Alternate relaying and the Degrees of Freedom of One-Way Cellular Relay Networks Aya Salah, Amr El-Keyi, and Mohammed Nafie Wireless Intelligent Networks Center (WINC) Nile University, Cairo, Egypt 12677 {
[email protected],{aelkeyi, mnafie}@nileuniversity.edu.eg}
Abstract—In this paper, a cellular relaying network consisting of two source-destination pairs, and four decode-and-forward relays operating in half-duplex mode is considered. Each source is assisted by two relays and all nodes are equipped with N antennas. In order to compensate for the loss of capacity by a factor of half due to the half-duplex mode, an alternate transmission protocol among the two relays is proposed. An outer bound on the degrees of freedom (DoF) of this system is developed. A constructive proof of achievability based on two different schemes is provided. Aligning the inter-relay interference due to alternate transmission and the interference received at the destination nodes is utilized to achieve the maximum DoF and recover the pre-log factor loss. For the K-user case, it is shown that the resulting system of interference alignment equations is proper when the number of streams transmitted by each source 3N . Numerical simulations show that is less than or equal to 2(K+1) for proper systems these DoF are achievable. Index Terms—Relaying networks, decode-and-forward relays, half-duplex relays, degrees of freedom, interference alignment.
I. I NTRODUCTION Relaying systems can be used for cellular networks to enhance throughput and extend coverage to cell-edge users [1]. In addition, they can be used for interference-limited cellular systems to manage interference among multiple cells. The classical relay channel model can be traced back to the groundbreaking work of Cover and El Gamal [2]. This work analyzed the capacity of a three-node network consisting of one source, one destination, and a single relay. Capacity results for MIMO relay channels with a single relay can also be found in [3]. In this work, our focus is on the spatial degrees of freedom (DoF) of cellular relay networks. The DoF are one of the simple ways to characterize the network capacity. They represent the rate of growth of the sum rate with the log of the signal to noise ratio. In most cases, the spatial DoF can be defined to be the number of non-interfering paths created in the wireless channel. Relays typically operate in half duplex mode, i.e., the relay cannot transmit and receive at the same time. This results in a loss of capacity by a factor of half. One of the recent approaches to compensate for the half-duplex loss factor is the use of alternate relaying. In this relaying protocol, the source alternately transmits the signals to the destination via different relays. However, this leads to inter-relay interference between relays in transmit and receive modes causing degradation in the This paper was made possible by NPRP grant #4-1119-2-427 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors.
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DoF. Previous work for two-path relaying in [4] focused on a single-antenna environment and has been extended to multipleantenna scenarios in [5] and [6]. In addition [7] proposed an alternate relaying protocol with one source, one destination, and three amplify and forward (AF) relays equipped with N antennas. Using this protocol, the loss of DoF was compensated by aligning the inter-relay interference from different DoF, while the nodes. This proposed scheme can achieve 3N 4 conventional AF relaying schemes provide N2 DoF. In this paper, we consider the downlink of a cellular oneway relaying (OWR) system. We start with a two-cell setting which is composed of two sources (transmitting base stations (BSs)) where each source is transmitting to a single destination (receiving mobile stations (MSs)). Each cell has two decodeand-forward relays associated with it. Each node is equipped with N antennas that are used for transmitting and/or receiving multiple data streams. We assume that there is no direct link between the source and destination terminals, and hence, the communication occurs through the relays. We also assume that the relays are located near the cell edge, and hence, the relays of one cell cause interference to the MSs of the other cell. A transmission protocol is proposed where each source alternately transmits to one of the two relays associated with it in each time slot. At the same time, the other relay transmits the re-encoded message obtained from the source in the previous time slot to the destination. The relays and destination nodes employ linear processing matrices to align the interference. By this protocol, we not only extend the cell coverage but also achieve a total DoF equal to N . We also show that this is the maximum achievable DoF for the two user setting. by developing an outer bound on the DoF of this setting. The outer bound is used to prove a converse result for the maximum number of DoF. We provide a constructive proof of achievability based on two different schemes. We also extend our results to the K-cell case, where K MSs seek communication with K BSs and each source-destination pair is supported by two relays. Each node in the network has N antennas. We show that the resulting interference alignment 3N or less equations are proper if each terminal transmits 2(K+1) streams. This can be shown by the comparison of the number of variables and constraints in the system of IA equations. We validate the number of achieved DoF numerically for proper systems using a distributed iterative algorithm that minimizes the interference leakage as in [8].
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Fig. 1: Equivalent system when the relays transmit and receive simultaneously.
Fig. 2: Equivalent system after cooperation in the alternate relaying case.
II. SYSTEM MODEL
dedicated for each BS. As a result, each BS has one dedicated half duplex RS with 2N antenna. The resulting system consists of two sources, two relays and two destination nodes where each source and destination have N antennas and each relay has 2N antennas as shown in Fig. 1(b). Communication in the resulting network occurs in two hops; the first hop consists of two parallel (non-interfering) point-to-point channels (from each BS to the its assigned combined RS) while the second hop is a two-user MIMO interference channel. Note that allowing the nodes to cooperate cannot reduce the DoF, and hence, the DoF of the original system are less than or equal to the DoF of the system in Fig. 1(b). Therefore, the DoF of the 2-user MIMO-OWR network is 1 η ≤ min{ηh1 , ηh2 } (1) 2 where ηh1 and ηh2 are respectively the DoF of the first and second hops and the factor half is due to the two hop transmission. Since the DoF of a point-to-point link with K transmit antennas and L receive antennas is given by min{K, L} [9], we have ηh1 = 2N . Also, the DoF of a 2-user interference channel with K antennas at each transmitter and L antennas at each receiver is given by min{2K, 2L, max{K, L}} [9], we have ηh2 = 2N . Therefore, using (1) we have the following upperbound on the DoF of the 2-user MIMO-OWR network
We consider the downlink of a two-user cellular MIMOOWR network. In this network, there is two N -antenna BSs and two N -antenna MSs where each BS has d data streams to deliver to its intended MS. There is no direct link between the BSs and the MSs and all communication occurs through the relays. Each BS is assisted by two decode-and-forward relay stations (RSs) equipped with N -antenna each. All network nodes operate in half-duplex mode. The relays are placed near the cell edge, i.e., in close proximity of the MSs, and hence, the two MSs can hear the transmission of the four RSs. The relays are also placed such that they have line-of-sight communication with their serving BS. Furthermore, each RS is equipped with directional antennas, and hence, it can only listen to the transmission of its serving BS. For the given system, we pose the following question. How can we efficiently use the relays to maximize the DoF of the system, i.e., transmit the maximum number of non-interfering streams from each BS to its assigned MS? Since the RSs are half-duplex, we have only two possible choices for operating the RSs. In the first choice, each BS transmits to both of its assigned RSs in the first time slot. The RSs then decode the message of the BS and retransmit it to the MSs in the second time slot. The second choice is to use alternate relaying where each BS sends a message to one of its relays which forwards it to the MS in the next time slot. The BS alternately transmits to one RS while the second is transmitting to the MS. III. U PPERBOUND ON THE D O F OF THE 2- USER SYSTEM In this section, we show that the maximum number of DoF of the 2-user MIMO-OWR system is less than or equal to N DoF if we use any of the two possible relay layouts discussed in the previous section. A. Case 1: The two RSs receive and transmit simultaneously Let us assume that the two relays associated with each BS simultaneously receive the transmission of the BS in one time slot and then simultaneously transmit to the MSs, as shown in Fig. 1(a). In order to obtain an upper bound on the DoF of the system, let us allow full cooperation among the two RSs
η≤N
(2)
B. Case 2: The two RSs alternately receive and transmit Let us assume that the relays perform alternate relaying. In order to obtain an upperbound on the DoF, we allow full cooperation among the two BSs to form a single 2N -antenna node. Similarly, the two transmitting RSs (one associated with each BS), the two receiving RSs, and the two MSs are also allowed to perfectly cooperate so that they form three 2N antenna nodes. The resulting system is shown in Fig. 2 where we have a Z-channel with two sources (the cooperating BSs and the cooperating transmitting RSs) and two destination nodes (the cooperating MSs and the cooperating receiving RSs) and each node is equipped with 2N antennas.
Let di,j denote the number of non-interfering streams transmitted from terminal i to terminal j in Fig. 2. Since cooperation cannot decrease the DoF, the DoF of the resulting Z-network are an upper bound of the DoF of the original system. The DoF of the 2-user cellular MIMO-OWR network is upperbounded by η≤
max
d1,1 ,d2,2 ∈Sz
min{d1,1 , d2,2 }
(3)
where Sz is the DoF region of the resulting Z-channel shown in Fig. 2. Outer bounds on the DoF for the MIMO Z-channel can be obtained from [10]. The following lemma summarizes the results on the DoF of the MIMO Z-channel. Lemma 1: The total number of achievable DoF region of the MIMO Z-channel, i.e., a system with two transmitters, two receivers, each equipped with multiple antennas, where one transmitter (with T2 antennas) has an independent messages to each receiver while the other transmitter has only one message to one of the receivers (with S1 antennas), can be defined as ηz∗ = d1,1 + d2,1 + d2,2 where ηz∗ ≤ max(S1 , T2 ). Proof: The proof of Lemma 1 can be found in [10]. Applying Lemma 1 on the Z-channel in Fig. 2 where S1 = T2 = 2N , we have ηz∗ ≤ 2N . In order to obtain a bound on the total number of DoF of the 2-user cellular MIMO-OWR network, we set d2,1 = 0, i.e., there is no intended message from the transmitting RSs to the receiving ones. In addition, we can see that setting d1,1 = d2,2 yields the maximum of (3). Therefore, we have the same bound in (2) on the DoF for the 2-user cellular MIMO-OWR network, i.e., η ≤ N , which is obtained by setting d1,1 = d2,2 = N and d2,1 = 0. IV. ACHIEVABLE SCHEME A. Interference neutralization scheme The key to this scheme is providing a way to cancel interference terms over the air at the second hop in the first layout shown in Fig. 1(a). In this scheme, each base station sends N symbols simultaneously to its two serving relays in the first slot. Each relay then decodes its N symbols. The received signal at RSs is given by yRi=HRiBj xBj+zRi
(4)
where j = 1 when i = 1, 2 and j = 2 when i = 3, 4, the matrix H ij ∈ CN ×N is the MIMO channel matrix from terminal j to terminal i, and the vectors xj ∈ CN and z i ∈ CN are respectively the transmitted signal from terminal j and the complex zero-mean white Gaussian noise vector received at terminal j. At the ith relay, the received ˆ Ri . In the second time slot, the signal is decoded to yield x decoded signal is encoded using the transmit precoding matrix W Ri ∈ CN ×N , and then transmitted. The transmitted signal from the ith relay is given by ˆRi xRi = WRi x
Fig. 3: System model for even time slots. where the vector z M1 ∈ CN is the complex Gaussian noise 2 vector M1 and is distributed as CN (0, σM I ). Since R3 1 N ˆ R3 = x ˆ R4 , they can and R4 have the same symbols, i.e., x neutralize these N symbols at M1 . We focus on perfectly cancelling the interference terms from R3 and R4 over the air at M1 by exploiting the precoding matrices at the relays R3 and R4 , {W Rk }k=3,4 . In order to do that, we have H M1 R 3 W R 3 = − H M1 R4 W R4
The N columns of the precoding matrix W R4 can be picked randomly according to isotropic distribution so that they are linearly independent with probability one. Then, we determine W R3 from (7). Similarly, The precoding matrices at the relays R1 and R2 , {W Rk }k=1,2 can be designed to neutralize their symbols at M2 . So, each cell achieves N DoF over two slots. Thus, the total DoF value is given by N per slot. B. Alternate transmission scheme In the previous section, we have shown that the number of DoF of the 2-user cellular MIMO-OWR network is less than or equal to N . In this section, we present a alternate relaying scheme for achieving these DoF. In this scheme, communication between the BSs and MSs occurs in two phases, where the duration of each phase is one time slot. For the even time slots, B1 and B2 transmit d data streams to R1 and R3 , respectively, while R2 and R4 forward the received signal in the previous time slot to M1 and M2 , respectively. In the odd time slots, B1 and B2 send d data streams to R2 and R4 , respectively, while R1 and R3 forward the received signal of the previous time slot to M1 and M2 , respectively. Fig. 3 shows the system model for the even time slots. Due to the symmetry of the alternate transmission scheme, we will consider the even time slots only. The received signal at R1 and R3 at the mth time slot is given respectively by HR1Ri xRi(m)+zR1(m) (8) yR1(m)=HR1B1 xB1(m)+ i=2,4
(5)
yR3(m)=HR3B2 xB2(m)+
Let us consider the received signals at M1 which is given by yM1=
4
i=1
HM1Ri xRi+zM1
(6)
(7)
HR3Ri xRi(m)+zR3(m) (9)
i=2,4
where m = 2, 4, 6, . . ., the matrix H ij ∈ CN ×N is the MIMO channel matrix from terminal j to terminal i and is modelled
as independent block fading, and the vectors xj (m) ∈ CN and z i (m) ∈ CN are respectively the transmitted signal from terminal j and the complex zero-mean white Gaussian noise vector received at terminal j at the mth time slot. At the ith relay, the received signal at time slot m is decoded by an interference suppression matrix U Ri ∈ CN ×d , and then ˆ Ri (m) = U H decoded to yield x Ri y Ri (m). The decoded signal is encoded using the transmit precoding matrix W Ri ∈ CN ×d , and then transmitted. The transmitted signal from the ith relay at time slot m is given by ˆRi (m − 1) xRi (m) = WRi x
yMi(m)=HMiR2 xR2(m)+HMiR4 xR4(m) +zMi(m) (11) where m = 2, 4, 6, . . ., the vector z Mi (m) ∈ CN is the complex Gaussian noise vector at the ith MS at the mth 2 I ). At each MS, time slot and is distributed as CN (0, σM i N the received signal is linearly processed by the interference suppression matrix U Mi ∈ CN ×d to yield the output signal ˆ Mi (m) = U H x Mi y Mi (m). In (11), the first term of the received signal at the mobile station, HMiR2 xR2(m), represents the desired signal for M1 and an interference for M2 , while the second term, HMiR4 xR4(m), represents the desired signals for M2 and an interference for M1 as shown in Fig. 3. Note that the matrix U Mi is different for the odd and even time slots as R1 and R3 are transmitting in odd time slots while R2 and R4 are transmitting in even time slots. Theorem 1: For the two-user cellular MIMO-OWR setting shown in Fig. 3, the total system DoF using alternate relay transmission is given by N . Proof: We have shown in the previous section that the DoF of the two-user cellular MIMO-OWR network is less than or equal to N . Next, we will show that the proposed alternate relaying scheme achieves N DoF where each BS-MS pair can have a maximum of d = N2 DoF. Let us consider the even time slots as shown in Fig. 3. In this case, the relays R1 and R3 are receiving while R2 and R4 are transmitting. Note that the desired signal for the relays R1 and R3 are those transmitted by their serving BSs while the transmission of the relays R2 and R4 is interference. Also, the relay R2 causes interference at M2 while the relay R4 causes interference at M1 . In order to achieve N DoF, we set the number of streams transmitted by each terminal to N/2, i.e., d = N/2. We focus on perfectly cancelling the interrelay interference signal and the interference at the MSs by exploiting the decoding matrices at the relays R1 and R3 , {U H Rk }k=1,3 , the decoding matrices at the mobile stations, {U Mk }k=1,2 and the precoding matrices at the relays R2 and R4 , {W Rk }k=2,4 . In order to cancel the inter-relay interference at R1 and R3 in the even time slots, it is required that UH Rk H Rk R2 W R2 = 0
k = 1, 3
(12)
UH Rk H Rk R4 W R4
k = 1, 3.
(13)
UH M1 H M1 R 4 W R 4 = 0
(14)
UH M2 H M2 R 2 W R 2
(15)
= 0
We follow the closed-form solution for IA presented in [11] to prove the achievability of the DoF by the proposed scheme. At relay R1 , the interference from R2 and R4 is aligned along the same direction, i.e., in the interference subspace. The receive filter U R1 is then designed such that it is orthogonal to the interference subspace. The IA condition is given by
(10)
In addition, in even time slots, the received signals at the ith destination node, Mi , at time slot m is given by
= 0
Also, in order to cancel the interference at the MSs, we have
span {H R1 R2 W R2 } = span {H R1 R4 W R4 }
(16)
where span{·} denotes the subspace spanned by the columns of a matrix. Therefore the condition required to satisfy (16) is span H −1 (17) R1 R4 H R1 R2 W R2 = span {W R4 } Similarly, at relay 3, the interference from R2 and R4 is aligned in the interference subspace of R3 , i.e., span H −1 (18) R3 R4 H R3 R2 W R2 = span {W R4 } and the receive filter U R3 is then designed such that it is orthogonal to the interference subspace of the relay R3 . We design the relay beamforming matrices {W Rk }k=2,4 in order to satisfy (17) and (18) which yields −1 span{W R2} = span H −1 R1 R2H R1 R4H R3 R4H R3 R2W R2 (19) Therefore, selecting the columns of W R2 as any N/2 −1 eigen vectors of the matrix H −1 R1 R 2 H R 1 R 4 H R 3 R 4 H R 3 R 2 solves (19). Given W R2 , we can determine W R4 from (18) and the receive filters {U H Rk }k=1,3 , {U Mk }k=1,2 from (12), (14), and (15) respectively. Since the matrix −1 H −1 R1 R2 H R1 R4 H R3 R4 H R3 R2 is full rank as the channel matrices are independent, we can show that the signal space at R1 , R3 , M1 , and M2 are of rank N/2, and hence, the number of achievable DoF is given by N . While the two achievable schemes achieve N DoF for the 2-user cellular MIMO-OWR network, it is clear that, for generic channels, almost surely, interference neutralization equations cannot be simultaneously satisfied at more than one destination. Thus, the interference neutralization scheme cannot be extended for K > 2. We finally extend the alternate relaying scheme to the K-user cellular MIMO-OWR setting. V. D O F OF THE K-U SER S ETTING In this section, we extend the alternate relaying scheme to the K-user cellular MIMO-OWR setting. In this setting, K MSs seek communication with K BSs. It is assumed that each source-destination pair is supported by two relays and each node in the network has N antennas. The alternate relaying scheme can be easily extended to this case where each BS alternately transmits to one of its relays while the second relay forwards the message received in the previous time slot to the MS. Unfortunately, closed form solutions for IA are difficult to derive in this case. Instead, we derive conditions for the properness of the resulting system of equations. We also verify
the results using a distributed IA algorithm that minimizes the interference leakage at the receiving RSs and MSs. Definition 1. A system of equations is proper if and only if, for all subsets of equations, the number of variables involved must be at least large as the number of equations in that subset, i.e., Nv ≥ Ne As outlined in [11], in a proper system, and in the absence of particular structure in the channel coefficients, IA is achievable almost surely at least for the case d = 1. Also, For d > 1, the connection between feasible and proper systems can be further strengthened by including standard information theoretic outer bounds in the feasibility analysis. It is therefore important to characterize whether a system is proper in order to assess the achievability of IA.1 Theorem 2: The system is proper if and only if 3N ≥d 2(K + 1)
(20)
Proof: The number of distinct matrix equations can be easily shown to be equal to 2K(2K − 1) where one contains d2 scalar equations. This yields a total of Ne = 2Kd2 (2K − 1) equations. The total number of variables Nv in the U Rk , W Rk , and U Mi matrices can also be shown to be equal to 6Kd(N − d). Plugging in the values of Nv and Ne computed earlier, we have the result of Theorem 2. We use an iterative distributed IA similar to that in [8] to verify the results of Theorem 2. In this algorithm we start with arbitrary precoding matrices W Ri and interference suppression matrices U Ri and U Mi and iteratively update these filters to to reduce the residual interference at each step. The quality of IA is measured by the power in the leakage interference at each receiver. Fig. 4 shows the leakage interference versus iteration number for a 5-user system with one stream per user. From Theorem 2 the minimum number of antennas required to achieve IA is 2d(K+1) = 4 antennas 3 per node. We can see from Fig. 4 that the leakage interference approaches zero for the cases of N = 4 and N = 5 antennas per user. However, when number of antennas is reduced to 3 the leakage interference is high and IA is not possible. VI. CONCLUSION We have investigated a two-hop decode and forward MIMO relaying network where two half-duplex relays forward the message of the BS to the destination MSs. An alternate relaying protocol was proposed and IA scheme were adopted to compensate for an inherent penalty of capacity pre-log factor 1 2 due to half duplex relaying. The inter-relay interferences incurred by an alternate protocol and the interference at the destinations were aligned to the reduced spatial dimensions and completely cancelled at the relay and the destination. We have developed an outer bounds on the DoF of this setting. The outer bound has been used to prove a converse result for the maximum number of DoF. We have also provided a 1 Although generally properness does not imply achievability, our simulations show that in the cases we have simulated a system was achievable when it was proper.
Fig. 4: The leakage interference for the iterative IA algorithm.
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