Globecom 2012 - Communication Theory Symposium

Degrees of Freedom of MIMO Two-Way X Relay Channel Zhengzheng Xiang, Jianhua Mo and Meixia Tao Dept. of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China Emails: {7222838, mjh, mxtao}@sjtu.edu.cn

Abstract—In this paper, we study the degrees of freedom of a multiple-input multiple-output (MIMO) two-way X relay channel, i.e., a system with two groups of source nodes and one relay node, where each of the two source nodes in one group wants to exchange independent messages with both the two source nodes in the other group via the relay node. We only consider the symmetric case where each source node is equipped with M antennas while the relay is equipped with N antennas. We first show that the upper bound of the degrees of freedom is 2N when N ≤ 2M . Then by applying physical layer network coding and joint interference cancellation, we propose a novel transmission scheme for the considered network. We show that  this  scheme can always achieve this upper bound when N ≤ 4M . 3 Index Terms—MIMO X channel, relay, degrees of freedom, network coding, signal alignment, joint interference cancellation.

I. I NTRODUCTION Wireless communication has been advancing at an exponential rate, due to the increasing expectations for multimedia services. This, in turn, necessitates the development of novel signaling techniques with high spectral efficiency and capacity. Among those factors limiting the capacity of wireless networks, interference has been seen as a key bottleneck. Recently, two novel signaling schemes have been proposed to deal with the interference problem: interference alignment and network coding. Interference alignment was first proposed in [1] to achieve the maximum degrees of freedom (DoF) for the MIMO X channel. The key idea behind this technique is that we shall align these interference signals so that they occupy the smallest signal space, leaving more free space for the useful signals. With this powerful technique, authors in [2] have shown that the capacity of the K-user time varying interference channel is characterized as K log(SNR) + o(log(SNR)). (1) C(SNR) = 2 Thus, no matter the size of the interference network, it is theoretically possible that each user may be able to achieve half the DoF as if there were no interference at all. This surprising result reveals that interference is not a fundamental limitation for such networks. Based on the concept of interference alignment, an increasing variety of interference alignment This work is supported by the Joint Research Fund for Overseas Chinese, Hong Kong and Macao Young Scholars under grant 61028001, the Innovation Program of Shanghai Municipal Education Commission under grant 11ZZ19, and the NCET program under grant NCET-11-0331.

schemes have emerged such as distributed interference alignment, ergodic alignment and blind interference alignment [3][5]. Also interference alignment can be implemented in time, frequency and spatial dimensions. A review of the current status of interference techniques is presented in [6]. Network coding is also a promising transmission technology to improve spectral efficiency and system throughput [7]. The key idea of network coding is to ask an intermediate node to mix the messages it received and forward the mixture to several destinations simultaneously. Compared with the conventional time sharing based schemes where destinations are served in different time slots, the implement of network coding can increase the overall throughput significantly. The first wireless communication scenario where the network coding was applied to is two-way relaying channel (TWRC), where two source nodes exchange information with the help of a relay (sometimes referred as physical layer network coding) [8] and [9]. By applying physical layer network coding at the relay, the rate of information exchange between the two source nodes can be increased by two times compared with conventional schemes such as 802.11. Beyond this basic TWRC, physical layer network coding has been applied to several generalized relay-aided wireless networks such as multiuser two-way relay networks, multipair two-way relay networks and multiway relay networks [10]- [14]. In this paper, we consider the network information flow problem for a multiple-input multiple-output (MIMO) twoway X relay channel and analyze its total DoF. In this network, all the four source nodes are equipped with M antennas and the relay is equipped with N antennas. Each of the two source nodes in one group exchanges two independent messages with both the two source nodes in the other group with the help of relay. We first derive an upper bound on DoF for such a network when N ≤ 2M . By combining the physical layer network coding and joint interference cancellation techniques, we then propose a novel transmission scheme and show that our proposed can always achieve this upper bound   scheme . The relay-aided MIMO X channel has been when N ≤ 4M 3 considered in [15] to analyze the diversity and multiplexing tradeoff and in [16] [17] to study its DoF. However, all these works considered one-directional transmission, which we refer to as the MIMO one-way X relay channel. For the MIMO two-way X relay channel, it has been considered in [18] for a special case of M = 3, N = 4. In this paper, we consider the general case with arbitrary M and N and derive an upper

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Tx1

W3,1, W3,2

W1,3, W1,4



 W1,3, W1,4

Tx1

Tx3

 W3,1, W4,1

W1,3, W2,3

W1,3, W2,3

R

R W1,4, W2,4

W1,4, W2,4

W3,2, W4,2





Tx2

W4,1, W4,2

W2,3, W2,4

Fig. 1.

Tx3 

Tx4

Consider an MIMO two-way X relay channel as shown in Fig. 1. The channel consists of four source nodes with M antennas each and a relay with N antennas. Each of the two source nodes on the left-hand-side wants to convey two independent messages to both the two source nodes on the right-hand-side via the relay. So are the two source nodes on the right-hand-side. In the multiple access (MAC) phase, all the four source nodes transmit their signals to the relay. The received signal at the relay is given by (2)

i=1

where yr and nr denote the N × 1 received signal vector and additive white Gaussian noise (AWGN) vector with unit variance at the relay, respectively. Here, x i is the M × 1 transmitting vector at source node i with the power constraint,  ≤ Pi . Hi,r is the N × M channel i.e., E Tr{xi xH i } matrix from source node i to the relay. All the entries of the channel matrices H i,r for i = 1, 2, 3, 4 are independently and identically distributed (i.i.d.) zero mean complex Gaussian random variables with unit variance, i.e. CN (0, 1). Hence, all channel matrices will be of full rank with probability 1. When receiving these signals successfully, the relay then generates new transmitting signals and broadcasts them to all the source nodes. This is called the broadcast (BC) phase. The received signal at the ith source node is as below yi = Hr,i xr + ni

cut 2

Tx4

where yi and ni denote the M × 1 received signal vector and additive white Gaussian noise (AWGN) vector at the ith source vector node, respectively. x r is the N × 1 transmitting   at the relay with the power constraint, i.e., E Tr{xr xH } ≤ Pr . r Hr,i is the M × N channel matrix from the relay to source node i. Throughout this paper, it is assumed that perfect channel state information (CSI) is available at all the source nodes and the relay. Additionally, we assume that the source nodes and the relay operate at full-duplex mode. We then define the total DoF of the above network as d = d13 + d14 + d23 + d24 + d31 + d32 + d41 + d42 R (SNR) (4) = lim SNR→∞ log (SNR)

II. MIMO T WO -WAY X R ELAY C HANNEL

Hi,r xi + nr

W2,3, W2,4

Fig. 2. One direction network information flow for the MIMO two-way X relay channel.

bound on DoF can be achieved by our proposed scheme   which . when N ≤ 4M 3 Notation: Boldface uppercase letters denote matrices and boldface lowercase letters are used for vectors. R and C denote the real and complex spaces. Z + stands for the positive integer. (·)T , (·)H , (·)† and Tr{·} stand for transpose, Hermitian transpose, Moore Penrose pseudoinverse and the trace, respectively. E(·) denotes the expectation operator. Span(H) and Null(H) stand for the column space and the null space for the matrix H, respectively. dim(H) denotes the dimension of column space of H. IN denotes the N × N identity matrix and ⊕ denotes the exclusive-OR operation.

4 

cut 1

Tx2

MIMO two-way X relay channel.

yr =





where R (SNR) is the sum capacity at the SNR for the network and dij is the DoF from the source node i to source node j. III. A N U PPER B OUND ON D O F In this section, we derive an upper bound of the DoF for the MIMO two-way X relay channel when N ≤ 2M . Theorem 1: Consider a MIMO two-way X relay channel with M antennas at every source node and N antennas at the relay. When N ≤ 2M , the total number of DoF is upper bounded by 2N , i.e., d ≤ 2N. (5) Proof: Without loss of generality, we assume that the power constraints at the source nodes and the relay are the same, i.e., Pi = Pr = P , for ∀i ∈ {1, 2, 3, 4}. We first consider the network information flow of one direction, i.e., from source nodes 1, 2 to the source nodes 3, 4 via the relay. Then the signal model in (2) for the MAC phase can be written as follows yr

= H1,r x1 + H2,r x2 + nr   x1 = [H1,r , H2,r ] + nr x2 = H12,r x12 + nr

(6)

Similarly, the BC phase in (3) can be written as follows

(3)

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y3 = Hr,3 xr + n3 y4 = Hr,4 xr + n4 .

(7) (8)

Applying the cut-set theorem in [19] on each phase, as shown in Fig. 2, the cut-set bound of the rate is presented as below

v1,3 s1,3

v 3,2 s3,2 TX 1

R13 + R14 + R23 + R24 ≤ min {I (yr ; x1 , x2 ) , I (y3 , y4 ; xr )} (9) where R13 , R14 , R23 , R24 stand for the information rates from the source nodes 1, 2 to the source nodes 3, 4, respectively; and I(A; B) stands for the mutual information between variables A and B. For the first term on the right-hand-side of (9), we have I (yr ; x1 , x2 ) = h (H12,r x12 + nr ) − h (nr ) 

 2P H12,r HH ≤ log2 det IN + 12,r min {2M, N }  min{2M,N }  2P · λ12,r i = log2 1 + min {2M, N} i=1

 where y34 and n34

T T T n3 , n4 , respectively; and H r,34 denotes the matrix  T  T T Hr,3 , Hr,4 . Then we can similarly have that  min{2M,N } r,34  P · λ i I (y3 , y4 ; xr ) ≤ (14) log2 1 + min {2M, N } i=1

where λr,34 is the ith eigenvalue of the matrix H r,34 HH r,34 . i Based on the above results, we have that (15) (16) (17) (18) (19)

Fig. 3.

(20)

Combining (15)-(20), we conclude that d ≤ 2N

(21)

which completes the proof of Theorem 1. From the above theorem, we can see that when N ≤ 2M , the total DoF for the above network is exactly bounded by the number of the relay’s antennas.

g r2 ( s1,4  s4,1 )

g 3r ( s2,3  s3,2 )

g r4 ( s2,4  s4,2 ) R

TX 3

TX 4 v 4,1 s4,1

v 4,2 s4,2

Signal alignment for network coding during the MAC phase.

IV. N OVEL TRANSMISSION SCHEME In this section, by applying the physical layer network coding and joint interference cancellation, we propose a novel transmission scheme for the considered network to maximize its total DoF. To explain our scheme and its benefits clearly, a system where each source node has M = 3 antennas and a relay has N = 4 antennas is assumed in this subsection. Each of the two source nodes in one group will transmit two independent data streams to both the source nodes in the other group. Source node 1 transmits codewords s 1,3 , s1,4 for messages W1,3 , W1,4 by using beamforming vectors v 1,3 , v1,4 to source nodes 3, 4 via the relay, respectively. The other three source nodes are in a same manner. Step 1: Signal alignment during the MAC phase During the MAC phase, there are totally 8 data streams arriving at the relay. Since the relay has only 4 antennas, it is impossible for it to decode all the 8 data streams. However, based on the idea of physical layer network coding, the relay node only needs to decode the mixture of the symbols. Thus the key point of the proposed scheme is to obtain the network coded messages W1,3 ⊕ W3,1 , W1,4 ⊕ W4,1 , W2,3 ⊕ W3,2 and W2,4 ⊕ W4,2 at the relay. Inspired by the signal alignment for network coding [14], we should carefully design the beamformers so that two desired signals for network coding are aligned within the same spatial dimension. Taking source node 1 as an example, we should align its transmitted data streams with these streams from source node 3, 4 as follows

Considering the other direction of the network information flow, we can similarly obtain that d31 + d32 + d41 + d42 ≤ N.

g1r ( s1,3  s3,1 )

v 2,3 s2,3

(12)

(13) T

T T denote the vectors y3 , y4 and

= N.

v 2,4 s2,4

(11)

 y34 = Hr,34 xr + n34

d13 + d14 + d23 + d24 R13 + R14 + R23 + R24 = lim SNR→∞ log (SNR)   I (yr ; x1 , x2 ) I (y3 , y4 ; xr ) ≤ min lim , lim P →∞ P →∞ log(2P ) log(P ) = min {min {2M, N } , min {2M, N }}

TX 2

(10)

where h(A) stands for the differential entropy of the continis the ith eigenvalue of the matrix uous variable A and λ 12,r i H12,r HH . For the second term in the right-hand-side of (9), 12,r we first rewrite (7) and (8) as follows

v 3,1 s3,1

v1,4 s1,4

span (H1,r v1,3 ) = span (H3,r v3,1 )  gr1

span (H1,r v1,4 ) = span (H4,r v4,1 )  gr2 .

(22) (23)

where gr1 , gr2 are the signal vectors seen by the relay. Fig. 3 illustrates the concept of the signal alignment in the MAC phase. Then the relay can obtain the above four network coded messages. Step 2: Joint interference cancellation during the BC phase For the BC phase, the relay broadcasts these four network coded messages using beamformers u 1r , ..., u4r . Then we can

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u1

u1r



 null H r ,4



u

3

Then the transmitted signals for source node 1 are s4,1

u2

u4

u 2r  null  H r ,2

s3,1

u1 u2

u

4

u3

u4

TX1

x1 = V1,3 s1,3 + V1,4 s1,4 N 4

s4,2

s3,2

=

TX2

u 4r

u3r  null  H r ,1



 null H r ,3



TX3

R u

2

u4

s1,4

u2 u4

u3

Fig. 4.

s1,3

u1

u2

TX4 s2,4

i i H1,r v1,4 = H4,r v4,1

see that each source node suffers from two parts of interference. In order to handle these interference properly, we let the relay and the source nodes jointly cancel these interference streams, which is another key point in the scheme. More specifically, the relay helps cancel one part of interference for each source node; Upon receiving the signals from the relay, each source node then nulls the remaining part of interference by itself. For source node 1, interference streams are along with the vectors u 3r and u4r . The relay can choose u3r ⊆ Null (Hr,1 ) to cancel this interference for it. Since source node 1 has 3 antennas and the number of its useful data streams is 2, then it can successfully null the remaining one interference stream with probability 1. In a similar way, other beamformers are constructed so that selective interference streams are nulled at the relay for the remaining source nodes. Fig. 4 illustrates the process of the joint interference cancellation between source nodes and the relay in the BC phase.

i i H2,r v2,3 = H3,r v3,2 i i H2,r v2,4 = H4,r v4,2

(24)

Proof: We first consider the nontrivial case 4M = 3k, k ∈ Z+ and that N = 3 , ∀M = show that (d13 , d14 , d23 , d24 , d31 , d32 , d41 , d42 ) ( N4 , N4 , N4 , N4 , N4 , N4 , N4 , N4 ) is achieved by using the proposed scheme. During the MAC phase, the ith source node sends message encoded Wi,j to the jth source node using N4 independently 

streams along beamforming vectors V i,j =

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

For each pair of source nodes 1 , according to dimension theorem [20], we obtain that dim (span(Hi,r ) ∩ span(Hj,r ))

(35)

= dim (span(Hi,r )) + dim (span(Hj,r )) −dim (span([Hi,r Hj,r ])) .

In this section, we show that our proposed can   scheme . always achieve this upper bound when N ≤ 4M 3 Theorem 2: The upper bound of the DoF for the considered network can be achieved when N ≤ 4M 3 , i.e.,  4M . 3

1≤i≤

where gr1 , ..., grN are N transmitting vectors seen by the relay. The above conditions imply that   N ⊆ span (H1,r ) ∩ span (H3,r ) (31) span gr1 , ..., gr4   N N +1 ⊆ span (H1,r ) ∩ span (H4,r ) (32) span gr4 , ..., gr2   N 3N +1 ⊆ span (H2,r ) ∩ span (H3,r ) (33) span gr2 , ..., gr 4   3N +1 span gr 4 , ..., grN ⊆ span (H2,r ) ∩ span (H4,r ) (34)

V. ACHIEVABILITY OF THE U PPER B OUND



(26)

N 4 N N 4 +i  gr , 1 ≤ i ≤ 4 N N 2 +i  gr , 1 ≤ i ≤ 4 3N N 4 +i  gr , 1≤i≤ 4

i i = H3,r v3,1  gri , H1,r v1,3

Joint interference cancellation during the BC phase.

dmax = 2N, ∀N ≤



where s1,3 and s1,4 are the N4 × 1 encoded symbol vectors for W1,3 and W1,4 , respectively. The transmitted signals for the other source nodes are in a similar form. In order for the relay to obtain the network coded messages W 1,3 ⊕ W3,1 , W1,4 ⊕ W4,1 , W2,3 ⊕ W3,2 and W2,4 ⊕ W4,2 , we should carefully choose the beamforming vectors to satisfy the signal alignment conditions as below

s2,3

u3

u3

i i v1,3 si1,3 + v1,4 si1,4

i=1

u1

u1



(25)

N

1 4 . vi,j , ..., vi,j

(36)

Since all the entries of the channel matrices are i.i.d. zero mean complex Gaussian random variables, there exists a  2M − N = N2 -dimensional intersection subspace constituted by the column space of channel matrices for each user pair N with probability 1. Then we can always  choose 4 linearly i independent transmitting vectors gr for each source node pair. As a result, the received signal in (2) is rewritten as yr = Gr sr + nr





(37)

where Gr denotes the N × N matrix gr1 , ..., grN , and N 4 sr denotes the N × 1 vector [s 11,3 + s13,1 , ..., s1,3 + N N N N N 1 1 1 1 1 4 4 4 4 4 s3,1 , s1,4 + s4,1 , ..., s1,4 + s4,1 , s2,3 + s3,2 , ..., s2,3 + s3,2 , s2,4 + N N 4 4 + s4,2 ]T . Also due to the entries of all the channel s14,2 , ..., s2,4 1 Here, all the user pairs for the MIMO two-way X relay channel are (1,3), (1,4), (2,3) and (2,4).

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matrices are independently drawn from the Gaussian distribution, the probability that a basis vector in the intersection space of one pair of source nodes’ channel matrices lies in another intersection space of another pair is zero. Thus G r is full-rank with probability 1, which guarantees the decodability of sr at the relay. The four network coded messages ˆ 13 = W1,3 ⊕W3,1 , W ˆ 14 = W1,4 ⊕W4,1 , W ˆ 23 = W2,3 ⊕W3,2 W ˆ and W24 = W2,4 ⊕ W4,2 are then obtained by applying the physical layer network coding modulation-demodulation mapping principle [8] into each entry of s r . For the BC phase, the relay broadcasts the network codˆ 13 , W ˆ 14 , W ˆ 23 and W ˆ 24 to all the source ed messages W nodes using encoded symbols q r = [qr1 , ..., qrN ]T along specifthe beamforming vectors U r = [u1r , ..., uN r ]. More N N N N 3N 1 4 T 4 +1 2 T 2 +1 4 T ically, [qr , ..., qr ] , [qr , ..., qr ] , [qr , ..., qr ] and 3N +1 [qr 4 , ..., qrN ]T are the N4 × 1 encoded symbol vectors for ˆ 13 , W ˆ 14 , W ˆ 23 and W ˆ 24 , respectively. Then the transmitted W signal at the relay in (3) is rewritten as xr =

N 

uir qri .

(38)

i=1

  Source node 1 has total M = 3N 4 -dimensional space and N space of useful the  signal is 2 -dimensional. So it has exactly 3N N N -dimensional free space for the interference − = 4 2 4 signal whose dimension is also N4 . Thus source node 1 can cancel the other part of the interference and decodes the useful ˆ 14 . Using its ˆ 13 and W signals which contains the messages W side information, source node 1 can obtain the messages from source node 3 and source node 4 as follows ˆ 13 , W4,1 = W1,4 ⊕ W ˆ 14 . W3,1 = W1,3 ⊕ W

(45)

In the same manner, the other source nodes can also obtain the messages intended for themselves. Therefore, 2N DoF is achieved by using the proposed scheme on the MIMO twoway X relay channel. For the other that M is not multiple of 3 or N is  cases  smaller than 4M , we can choose the DoF for each pair as 3 below2     N N d13 = d31 = , d14 = d41 = (46) 4 4     N N , d24 = d42 = N − 3 . (47) d23 = d32 = 4 4

The received signal for source node 1 can be expresses as

Then we can similarly apply the above scheme to achieve the upper bound 2N and the details are omitted here.  y1 = Hr,1 uir qri + uir qri + uir qri + n1 (39) The total DoF of the considered network for typical values M and N are summarized in Table 1. From the table, we can N i=1 i= 3N 2 +1 4 +1    i=    see that in some cases, the DoF for each source node may signal interference not be the same. However, we can apply time slot extension N where the first term in the bracket represents the combination to make the average DoF for each source node is equal to 2 ˆ 14 , while [2]. ˆ 13 and W of the desired network-coded messages W ˆ 23 For the conventional time division multiple access (TDMA)the remaining two terms are the unwanted interference W ˆ 24 . The received signals for the other source nodes are in based scheme, each source node transmits signals in orand W a similar way. We can see that each source node suffers from thogonal time slots via the relay. We can easily see that it two parts of interference,i.e., the signals intended for the other can achieve at most min{M, N } DoF. Thus, our proposed two pairs. By applying the joint interference cancellation, the transmission scheme outperforms the conventional TDMArelay chooses beamformers which shall satisfy the following based scheme significantly. Remark 1: The proposed transmission scheme only achieves conditions:   N the upper bound when N ≤ 4M 3 . As an extension of this ⊆ Null(Hr,4 ) (40) work, we show in [21] that by considering span u1r , ..., ur4 the joint transceiver   N N +1 design for interference cancellation in the BC phase, the upper ⊆ Null(Hr,2 ) span ur4 , ..., ur2 (41) 8M

can be achieved. bound when N ≤ 5   N 3N +1 ⊆ Null(Hr,1 ) span ur2 , ..., ur 4 (42) VI. S IMULATION R ESULTS  3N  N 4 +1 span ur , ..., ur (43) ⊆ Null(Hr,3 ). In this section, we provide numerical results to show the ergodic sum rate performance of the proposed scheme for For matrix H r,i , i = 1, ..., 4, there exists a the proposed transmission scheme. Each entry of the channel  each channel  N − M = N4 -dimensional null space with probability 1; and matrices is drawn from CN (0, 1). For simplicity, we assume Null(Hr,i ) ∩ Null(Hr,j ) = span(0), ∀i = j. So we can always that the transmitted power for the source nodes and the relay choose N linearly independent beamformers which satisfy the are the same, i.e., P = P = P = P = P = P , and equal 1 2 3 4 r above conditions. Thus received signals for source node 1 can power allocation is employed for each data stream. A common be rewritten as below noise variance is set to be σ 12 = σ22 = σ32 = σ42 = σr2 = σ 2 . N 

The numerical results are illustrated with respect to the ratio N 2  uir qri + uir qri + n1 . (44) of the total transmitted signal power to the noise variance y1 = Hr,1 N  2

3N 4

i=1

   signal

N 

i= 3N 4 +1

   interference



2 Note that the assignment of the DoF is not unique, and d = d ij ji = 0 means that there is no information exchange for this pair of users.

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TABLE I T HE TOTAL D O F FOR TYPICAL VALUES M M = 3, N = 1 M = 3, N = 2 M = 3, N = 3 M = 3, N = 4 M = 4, N = 1 M = 4, N = 2 M = 4, N = 3 M = 4, N = 4 M = 4, N = 5

d13 d23 d13 d23 d13 d23 d13 d23 d13 d23 d13 d23 d13 d23 d13 d23 d13 d23

= d31 = d32 = d31 = d32 = d31 = d32 = d31 = d32 = d31 = d32 = d31 = d32 = d31 = d32 = d31 = d32 = d31 = d32

= = = = = = = = = = = = = = = = = =

0, d14 0, d24 0, d14 0, d24 0, d14 0, d24 1, d14 1, d24 0, d14 0, d24 0, d14 0, d24 0, d14 0, d24 1, d14 1, d24 1, d14 1, d24

= = = = = = = = = = = = = = = = = =

AND

d41 d42 d41 d42 d41 d42 d41 d42 d41 d42 d41 d42 d41 d42 d41 d42 d41 d42

R EFERENCES

N.

=0 =1 =0 =2 =0 =3 =1 =1 =0 =1 =0 =2 =0 =3 =1 =1 =1 =2

160 140

Sum rate (bps/Hz)

120

M=3, N=4 proposed M=6, N=8 proposed M=3, N=4 TDMA M=6, N=8 TDMA

DoF= 16

100 DoF= 8

80 60 40

DoF= 6

20 DoF= 3

0 5

10

15

20 25 SNR (dB)

30

35

40

Fig. 5. Total DoF for the MIMO two-way X relay channel under different network architectures .

at each receive antenna in decibels (SNR = σP2 ). In total 10000 channel realizations are simulated for each network architecture. From Fig. 5, we can see that the proposed scheme indeed achieves the DoF upper bound. In specific, we can always observe a sum-rate increase of 2N bps for every 3 dB increase in SNR. For instance, when M = 6, N = 8, the curve has a slope of 2N = 16. Also we can see that our proposed scheme outperforms the conventional TDMA-based scheme significantly.

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VII. C ONCLUSION This paper considered the total DoF for the MIMO two-way X relay channel. We analyzed the upper bound of the DoF for such a network when N ≤ 2M . Then by exploiting physical layer network coding and joint interference cancellation, we proposed a novel transmission scheme to achieve the upper   . bound when N ≤ 4M 3

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