Interference Alignment in Two-Way Relay Networks via Rank Constraints Rank Minimization

Xue Jiang, Baoyu Zheng＊, Yuelin Du

College of Telecommunications & Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing, China

* The corresponding author, email： zby@njupt.edu.cn

I. INTRODUCTION

Interference alignment (IA) [1-3] designs the transmitting and receiving strategies for each transmitter-receiver pair to obtain as many interference free dimensions for communication as possible. In two-way relay networks [4-6],based on feasibility conditions, IA strives to maximize the signal to interference noise ratio(SINR) by devising pre-coding matrixes and interference suppression matrixes with perfect channel knowledge. However, the closed-form solution of the pre-coding and interference suppression matrixes has been found only for a few special cases. In order to improve the performances of two-way relay networks, rank constraints rank minimization (RCRM)[7] has been introduced recently, the rank constraints account for the useful signal subspaces spanning all spatial dimensions available, and the rank minimization corresponds to interference alignment maximization so that interference spans the dimensional subspace as low as possible.

Due to the non-convexity and NP-hardness of the RCRM problem, and inspired by compressed sensing and low-rank matrix completion theory, the reweighted nuclear norm is used to approximate the rank which is the convex relaxation of the RCRM problem. In K user MIMO interference channel the nuclear norm minimization [7] and a reweighted nuclear norm minimization [8] have been developed as the effective improvements of the rank function by adopting a convex envelope of the rank function. In multi-antenna cellular networks, coupling reweighted nuclear norm minimization [8] has been proposed, which is another different way of reweighted method.

In two-way relay networks, the interference alignment should also consider signal alignment and channel alignment [4-5] conditions,in order to improve the performance of interference alignment via rank constraints rank minimization in two-way relay networks, this paper proposes left reweighted nuclear norm minimization -γalgorithm and selective coupling reweighted nuclear norm minimization algorithm. The left reweighted nuclear norm minimization -γalgorithm has a novelγchoosing rule, and the selective coupling reweighted nuclear norm minimization algorithm selects weighting methods according to singular value of interference matrixes. Simulation results show that the proposed algorithms have obviously improved performance of sum rate and average achievable multiplexing gain.

II. SYSTEM MODEL

In this paper, we consider two-way relay channels, which a bidirectional communication between K node pairs. Each of the 2K users wants to transmit d data streams to its partner.Each user has N antennas and there is no direct path between any two users. A half-duplex relay node with R antennas is used to enable their communication. In the first phase, all the users transmit their signals to the relay node,which is called multiple access (MAC) phase.In the second phase, the relay node broadcasts the signals to the destination users, which is called broadcast (BC) phase, as shown in Fig.1.

The relay strategy is assumed to be amplify-and-forward. After the BC phase, the received signal of the j-th user can be expressed as：

In the two-way relay networks, the signal and interference matrices of the i-th user are de fined as：

In this paper, the authors propose a new method to implement interference alignment in the two-way relay networks based on reformulation of the conditions for interference alignment via rank constrains rank minimization.

And for all users the perfect interference alignment requires that：

Since all the 2K optimization problems defined above cannot be solved at the same time,we can transform the aim to maximize the sum of interference free dimensions through the RCRM [7]. So the optimal problem is

We could find the above optimal problem is non-convex and intractable. So the nuclear norm [7] is chosen as a convex surrogate for the cost function and the rank constraints replaced by Hermitian positive semi-de finite matrix, that is：

Fig. 1 System model of two-way relay channels

In two-way relay networks [4-5], interference alignment is achieved by signal alignment and channel alignment：

The number of users antennas is N and relay node is R. In the two-way relay networks,N and R should meet the follow conditions[4]：

Then by using the normalized signal values[10], the channel alignment (8b) can be rewritten as：

By using the equation (11) to simplify optimal problem (7), we can get

III. INTERFERENCE ALIGNMENT ALGORITHM VIA RCRM IN TWO-WAY RELAY NETWORKS

3.1 Left reweighted nuclear norm minimization-γ algorithm

The nuclear norm minimization in above optimal problems (7) and (12) has been developed as an effective relaxation of the rank function.However, the nuclear norm accounts for the sum of the magnitudes of the singular values,while the rank operator sums the number of positive values. As a result, if the perfect IA cannot be achieved, the rank of the interference subspace is larger than zero. Then it will cause the unbalanced penalization induced by the nuclear norm approximation, so we use weighting methods to solve this problem,e.g. reweighted nuclear norm minimization algorithm [8][9]. In these algorithms, the parameterγacts as a regularization constant that makes sure the weighting matrices are positive de finite. It limits the penalty imposed on small non-zero singular values and is typically reduced with each iteration to prevent the algorithm from prematurely converging to local minimal. So the value ofγis a crucial aspect of these algorithms. Usually, these algorithms set fixed value ofγin each iterative, which is not optimum choice. So in order to improve the reweighted nuclear norm minimization algorithms in two-way relay networks, we proposes a novelγchoosing rule that requires prior knowledge of the singular value of interference matrix.

From above discussion, it is clear that the required results should be：

So in the left reweighted nuclear norm minimization -γalgorithm, we update theγfor the(s + 1)-th iteration as follow：

3.2 Selective coupling reweighted nuclear norm minimization algorithm

When coupling reweighted [9] methods are used to implement interference alignment in two-way relay networks, the performance of the network is worse. This is mainly because of the differences between the notion of the rank and nuclear norm operations, if the perfect IA cannot be achieved and the minimum singular value of the interference subspace is larger than one, it will result in the unbalanced penalization induced by the weight matrixes.So we consider weighting the nuclear norm of

Similarly, we set the same weight matrices for the users and their partner users. The proposed weighting methods are as follows：

Table I Left reweighted nuclear norm minimization-γ algorithm.

Table II Selective coupling reweighted nuclear norm minimization algorithm

Fig. 2 Sum rate of NNM algorithm with the different number of iterations

whereγis set arbitrarily small positive value.

IV. SIMULATIONS

The sum rate is computed using the following formula：

Fig. 3 Sum rate with the different value of γ

In Fig. 3 (a) we run LR-NNM algorithm seen from Fig. 3(a) and Fig. 3(b), the relationship betweenγand the sum rate performance is nonlinear and choosingγis critical to the performance of two algorithms.

Fig. 5 (a) and Fig. 5 (b) compare the sum rate performance of LR-NNM and LR-NNM-γnetworks, the proposed LR-NNM-γalgorithm has the obviously better sum rate performance.

Fig. 4 Comparison between the sum rate of NNM, LR-NNM and CR-NNM algorithm

The previous four figures indicate that the performance of the LR-NNM-γalgorithm is always better than LR-NNM algorithm,which bene fits from the newγchoosing rule,which provides a tighter approximation for the rank function than LR-NNM algorithm. One drawback of LR-NNM-γalgorithm is that the computational complexity increases less due to newγchoosing rule.

Fig. 5 Comparison between the sum rate of LR-NNM and LR-NNM-γ algorithm

Fig. 6 Comparison between the average multiplying gain per user of LR-NNM and LR-NNM-γ algorithm

Fig. 7 Comparison between the sum rate of CR-NNM and SCR-NNM algorithm

Fig. 8 Comparison between the average multiplying gain per user of CR-NNM and SCR-NNM algorithm

V. CONCLUSION

In this paper we propose a new method to implement interference alignment in the twoway relay networks based on reformulation of the conditions for interference alignment via rank constrains rank minimization. Using these alternate conditions, we formulate a rank constrains rank minimization problem to design pre-coding and interference suppression matrix, then propose left reweighted nuclear norm-γalgorithm and selective coupling reweighted nuclear norm algorithm. A crucial aspect of the left reweighted nuclear norm-γalgorithm is a novelγchoosing rule,and the crucial aspect of the selective coupling reweighted nuclear norm minimization is novel weighing method which is according to singular value of interference matrixes. All these will lead objective function closer to the rank function. Simulation results indicate that the proposed algorithms are very effective.It significantly improves the performance of the sum rate and average multiplying gain per user.

ACKNOWLEDGEMENT

This research was supported by the National Science Foundation of China (NO.61271240,61671253).

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