels indicate the path-loss Ai,j between nodes i and j.

Subtraction (DIS). DIS also applies the ideas of SW-coding and is illustrated in Fig. 3(a). At first, BS 1 detects and decodes X3 of UT 3 treating the transmission of UT 4 as interference. Then, BS 1 encodes X3 with rate RW1 such that β ≥ RW1 ≥ H(X3 |Y2 ). The resulting message is then forwarded to BS 2 which uses this support information to detect and decode the interference caused by UT 3. After the interference has been subtracted using the estimated channel ˆ 3,2 , BS 2 detects and decodes the transmission of its assigned H UT, i. e. X4 . This cooperation method can be applied, of course, also vice versa such that BS 2 supports the decoding process at BS 1. We further apply the additional constraint that each BS must decode the message of its assigned UT. B. Downlink CoMP The applied downlink CoMP approach has been described in [26, Section VI.A], where it is referred to as Unquantized Message based Cooperation (UMC). In this case, each BS may be provided with a copy of the message of the other BS in order to perform linear pre-coding. Side-information cannot be exploited and therefore the rate of each support-message must not exceed the backhaul-capacity β. In our analysis, we allow for time-sharing between four cases: non-cooperative transmission, BS 1 receives a copy of X2 , BS 2 receives a copy of X1 , and both BSs receive a copy of the other BS’s message. V. R ESULTS AND D ISCUSSION A. Setup In the following, we evaluate the previously discussed approaches based on the setup in Fig. 4. This setup comprises two UTs which are placed symmetrically at distance dUT between two BSs. Both BSs are equipped with NBS = 2

Sym, Non-coop Sym, DIS-UMC Downlink

rs ut rs ut ut ut ut bc bcrs bc

bcrs

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Similarly, the compound channel matrix for the case of symmetric UL-DL-assignment is chosen as [25]   1/2 1/2 A1,3 A1,4 ej(ϕA +ϕB ) 1/2 1/2  A1,3 A1,4 ej(−ϕA +ϕB )    H =  1/2 −j(ϕ . 1/2 A −ϕB ) A2,3 e  A2,4

3.0 ut

antennas and both UTs are equipped with one antenna (NUT = 1). We apply an exponential path-loss model with path-loss exponent θUT = 3.5 between BSs and UTs as well as between UTs, and θBS = 2.5 between both BSs. The path loss between nodes i and j is denoted by Ai,j . In order to account for the directivity of BS antennas, we further apply a directional antenna-gain of ABS = 10 dB. The compound matrix for the case of asymmetric UL-DLassignment is given by   1/2 1/2 1/2 A1,3 A1,3 A4,3 ej(ϕA +ϕB ) 1/2 1/2 1/2   H= A1,2 A1,2 ej(ϕA −ϕB ) A4,2 . 1/2 1/2 1/2 −j(ϕA +ϕB ) A1,2 e A1,2 A4,2

rs

0.5 150

175

200 Distance dUT [m]

225

250

Fig. 5. Common rate performance depending on BS-UT-distance dUT and limited backhaul-capacity β = 4.

1/2

A2,4

Results are averaged over uniformly distributed angles ϕA and ϕB . We evaluate the maximum common rate in downlink and uplink (denoted by RDL and RUL ), which is defined by RDL = max min(τ1 R1 , τ2 R2 ) and RUL = max min(τ2 R3 , τ1 R4 ) for asymmetric assignment, and RDL = τ1 max min(R1 , R2 ) and RUL = τ2 max min(R3 , R4 ) for symmetric assignment. Furthermore, we optimize for η = 2 in the case of asymmetric assignment. For the channel estimation error we assume Np = 4 pilots, which corresponds to 6 dB SNR-improvement on pilot signals. Finally, the SNR at the cell-edge is given by γDL = 10 dB in the downlink and γUL = 3 dB in the uplink. Both values result from a typical urban micro scenario as defined by IMT-Advanced [53] and used in 3GPP LTE for calibration. Specifically, we obtain these SNR values by applying non-line-of-sight condition, 46 dBm transmission power at the BS, 23 dBm transmission power at the UT, uplink power control as in 3GPP LTE, 1024 subcarriers, −174 dBm/Hz thermal noise density, and 15 kHz subcarrier spacing as it is applied in 3GPP LTE. B. Performance depending on distance

to be ignored. In this regime of weak interference, the symmetric assignment outperforms the asymmetric assignment, particularly at around dUT = 175 m where the performance advantage is about 20 %. In this case, the inter-cell interference for the symmetric assignment is significantly lower than the inter-UT interference in case of the asymmetric assignment. However, in this region, CoMP will not be applied anyway as the performance advantage over non-cooperative transmission is negligibly small (only 5 % in DL and 4 % in UL). At larger BS-UT-distances (towards the cell-edge), the inter-UT interference becomes stronger and the asymmetric assignment resembles a (very) strong IC [10]–[12]. In this region, the asymmetric assignment outperforms the symmetric assignment and provides about 25 % gain over non-cooperative symmetric transmission, and 10 % gain compared to CoMP. Furthermore, non-cooperative transmission in the asymmetric setup would provide the same benefits as cooperative transmission if the inter-BS interference was higher, e. g. ABS = 0 dB.

At first, we analyze the performance if the backhaulcapacity is limited by β = 4 and depending on the BS-UTdistance dUT . The results are shown in Fig. 5 and allow for the following general observations: • The symmetric assignment outperforms the asymmetric assignment by more than 20 % towards the cell-center, • Towards the cell-edge, the asymmetric assignment outperforms the symmetric assignment by about 10 % in the case of CoMP and 25 % in the case of non-cooperative transmission. At small and medium distances, the asymmetric assignment resembles a weak IC [5], [54] where the inter-UT interference is not strong enough to be decoded reliably and not weak enough

Fig. 6 shows a contour plot of the relative performance benefit of the asymmetric assignment over CoMP for different, asymmetric UT positions. The figure emphasizes that the gains increase as at least one of both UTs gets closer to the celledge. The benefits are highest if both UTs are closer to each other because in this case, a very strong interference channel is resembled. This result shows again that the asymmetric assignment is not suitable for all cases but only for those with high inter-cell interference which is usually also the area where CoMP is applied. In these typical CoMP scenarios, the asymmetric assignment is able to provide most of its benefits. In the remaining scenarios, it may even be preferable to apply no cooperation technique due to the low inter-cell interference.

2.5

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qp

Fig. 8. Performance depending on the sum-backhaul-capacity β and at distance dUT = 225 m

VI. A PPLICABILITY

bcrs

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become less if more backhaul resources are provided. This is illustrated in Fig. 8, which shows the performance for different values of β and at distance dUT = 225 m. It confirms that in comparison to the symmetric approaches, the asymmetric approach achieves its maximum performance faster, i. e., at β = 2. This emphasizes that particularly towards the celledge an asymmetric assignment is more backhaul-efficient compared to CoMP.

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Fig. 6. Rate-improvement in percent of asymmetric assignment over CoMP in DL for backhaul capacity β = 4 depending on the distance of both UTs, d1 and d2 , from their assigned BSs

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Performance depending on number of pilots at dUT = 225 m and

C. Performance depending on number of pilots Fig. 7 illustrates the performance depending on the number of pilots and therefore the quality of CSI. It shows that the performance benefit of the asymmetric assignment increases with the number of pilots due to the susceptibility of CoMP to imperfect CSI. For a smaller number of pilots, the asymmetric assignment suffers from stronger interference power caused by imperfect CSI. D. Performance depending on backhaul The previous part showed that an asymmetric assignment provides benefits over a symmetric assignment and CoMP towards the cell-edge. These benefits result among others from the limited backhaul capacity and the benefits may

In this paper, we discussed an asymmetric uplink-downlinkassignment approach based on a simple model involving two communication pairs. The interference due to different assignments of uplink and downlink in adjacent cells may result from a traffic adaptation in cells as considered currently by the 3GPP work item eIMTA [45]. In such a situation, smartphones with very different rate demands and channel conditions may require a re-distribution of uplink and downlink resources within a cell. Furthermore, UTs may be scheduled intentionally into those subframes with overlapping uplink and downlink in order to counteract strong inter-cell interference. In 3GPP LTE, CoMP is currently discussed as well. The preferred scenario for CoMP are BSs which are either physically co-located (at the same site) or which are connected through optical fiber to each other (remote radio heads). Both deployment scenarios may not always be implementable, particularly in small-cell deployments. By contrast, applying ICIC approaches which do not rely on a high-capacity backhaul will lower deployment requirements and allow for inter-cell interference mitigation. Fig. 9 gives a specific example how the partitioning in 3GPP LTE [55] may look and how an overlapping uplink-downlinkassignment may result. In the case of an asymmetric assign-

Cell 1:

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•

Backhaul resources are not only subject to bandwidthconstraints but also delay-constraints, which need to be considered by the design of cooperation approaches.

9

Fig. 9. Example for the different, overlapping regions with symmetric and asymmetric UL/DL assignment.

ment of UTs to overlapping uplink-downlink-subframes based on their rate demands (η) and channel condition, different challenges must be tackled which are listed in the following: • Multi-cell scenarios with more than two cells require more coordination efforts. Divide all BSs in cooperation sets of size L ≥ 2. The simplest way to apply the introduced technique is to consider only L = 2 which of course ignores the interference from other cells. Another solution is to have within each cluster at least one slot for each BS in which it is the only one receiving while all other L − 1 BSs are interfering (or vice versa). In this case, all transmitting BSs would then support the receiving BS in a similar way as explained before. Furthermore, the same asymmetric assignments may be applied to each cluster in order to have repeating pattern throughout the network. A dynamic solution may also be employed such that those BSs with with intercell interference receive asymmetric slots on demand. However, the more flexibility is offered and the higher L, we also have to cope with more complexity to find an optimal assignment. Therefore, a practical cluster size for heterogeneous deployments may be L = 3, e. g., to avoid interference between adjacent small-cells along a street canyon and to counteract inter-layer interference with the macro-BS which usually provides signal-coverage in those areas. • Frequency-selectivity may require to frequently change the assignment of users to individual subframes with symmetric or asymmetric assignment. Due to the possibly high latency on the backhaul, frequency-selectivity needs to be taken into account when scheduling UTs to either symmetric or asymmetric UL-DL-assignments. • Multi-user diversity offers additional degrees of freedom to avoid inter-UT interference, which requires new methods to schedule UTs based on their geographical properties. In particular, BSs may not be able to acquire CSI for the inter-UT link. However, large-scale fading may support the selection of UTs and their assignment to the individual subframes. • Inter-cell coordination is required to synchronize the UT scheduling of different, adjacent cells and to avoid strong inter-UT interference. Therefore, the required information, the granularity and the frequency of such inter-cell coordination needs to be determined. • Realistic models for the inter-UT and inter-BS interference need to be derived in order to evaluate the derived approach.

VII. C ONCLUSIONS This paper analyzed the performance of an asymmetric ULDL-assignment in TDD-systems over a symmetric assignment as it is applied in today’s networks. The asymmetric assignment allows for more flexibility as UTs may be scheduled to either symmetric or asymmetric assignments. In addition, the asymmetric assignment allows for a joint optimization of UL and DL rates. Depending on whether the UT is scheduled for symmetric or asymmetric assignment, the cooperation approach may be chosen and the interference channel with preferable properties can be used. Furthermore, an asymmetric assignment is less complex and requires less backhaulresources than a symmetric assignment. The asymmetric assignment is more robust and does not require an exchange of CSI over backhaul. This makes the asymmetric assignment of particular interest for heterogeneous deployments with macroand pico-cells where a high-capacity backhaul to pico-cells cannot be guaranteed and significant interference between macro- and pico-cells may be experienced. A PPENDIX A ACHIEVABLE RATES FOR THE DMC In order to facilitate the proof of Theorem 1, we first derive the achievable rates for the discrete memoryless channel. In n the following, we use X ∼ pX (x) to denote a random nlength sequence {X[t]}nt=1 whose elements X[t] ∈ X are i.i.d. distributed according to pX (x), i. e. pX (x) = Πnt=1 p(x[t]). The discrete memoryless channel as used in our paper is defined for phase 1 by p(y1 |u1 , u4 ) and for phase 2 by p(y2 |u2 , u3 ) over all possible channel inputs (u1 , u2 , u3 , u4 ) ∈ U1 ×U2 ×U3 ×U4 and channel outputs (y1 , y2 ) ∈ Y1 × Y2 × Y3 × Y4 . (n) In the following, we use (x, y) ∈ Aǫ (X, Y) to denote that an n-length tuple (x, y) ∈ X n ×Y n is ǫ-typical with respect to (n) a joint pdf p(x, y). We abbreviate Aǫ (X, Y) in the following (n) with Aǫ if it is clear from the context [56, Ch. 13.6]. Definition 1: A (2nR1 , 2nR2 , 2nR3 , 2nR4 , n, λn ) code for the described system model consists of the following •

• •

•

A set of equally probable indices Zk = [1; 2nRk ] with k ∈ [1; 4] and the corresponding random variable Zk over Zk , (n) A set of encoding functions fk : [1; 2nRk ] → Uk , (n) nRk′ ] with A set of decoding functions gk : Yk → [1; 2 k ′ being the transmitter assigned to node k, The maximum probability of error λn =

max

k,zk′ ∈Zk′

Pr {g(yk ) 6= zk′ |Zk′ = zk′ } .

(26)

Definition 2: A rate tuple (R1 , R2 , R3 , R4 ) is said to be achievable if a sequence of (2nR1 , 2nR2 , 2nR3 , 2nR4 , n, λn ) codes exists such that λn → 0 as n → ∞ [56, Ch. 8.5].

Theorem 2: In the case of the asymmetric UL/DL assignment and the protocol structure as described in Section III, any achievable rate tuple (R1 , R2 , R3 , R4 ) ∈ Rasym must satisfy β R1 ≤ I (U1 ; Y3 |V1 , W1 , V4 )

+ min I (V1 ; Y3 |W1 , V4 ) ,
β − β2′ + I(V1 ; Y2 |W1 )

R4 ≤ I (U4 ; Y2 |V4 , V1 , W1 ) + min (R4′ , I (V4 , W1 ; Y3 )) R4′ ≤ I (V4 ; Y2 |V1 , W1 )

I (W1 ; Y3 ) ≤ min (β1′ , R4′ )

(27) (28) (29) (30)

and R2 ≤ I (U2 ; Y4 |V2 , W2 , V3 )

+ min I (V2 ; Y4 |W2 , V3 ) ,
β − β1′ + I(V2 ; Y1 |W2 ) R3 ≤ I (U3 ; Y1 |V3 , V2 , W2 )

+ min (R3′ , I (V3 , W2 ; Y4 )) ≤ I (V3 ; Y1 |V2 , W2 ) I (W2 ; Y4 ) ≤ min (β2′ , R3′ ) . R3′

(31) (32) (33) (34)

The set of all achievable rate pairs is given by the convex hull [ Rasym = (R1 , R2 , R3 , R4 ) (35) β ∈Pβasym pasym β

over all PDFs pasym ∈ Pβasym of the form β pasym = p(y1 |u1 , u4 )p(u1 |v1 , w1 )p(v1 , w1 ) · p(u4 |v4 )p(v4 ) β

· p(y2 |u2 , u3 )p(u2 |v2 , w2 )p(v2 , w2 ) · p(u3 |v3 )p(v3 ). (36)

Proof: The following proof is performed for phase 1 of the protocol which involves nodes 1 and 4 as transmitters and nodes 2 and 3 as receivers. The proof for the second phase is equivalent. a) Messages: Node 1 uses the following messages: n • Common message V1 ∼ p(v1 ), n • Support message W1 ∼ p(w1 ) for node 4, and n • Private message U1 ∼ p(u1 |v1 , w1 ). Furthermore, node 1 groups randomly all messages V1 to ′ 2n(β−β2 ) buckets. Node 4 uses the following messages n • Common message V4 ∼ p(v4 ), and n • Private message U4 ∼ p(u4 |v4 ). Furthermore, all messages V4 are grouped randomly and ′ assigned to 2nβ1 buckets. b) Encoding: The encoding at the individual nodes is done as detailed in the following:: • Node 1 divides its message Z1 into two parts which are mapped to (U1 , V1 ). • Node 1 forwards the bucket index, to which message V3 (b) is assigned to, to node 2 over the error-free backhaul (with rate β2′ ).

Node 1 further uses the message W1 which is assigned to the bucket index received from node 2 in block b − 1 (with rate β1′ ). • Node 1 forwards the bucket index of message V1 to node 1 with rate β − β2′ . • Node 4 (transmitting UT) maps its message Z4 to a tuple (U4 , V4 ). c) Decoding: At node 3 (receiving UT) in block b: • First, it decodes the common message from node 4 submitted in block b − 1, V4 , which is supported by message W1 in block b. This is done by searching for exactly one (n) jointly typical sequence (w1 (b), y3 (b)) ∈ Aǫ and then (n) for a sequence (v4 (b − 1), y3 (b − 1)) ∈ Aǫ which is in the bucket identified by the decoded sequence w1 (b). This is successful if and only if R(W1 ) ≤ I (W1 ; Y3 ) and R(V4 ) ≤ I (V4 , W1 ; Y3 ). • After the interference from the other UT has been decoded, the message from its assigned BS is decoded, i. e. (U1 (b − 1), V1 (b − 1)) by searching for a jointly typical sequence (u1 (b−1), v1 (b−1), y3 (b−1), v4 (b−1), w1 (b− (n) 1)) ∈ Aǫ . This is successful if and only if R(U1 ) ≤ I (U1 ; Y3 |V1 , W1 , V4 ) and R(V1 ) ≤ I (V1 ; Y3 |W1 , V4 ). At node 2 (receiving BS) in block b: • It first detects and removes message V1 (b) by searching (n) for the set (v1 (b), w1 (b), y3 (b)) ∈ Aǫ and then by searching for a unique sequence that is also in the bucket identified by v ˆ1 . This is possible if and only if R(V1 ) ≤ β − β2′ + I(V1 ; Y2 |W1 ). • Finally, node 2 decodes the message sent by node 4 which is done by searching for a jointly typical sequence (n) (u4 (b), v4 (b), w1 (b), v1 (b), y2 (b)) ∈ Aǫ . This is successful if and only if R(U4 ) ≤ I (U4 ; Y2 |V4 , V1 , W1 ) and R(V4 ) ≤ I (V4 ; Y2 |V1 , W1 ). d) Bringing all together: • Let R1 = RU1 + RV1 . From the above description we know that •

RU1 ≤ I (U1 ; Y3 |V1 , W1 , V4 ) , RV1 ≤ I (V1 ; Y3 |W1 , V4 ) .

•

Furthermore, message V1 is decoded by the second BS, therefore RV1 ≤ β − β2′ + I(V1 ; Y2 |W1 ). Combining the above expressions gives (27). The message rate of node 4 is given by R4 = RU4 +RV4 . From the description above we know that RU4 ≤ I (U4 ; Y2 |V4 , V1 , W1 ) , RV4 ≤ I (V4 ; Y2 |V1 , W1 ) .

•

(37) (38)

(39) (40)

Message V4 is further decoded by node 3 and therefore RV4 ≤ I (V4 , W1 ; Y3 ). Both combined give (28) and (29). Finally, the constraint in (30) is explained above (β1′ ) and results from the fact that the support message rate cannot exceed the message rate of the message that is supported.

A PPENDIX B P ROOF FOR T HEOREM 1 In this Section, we provide a proof for Theorem 1 based on the system model presented in Section II and the achievable rates derived in the previous section for the DMC. Proof: First, we consider the following messages n • U1 ∼ CN (0, αU1 P1 ), n • V1 ∼ CN (0, αV1 P1 ), n • W1 ∼ CN (0, αW1 P1 ), n • U4 ∼ CN (0, αU4 P4 ), and n • V4 ∼ CN (0, αV4 P4 ). The factors α· ≥ 0 must fulfill αU1 + αV1 + αW1 ≤ 1 and αU4 + αV4 ≤ 1 to preserve the power constraint at each transmitter. Again, we consider here only the first phase as the second phase is symmetric to the first phase. First, consider eq. (27): R1 ≤ I (U1 ; Y3 |V1 , W1 , V4 ) + min I (V1 ; Y3 |W1 , V4 ) ,
β − β2′ + I(V1 ; Y2 |W1 )

with the first term given by

I (U1 ; Y3 |V1 , W1 , V4 ) = H(Y3 |V1 , W1 , V4 ) − H(Y3 |U1 , V1 , W1 , V4 ).

(41)

The second term on the right hand side determines the noise plus interference part, which is divided into the Gaussian noise Rnn , the self-interference part caused by imperfect CSI given by R3vv (which follows from its definition in Section II). Furthermore, interference from the transmission of node 4 must be considered (caused by U4 ), i.e. He3 αU4 P4 (He3 )H . The first term on the right hand side of (41) determines the interference part plus the useful signal, which is given by He3 αU1 P1 (He3 )H . After applying the known entropy equation for multi-dimensional Gaussian signals [56, eq. (9.34)] we obtain (12). Now consider the first part of (13). In this case, message V1 is decoded by node 3 after the interference from node 4 caused by V4 has been canceled. Hence, the received power of message V1 is given by He2 αV1 P1 (He2 )H , interfered by R3vv (imperfect CSI), and the interference caused by messages U1 and U4 , i. e., He3 (αU1 P1 + αU4 P4 )(He3 )H . Finally, consider the second part of (13). In this case, node 2 decodes message V1 of received power He2 αV1 P1 (He2 )H . Message W1 is known. In addition, the transmission is interfered by node 4 with interference power He2 P4 (He2 )H . The term He2 αU1 P1 (He2 )H accounts for the private message from node 1 which is not decoded by node 2. Furthermore, an additional term R2vv interferes the transmission due to imperfect CSI. Now consider eq. (28): R4 ≤ I (U4 ; Y2 |V4 , V1 , W1 ) + min (R4′ , I (V4 , W1 ; Y3 )) . Similar to the previous description, we can again split the individual terms into individual entropy expressions. For the first term in (28), node 2 decodes the private message from

its UT, node 4. Since it already decoded part of the interference caused by node 1, the interference is given by one part caused by imperfect CSI, R2vv , and one part caused by node 1, He2 αU1 P1 (He2 )H . The useful part is given by He2 αU4 P4 (He2 )H . The second term in (28) refers to the common message part of node 4 decoded by node 3. In this case, the interference is the sum of the interference caused by imperfect CSI, R3vv , and the interference caused by nodes 1 and 4, He3 (αU4 P4 + αU1 +αV1 P1 )He3 . The useful signal is given by He3 (αW1 P1 + αV4 P4 )(He3 )H . Next, consider eq. (29) R4′ ≤ I (V4 ; Y2 |V1 , W1 ) . In this case, the interference power is again given by R2vv (imperfect CSI) and the interference caused by nodes 1 and 4, He2 (αU4 P4 + αU1 P1 )He2 . The useful signal part is given by message V4 , i. e., He2 αV4 P4 (He2 )H . Combining these terms gives (14)-(16). Finally, consider term I (W1 ; Y3 ) where the interference power is the sum of R3vv (imperfect CSI) and the interference caused by nodes 1 and 4, i. e., He3 (P4 + (αU1 + αV1 )P1 )He3 . The useful signal part is given by message W1 , i. e., He3 αW1 P1 (He3 )H , which gives (17) based on (30). R EFERENCES [1] IEEE Computer Society, “IEEE standard for local and metropolitan area networks - part 16: Air interface for broadband wireless access systems,” IEEE, Tech. Rep., May 2011. [2] 3GPP, “Work Item: Coordinated Multi-Point Operation for LTE,” 3GPP, Tech. Rep., March 2012. [3] P. Rost, “The two-way interference channel: Towards a redesign of mobile communication systems,” in IEEE Vehicular Technology Conference (VTC), San Francisco (CA), USA, September 2011. [4] 3GPP, “Evolved Universal Terrestrial Radio Access (E-UTRA); Mobility enhancements in heterogeneous networks,” 3GPP, Tech. Rep., March 2012. [5] C. Shannon, “Two-way communication channels,” in 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, University of California Press, 1961, pp. 611–644. [6] A. Dash and A. Sabharwal, “An outer bound for a multiuser two-way channel,” in Annual Allerton Conference on Communication, Control and Computing, Allerton (IL), USA, September 2006. [7] B. Rankov and A. Wittneben, “Spectral efficient protocols for halfduplex fading relay channels,” IEEE Journal on Selected Areas in Communications, vol. 25, no. 2, pp. 379–389, February 2007. [8] G. Kramer and S. Savari, “Cut sets and information flow in networks of two-way channels,” in International Symposium on Information Theory, Chicago (IL), USA, June 2004. [9] A. Carleial, “Interference channels,” IEEE Transactions on Information Theory, vol. IT-24, pp. 60–70, January 1978. [10] ——, “A case where interference does not reduce capacity,” IEEE Transactions on Information Theory, vol. IT-21, no. 5, p. 569, September 1975. [11] H. Sato, “The capacity of the Gaussian interference channel under strong interference,” IEEE Transactions on Information Theory, vol. IT-27, no. 6, pp. 786–788, November 1981. [12] T. Han and K. Kobayashi, “A new achievable rate region for the interference channel,” IEEE Transactions on Information Theory, vol. IT-27, no. 1, pp. 49–60, January 1981. [13] I. Maric, R. Yates, and G. Kramer, “Capacity of interference channels with partial transmitter cooperation,” IEEE Transactions on Information Theory, vol. 53, no. 10, pp. 3536–3548, October 2007.

[14] V. Cadambe, S. Jafar, and S. Shamai, “Interference alignment on the deterministic channel and application to fully connected AWGN interference networks,” in IEEE Information Theory Workshop, Porto, Portugal, May 2008, pp. 41–45. [15] J. Zander, “Distributed cochannel interference control in cellular radio systems,” IEEE Transactions on Vehicular Technology, vol. 41, no. 3, pp. 305 – 311, August 1992. [16] D. Lopez-Perez, I. Guvenc, G. de la Roche, M. Kountouris, T. Quek, and J. Zhanga, “Enhanced intercell interference coordination challenges in heterogeneous networks,” IEEE Wireless Communications Magazine, vol. 18, no. 3, pp. 22–30, June 2011. [17] C. You, G. Yoon, C. Seo, S. Portugal, G. Park, T. Jung, H. Liu, and I. Hwang, “Intercell interference coordination using thresholdbased region decisions,” Journal on Wireless Personal Communications, vol. 59, no. 4, pp. 789–806, August 2011. [18] S. Rangan and R. Madan, “Belief propagation methods for intercell interference coordination in femtocell networks,” IEEE Journal on Selected Areas in Communications, vol. 30, no. 3, pp. 631–640, April 2012. [19] S. Akoum and R. Heath, “Interference coordination: Random clustering and adaptive limited feedback,” October 2012, arXiv:1210.6095. [20] L. Liu, J. Zhang, J.-C. Yu, and J. Lee, “Intercell interference coordination through limited feedback,” International Journal of Digital Multimedia Broadcasting, 2010. [21] W. Yu, T. Kwon, and C. Shin, “Multicell coordination via joint scheduling, beamforming and power spectrum adaptation,” in IEEE International Conference on Computer Communications, Shanghai, China, April 2011. [22] A. Sahai, V. Aggarwal, M. Y¨uksel, and A. Sabharwal, “On channel output feedback in deterministic interference channels,” in IEEE Information Theory Workshop, Taormina, Italy, October 2009. [23] Z. Cheng and N. Devroye, “Two-way networks: when adaptation is useless,” June 2012, arXiv:1206.6145. [24] C. Suh, I. Wang, and D. Tse, “Two-way interference channels,” February 2012, arXiv:1202.5014v1. [25] P. Marsch and G. Fettweis, “Uplink CoMP under a constrained backhaul and imperfect channel knowledge,” IEEE Transactions on Wireless Communications, vol. 10, no. 6, pp. 1730 – 1742, June 2011. [26] ——, “On downlink network MIMO under a constrained backhaul and imperfect channel knowledge,” in IEEE Global Communications Conference, Hawaii, USA, December 2009. [27] D. Slepian and J. Wolf, “Noiseless coding of correlated information sources,” IEEE Transactions on Information Theory, vol. IT-19, no. 4, pp. 471–480, July 1973. [28] M. Medard, “The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel,” IEEE Transactions on Information Theory, vol. 46, no. 3, pp. 933–946, May 2000. [29] T. Yoo and A. Goldmsith, “Capacity and power allocation for fading MIMO channels with channel estimation error,” IEEE Transactions on Information Theory, vol. 52, no. 5, pp. 2203–2214, May 2006. [30] S. Shamai and B. Zaidel, “Enhancing the cellular downlink capacity via co-processing at the transmitting end,” in IEEE Vehicular Technology Conference (VTC), vol. 3, Rhodes, Greece, May 2001, pp. 1745–1749. [31] A. Sklavos, T. Weber, E. Costa, H. Haas, and E. Schulz, Multi-Carrier Spread Spectrum and Related Topics. Kluwer Academic Publishers, September 2002, ch. Joint Detection in Multi-Antenna and Multi-User OFDM systems, pp. 191–198. [32] S. Shamai, O. Somekh, and B. Zaidel, “Multi-cell communications: An information-theoretic perspective,” in Joint Workshop on Communication and Coding, Florence, Italy, October 2004. [33] A. del Coso and S. Simoens, “Distributed compression for the uplink channel of a coordinated cellular network with a backhaul constraint,” in IEEE Workshop on Signal Processing Advances in Wireless Communications, Recife, Brazil, July 2008. [34] S. Shamai, O. Simeone, O. Somekh, and V. Poor, “Joint multi-cell processing for downlink channels with limited-capacity backhaul,” in Information Theory and Applications Workshop, San Diego (CA), USA, January 2008. [35] R. Irmer, H. Droste, P. Marsch, M. Grieger, G. Fettweis, S. Brueck, H.-P. Mayer, L. Thiele, and V. Jungnickel, “Coordinated multipoint: Concepts, performance, and field trial results,” IEEE Communications Magazine, vol. 49, no. 2, pp. 102 – 111, February 2011.

[36] X. Tao, X. Xu, and Q. Cui, “An overview of cooperative communications,” IEEE Communications Magazine, vol. 50, no. 6, pp. 65–71, June 2012. [37] D. Gesbert, S. Hanly, H. Huang, S. Shamai-Shitz, O. Simeone, and W. Yu, “Multi-cell mimo cooperative networks: A new look at interference,” IEEE Journal on Selected Areas in Communications, vol. 28, no. 9, pp. 1380–1408, December 2010. [38] A. Ozgur, O. Leveque, and D. Tse, “Hierarchical cooperation achieves optimal capacity scaling in ad hoc networks,” IEEE Transactions on Information Theory, vol. 53, no. 10, pp. 3549–3572, October 2007. [39] A. Ozg”u, R. Johari, D. Tse, and O. Leveque, “Information-theoretic operating regimes of large wireless networks,” IEEE Transactions on Information Theory, vol. 56, no. 1, pp. 427–437, January 2010. [40] L.-C. Wang and Y.-C. Chen, “A novel link proportional dynamic channel assignment for TDD-CDMA systems with directional antennas,” in IEEE International Conference on Networking, Sensing and Control, vol. 1, Taipei, Taiwan, March 2004, pp. 164–169. [41] K. Mori, K. Naito, H. Kobayashi, and H. Aghvami, “Asymmetric traffic accommodation using adaptive cell sizing technique for CDMA/FDD cellular packet communications,” IEICE Transactions on Fundamentals, vol. E90, no. 7, pp. 1271 – 1279, July 2007. [42] H. Lee and D.-H. Cho, “Combination of dynamic-TDD and staticTDD based on adaptive power control,” in IEEE Vehicular Technology Conference (VTC), vol. 1, Calgary, Canada, October 2008, pp. 1–5. [43] Y. Wang, S. Frattasi, T. Sorensen, and P. Mogensen, “Network timesynchronization in TDD based LTE-advanced systems,” in IEEE Vehicular Technology Conference (VTC), vol. 1, Barcelona, Spain, June 2009, pp. 1–5. [44] H. Falaki, D. Lymberopoulos, R. Mahajan, S. Kandula, and D. Estrin, “A first look at traffic on smartphones,” in ACM Internet Measurement Conference, Melbourne, Australia, 2010. [45] 3GPP, “Evolved Universal Terrestrial Radio Access (E-UTRA); Further enhancements to LTE Time Division Duplex (TDD) for DownlinkUplink (DL-UL) interference management and traffic adaptation,” 3GPP, Tech. Rep., June 2012. [46] G. Primolevo, O. Simeone, and U. Spagnolini, “Effects of imperfect channel state information on the capacity of broadcast OSDMA-MIMO systems,” in IEEE Workshop on Signal Processing Advances in Wireless Communications, Lisbon, Portugal, July 2004, pp. 546–550. [47] T. Yoo and A. Goldsmith, “Capacity and power allocation for fading MIMO channels with channel estimation error,” IEEE Transactions on Information Theory, vol. 52, no. 5, May 2006. [48] S. Kay, Fudamentals of statistical signal processing: Estimation theory, 1st ed. Prentice-Hall, 1993. [49] B. Hassibi and B. Hochwald, “How much training is needed in multipleantenna wireless links,” IEEE Transactions on Information Theory, vol. 49, no. 4, April 2003. [50] T. Cover and A. E. Gamal, “Capacity theorems for the relay channel,” IEEE Transactions on Information Theory, vol. 25, no. 5, pp. 572–584, September 1979. [51] T. Cover, “Broadcast channels,” IEEE Transactions on Information Theory, vol. 18, no. 1, pp. 2–14, January 1972. [52] P. Bergmans, “Random coding theorem for broadcast channels with degraded components,” IEEE Transactions on Information Theory, vol. 19, no. 2, pp. 197–207, March 1973. [53] ITU-R, “Report ITU-R M.2135-1, Guidelines for evaluation of radio interface technologies for IMT-Advanced,” ITU, Tech. Rep., 2009. [54] H. Sato, “Two-user communication channels,” IEEE Transactions on Information Theory, vol. IT-23, no. 3, pp. 295–304, May 1977. [55] 3GPP, “Evolved Universal Terrestrial Radio Access (E-UTRA); Further advancements for E-UTRA physical layer aspects (Release 9),” 3GPP, Tech. Rep., Mar. 2010. [56] T. Cover and J. Thomas, Elements of Information Theory. John Wiley & Sons, Inc., 1991.