### Relay-aided interference alignment for bi-directional communications in wireless networks

In recent years, the number of wireless nodes increases exponentially and the interference between the communication links is the major limiting factor in wireless communication networks. If the interference signals power is considerably weaker than the useful signal power, then the interference signals can be treated as noise. If the interference signals power is considerably stronger than the useful signal power, then first the useful signal can be treated as noise and the interference signals can be decoded. Secondly, the interference signals can be subtracted from the received signal and the useful signal can be decoded. However, often the interference signals are of similar power as the useful signal. In this case, conventionally the nodes perform transmission using orthogonal resources. If there are K nodes, then each node gets only 1/K of the total bandwidth. Recently, interference alignment (IA) has been developed as an efficient technique to handle interference signals, especially at high signal to noise ratio (SNR). In IA, the receiver space is divided into two subspaces, namely, the useful subspace and the interference subspace. Each node precodes its data streams such that at the intended receiver, all the interference signals align with each other within the interference subspace and the useful signal is in the interference-free useful subspace. Through IA, each node is able to get more than 1/K of the total bandwidth. However, to perform IA, often precoding over multiple time slots is necessary which introduces large delays in the system. Furthermore, global channel knowledge is necessary at all the nodes and no generalized closed form solutions to perform IA are available. In this project, it is shown how relays can be utilized to reduce the delay to two time slots, to perform IA with local channel state information at the nodes, and to obtain closed form solutions.

In this project, the focus is on bidirectional communication. In contrast to the conventional use of relays, where the relays are used to improve the coverage, in this project, it is proposed to utilize the relays to manipulate the effective channel between the transmitters and the receivers in order to aid in the IA process. Q half-duplex relays with R antennas each aid in the bidirectional communication between K node pairs. Each node has N antennas and wants to transmit d data streams to its communication partner. For a bidirectional communication, two-way relaying is spectrally more efficient than one-way relaying and hence, two-way relaying is assumed as the underlying transmission protocol. It is assumed that the relays do not have enough antennas to spatially separate the data streams. It is derived that the relays need at least QR ≥ Kd antennas to aid in the IA process. Starting from this condition, depending on the number of relays and relay antennas, new algorithms to achieve IA are developed in this project. In terms of the sum rate achieved, IA is optimum at high SNR. For low and medium SNR, new algorithms to improve the sum rate performance are also developed in this project.

First, a single relay is considered. It is shown that R ≥ Kd is a necessary condition to perform IA. Initially, the case when the relay has the minimum required number R = Kd of antennas is considered. IA in a two-way relay network is a trilinear problem because the transmit, the relay and the receive filters have to be jointly designed. Two new concepts, namely signal alignment (SA) and channel alignment (CA), are proposed to decouple the design of the transmit and the receive filters, respectively, from the design of the relay filter. SA is the process through which the signals from the nodes align pair-wise at the relay. CA is the process of alignment of the effective channel including the channel between the relay and the receiver and the receive filter with the effective channel of its communication partner. It is shown that SA and CA are necessary steps to achieve IA. Thereby, the process of IA is decomposed into three linear steps, namely, SA, CA, and zero forcing (ZF). The number of antennas required at the nodes to perform SA and CA is derived. A closed form solution to achieve SA, CA, and ZF is proposed. It will be shown that, for the special case R = Kd, pair-wise channel knowledge at the nodes and global channel knowledge at the relays are sufficient to perform IA. Then, the case R ≥ Kd is considered. Now the relays have more antennas compared to the first case. New algorithms to use the additional antennas either to increase the number of interference free data streams defined as degrees of freedom or to reduce the minimum required number of antennas at the nodes to perform SA and CA have been proposed. For this purpose, SA and CA are generalized and new concepts namely, partial signal alignment (PSA) and partial channel alignment (PCA) are proposed. It is shown that IA is achieved through PSA, PCA, and ZF. In order to investigate how many antennas are needed at the nodes and at the relay to achieve IA, the properness conditions for the solvability of the IA equations are derived. If the number of variables is larger than or equal to the number of equations in the system, then the system is classified as proper, else as improper. It is shown that the derived properness condition is the generalization of the condition on the number of node antennas derived for the case R = Kd. PSA and PCA are bilinear problems. An iterative algorithm to perform PSA and PCA is proposed. Also, for a special case for which the conditions are given in this project, a closed form solution is also proposed. The sum rate achieved by the proposed IA algorithm is compared with a reference algorithm without IA. It is shown that the proposed IA algorithm has better sum rate than the reference algorithm at high SNR.

Secondly, multiple relays are considered. It is shown that QR ≥ Kd is a necessary condition to perform IA. Initially, the case QR = Kd is investigated. In this case, the concepts of SA and CA developed for the single relay scenario are extended to the multiple relay scenario. However, in multiple relay case, the relays do not share their received signals i.e., the signal received at one relay is not available at the other relays. Hence, the relay processing matrix is a block diagonal matrix. Therefore, ZF cannot be performed as in the single relay case. A new method named cooperative zero forcing (CZF) is proposed for this case. In CZF, the nodes cooperate with the relays and choose their SA and CA directions such that the relays can perform ZF with a block diagonal matrix. The properness condition is derived. It is shown that the derived properness condition is a generalization of the expression derived for the single relay case with Q = 1. An iterative algorithm to achieve IA is proposed. Then, the case QR ≥ Kd is considered. It is shown that the generalization of SA or PSA to multiple relays leads to a quad-linear problem and therefore, does not simplify the trilinear IA problem. Hence, a new iterative IA algorithm is proposed. The properness condition is derived. An iterative algorithm to achieve IA is proposed. During each iteration, each of the transmit, the relay and the receive filters are designed one after another while keeping the other filters fixed. The sum rate performance of the proposed IA algorithms are compared with the reference algorithm without IA and it is shown that the proposed IA algorithm achieves better sum rate than the reference algorithm at high SNR.

Finally, for all the scenarios considered above, interference management schemes which consider not only the interference signals but also the useful signals are proposed. IA is optimum at high SNR. At high SNR, noise is almost zero and the interference signals are the only limiting factor. IA completely suppresses the interference signals and is optimum at high SNR and, hence, achieves higher sum rate than the reference schemes at high SNR. However, at low and medium SNR, where the noise plays a significant role, it is beneficial to improve the useful signal power in comparison to the noise power. In this project, two algorithms are proposed to improve the sum rate performance at low and medium SNR. The first algorithm is based on IA. In this algorithm, the objective is as follows: out of all the available IA solutions, the one that maximizes the SNR is chosen. This algorithm is applicable whenever the IA solutions are obtained in closed form. A gradient based algorithm to find at least a local maximum is proposed. The second algorithm is based on minimization of the mean squared error between the transmitted and the estimated data symbols subject to the node power constraints and relay power constraint. An iterative algorithm to find at least a local minimum has been proposed. Through simulations it is shown that these two algorithms have better sum rate performance than the reference algorithm and the IA algorithms at low and medium SNR.