1. Introduction

In past decades, interference is a primary problem to restrict the development of wireless network. Various methods have been proposed to deal with such issue in the multi-user communication networks. Recently, the interference alignment [1]-[4] algorithm has been presented as an efficient technique. In the target to improve the system capacity, interference alignment is employed in the procedures of the pre-coding at the transmitters and the interference suppression at the receivers, which achieves a better multiplexing gain in interference networks. It is now widely used in two-way relay channels via amplify-and-forward (AF) [5]-[7]. The feasibility conditions of interference alignment are analysized in multi-user two-way relay networks [5]. Later, a more relaxed alignment conditions are introduced to optimize the sum rate of network in [6]. Based on this idea, the sum degrees of freedom for the two-way relay network is further discussed with different antenna number of users and relay node [7]. Reports also show the amplify-and-forward strategy leads to relays of both transmitted messages and noise. Meanwhile, a compute-and-forward(CF) [8]-[10] strategy is developed to enable relays to decode linear equations of transmitted message using the noise linear combinations provided by the channel, and then only forwarding the desired message, which is based on nested lattice. More specifically, the transmitting procedure of compute-and-forward can be described as follows. Firstly, each transmitter maps its message from the finite field into an element of nested lattice code. Then the lattice codes are transmitted over the channel, where each relay decodes a linear equation of the lattice codes. Finally, these lattice equations are re-mapped in the finite field to obtain the desired linear combination of message. Following the aforementioned process, a new interference alignment algorithm is proposed via a channel diagonalization [11] approach based on the measure of the MAX-SINR, Moreover, this paper also analyzes the performance of the interference alignment in both two-way single relay networks and two-way multiple relays networks via compute-and-forward.

2. SYSTEM MODEL

In the two-way relay network, each of the nodes wants to transmit data streams to its corresponding partner, which means that a bidirectional communication is assumed. There is no direct link between the nodes but there are two communication phases between them. In the first phase, all the nodes transmit their signals to the relay, which is called multiple access(MAC) phase, as shown in Fig. 1 (a). In the second phase, the relay broadcasts the signals to the destination nodes, which is called broadcast(BC) phase, as shown in Fig. 1 (b).

Fig. 1.System model of the two-way relay interference channels.

It assumed that 2K nodes are to transmit data streams xk to their corresponding partner. In the MAC phase, each transmitter maps its message into lattice codeword through codebook L.

Then add dither vector dk to lattice codeword, and transmit as wk :

where Λ is the coarse lattice.

All the transmitters transmit the codeword wk to the relay node. The j-th relay node receives the linear combination of codeword and noise rj , scales it by βj and then removes the dithers:

where αij is the coefficient vector of linear combination of the transmitted messages. Define . The last line of equation (3) is in the form of the nested codeword which is the linear combination of codeword and noise. Then relays use channel coefficient to decode. Let aij = g-1([αij]mod p) , and p is a prime number. Recall that g-1 maps finite messages to the corresponding element. The signals received at the j-th relay node are given by:

In the BC phase, the relay node firstly decodes yrj , then broadcasts the signals to the destination nodes. Each receiver firstly removes the self-interference signals, then suppresses interference and finally has the desired signals.

3. Proposed Interference Alignment algorithms via CF

In multi-user two-way relay networks [5] [6], interference alignment is achieved by signal alignment and channel alignment. In MAC phase, the transmitted signals are pair-wise aligned at the relays, the signal alignment is given by

where Vi denotes the pre-coding matrix at the i-th node and Hri denotes the channel between the i-th transmitter and the r-th relay. The i-th node and the i' - th node are communication partners.

In the BC phase, channel alignment followed by zero forcing is performed to achieve interference alignment at the destination nodes. With channel alignment, the effective channels of the communication partners are made to span the same subspace. The channel alignment is given by

where Ui denotes the zero forcing matrix at the i-th receiver , Gir denotes the channel between the r -th relay to the i-th receiver.

Assume each source node is equipped with L antennas to send L different messages, and the relay is equipped with N antennas. It is assumed that the antenna number of users and relay nodes satisfied the constraints of [3]. In this paper, the relaying strategy is compute-and-forward. Signal alignment means there exists linear relationship between coefficients aij . The main research of this paper is under the condition of channel alignment through channel diagonalization [11] to derive optimal pre-coding matrix in two-way single relay networks and two-way multiple relays networks respectively.

3.1 Interference alignment in two-way single relay networks

In BC phase, the relay node broadcasts the signal to all users after pre-coding. The signal received at the i-th user (i = 1,⋯,2K) is given by

where Di is interference suppression matrix at the i-th receiver, Gi is the channel matrix between the relay node and the i-th receiver, Pr is the pre-coding matrix at the i-th transmitter and ni is an additive Gaussian noise. aj denotes the linear coefficient of the signal which is decoded from the j-th transmitter.

After self-interference signals are cancelled, the signal to interference noise ratio (SINR) of the i-th user can be expressed as:

where σ2 is the variance of noise and i' is the partner user of i.

According to (7) and (8), assume that pre-coding matrix Pr is known, the interference suppression matrices Di and are used to maximize the total SINR of the i-th user. Di and can figured out by solving an optimization problem under the constraint of (6), which is described as

This equation is the multi-objective optimization problem. The KKT conditions are used to solve the optimization problem with weighting method, the Lagrange function of (9) is

The KKT conditions are:

Let then

But the close-form solution of (12) cannot be obtained. To solve this problem, the multi-objective optimization problem can be transformed into a single-objective one.

Let , then

The feasible solutions of the corresponding optimization problem (9) are

and

By substituting the equation (14) into (8)

Next, under interference alignment conditions, the optimal pre-coding matrix is derived. The relay node to the receiver channel can be reconstructed as virtual MIMO channels, which can be expressed as

Apply SVD [11] to G and G'

where Vi and are two unitary matrices. , Λ and Λ' are nonnegative diagonal matrix. TTT + T'TT' = I . U is unitary matrices.

Then, the relay pre-coding matrix can be designed as

where Wr is a diagonal matrix for relay power allocation. The ε-th diagonal element of Wr , (UUH)-1 and TTH is defined as wε, kε and tε. By substituting the equation (18) and (19) into (16),

and

From (20), it can be seen that both SINRi and are monotonic functions of . In order to maximize SINRi and , the optimal wε can be derived by simplifying the optimization problem as

And the Lagrange function of equation (21) can be expressed as:

Then according to the KKT conditions, the optimal solution can be derived as follows:

where is the maximum element in the set

By substituting wε into (20) , the maximum SINR of the i-th user and the partner are:

Similarly, according to the pre-coding matrix (15), the maximum SINR of the i-th user and the partner are :

3.2 Interference alignment in two-way multiple relays networks

There are 2K users in two-way multiple relays networks. The number of relay node is K. After MAC phase, the signal that the r-th relay node broadcasts is as follows:

where Pr denotes the pre-coding matrix of the r-th relay node and arj denotes the linear coefficient of the signal which is the r-th relay node decoding from the j-th transmitter.

At the BC phase, after interference suppression the signal of the i-th user can be expressed as:

where Di is interference suppression matrix at the i-th receiver, Gir is the channel matrix from the r-th relay node to the i-th receiver, ni is an additive Gaussian noise at the i-th receiver.

After self-interference signals are cancelled, the SINR of the i-th user can be expressed as:

where the i-th node and the i' - th node are communication partners.

According to (27) and (28), assume pre-coding matrix Pr is known, the interference suppression matrices Di and are used to maximize the total SINR of the i-th user. Then, under the constraint of (6) by solving an optimization problem, Di and can be figured out as follows:

and

By substituting (29) and (30) into (28), the SINR of the i-th user and the partner user are

The r-th relay node to the receiver channel can be reconstructed as virtual MIMO channels, which can be expressed as

Apply SVD [11] to Gir and

where Vir and are the unitary matrices, , Λr and Λ'r are nonnegative diagonal matrix. Then TrTTr + Tr'TTr' = I . Ur is unitary matrices.

The r-th relay node pre-coding matrix can be designed as Pr = (UH)-1Wr . Let wεr, kεr and tεr are the ε-th diagonal element of Wr, (UrUrH)-1 and TrTrH . Then (31) can be expressed as:

From (34), we can see that, the difference between SINRi and is only the noise term in the denominator. Therefore, SINRi and can achieve the maximum values under the same optimal wεr . Furthermore, when ρr is defined as the maximum transmit power of the r-th relay node, the Lagrange multipliers can be used to form the single-objective optimization problem as

The KKT conditions can be established as

then

If L=1, then k1rw21r = ρr , and

If L=2, then k1rw21r+k2rw22r = ρr ,

Let , then w1rt1r = w2rt2r, and

If L = n ,then k1rw21r+⋯+knrw2nr = ρr

let , then w1rt1r = w2rt2r =⋯= wnrtnr , and

Then the SINR of the i'-th transmitter is:

4. PERFORMANCE ANALYSIS

To evaluate the performance of the proposed interference alignment algorithm, the rate achieved by the i-th user and the sum rate [12] of the network are defined as

In this section, the paper compares the performance of the proposed interference alignment in two-way relay channel via compute-and-forward (CF) with that via amplify-and-forward (AF) .

Fig. 2 (a) and Fig. 2 (b) illustrate the relation between the sum rate and the SNR for AF and CF respectively in two-way single relay interference networks. In Fig. 2 (a), we set and in Fig. 2 (b), set and σ2 denote the variance of the noise at relays and receivers respectively. As shown in Fig. 2, the system performance for CF increases with the increment of SNR, and is always better than the one for AF, especially in the low SNR region.

Fig. 2.Comparison between sum rate of AF and that of CF in two-way single relay interference networks

Fig. 3 (a) and Fig. 3 (b) illustrate the relation between the sum rate and the SNR for AF and CF respectively in the two-way multiple relays interference networks. In Fig. 3 (a), we set and in Fig. 3 (b), set and σ2 denote the variance of the noise at relays and receivers respectively. As shown in Fig. 3, the system performance via CF increases with the increment of SNR, and is always better than that via AF, especially in the low SNR region.

Fig. 3.Comparison between sum rate of AF and that of CF in two-way multiple relays interference networks under the condition of

It is observed from experiment results that the sum rate performance of CF strategy always better than AF, no matter in the two-way single relay networks or in the multiple relays networks, and no matter how many the number of the users is, especially in the low SNR region. It is mainly due to CF that enables relays to decode linear functions of transmitted messages according to their observed channel, only forwarding the desired messages. The relay of AF simply acts as a repeater, forwarding both the desired messages and noise. When the SNR is in the low region, the noise is similar to signals. CF has better sum rate performance in removing the noise. When the SNR is in the high region, the noise can be ignored, so there is little difference in the performance between CF and AF.

Fig. 4 illustrates the relation between the sum rate and the SNR for two-way single relay networks and two-way multiple relays networks use of CF respectively. In Fig. 4, the number of users is set 4, 10 and 50 respectively. As shown in Fig. 4, the sum rate performance in the two-way multiple relays interference networks is better than single relay interference networks, especially when there are few users. The main reason is that every user receives both signals and interferences from multiple relays in the multiple relays networks . Therefor, compared to the single relay networks, the rate of single user increases when there are few users. But if the number of users is large, the number of relay nodes is also large in multiple relays networks. From equations (8) and (28), it can be seen that the user rate in multiple relays networks is almost the same as in single relay networks.

Fig. 4.Comparison between the sum rate for two-way single relay interference networks and that for two-way multiple relays interference networks.

5. CONCLUSION

This paper analyzes interference alignment in the two-way single relay networks and two-way multiple relays networks with compute-and-forward. The use of compute-and-forward allows the relays to remove receiver noise, only forwarding useful signals. The simulation results are that the CF can enhance the sum rate performance of networks, and the sum rate performance of two-way multiple relays networks is better than two-way single relay networks when there are few users.