## 1 Introduction
### Contribution  
This article considers the beamforming in the SR cooperative NOMA system, where multiple antennas are adopted at the two SR relaying UTs to improve the performance of the cooperative NOMA system. While this SR cooperation avoids SI, it generate cross-relay interference (CRI) between relaying UTs, and hence, the removal of this CRI becomes a new challenge, which is handled by the beamforming proposed in this work. To the author's best knowledge, the beamforming problem for the SR cooperative NOMA system has not been discussed, while it is essential to utilize the spatial dimension to improve transmission efficiency. We present two sub-optimal beamformer design algorithms for the SR cooperative NOMA system since the problem structure does not easily lend itself to the optimal approach. First, the decoding conditions at different NOMA UTs and the CRI levels at the relaying UTs lead us to consider a semi-definite relaxation (SDR)-based beamformer design, where two sets of thresholds on SINRs at UTs and CI power levels are utilized to form sets of constraints for the SDR. The first algorithm finds two sets of beamformer vectors for the two relaying UTs in an alternating fashion, where the considered ranges of thresholds are reduced in every alternation. This alternating search method of multi-dimensional thresholds is hinted from a popular one-dimensional search. In every alternation, the power levels of the found transmit beamformer vectors are compared to the relay transmit power levels, and the ranges of thresholds in the next alternation are determined based on the comparison, and while the relay transmit beamformer vectors are found from the SDR, the receive beamformer vectors are set to zero-force CRI. The first algorithm depends on iterative applications of the SDR algorithm and requires much computation. The second algorithm is a reference scheme with simplified computation without SDR and alternations. The relay UTs receive antennas to try to zero-force the RCI between relaying UTs, while the transmit beamformers are set to the primary UTs in the second hop.  
In the numerical experiments, these two beamforming schemes and half-duplex relay (HDR)-based cooperative NOMA systems are compared in their performance in three different environments. The AP power, the relay UT power and the number of relayUT antennas are used for the experiment parameters to discuss out the pros and cons of proposed schemes.  
Section 1 presents the system model of the proposed SR cooperative NOMA. In Section 2, two beamforming algorithms are explained. Section 3 presents numerical results for a few experiments. Finally, Section 4 concludes this paper.  
_Notations:_ The notations \(\mathbf{A}^{H}\) and \(\mathbf{A}^{T}\) are the Hermitian transpose and the transpose of a matrix \(\mathbf{A}\), respectively. \(\mathbf{Tr}[\mathbf{A}]\) takes the trace of the matrix \(\mathbf{A}\) and \(\text{rank}\{\mathbf{A}\}\) returns the rank of the matrix \(\mathbf{A}\). \(\|\mathbf{a}\|\) denotes the \(\ell_{2}\)-norm of a vector \(\mathbf{a}\). The notation \(\mathbf{A}_{ij}\) represents the element at the \(i\)-th row and the \(j\)-th column of \(\mathbf{A}\). The notation \(\text{diag}[\mathbf{v}]\) produces a diagonal matrix with the elements of the vector \(\mathbf{v}\) on its diagonal. The vector notations \(\mathbf{0}_{k}\) and \(\mathbf{1}_{j}\) represent all zero vector with \(k\) elements and all one vector with \(j\) elements, respectively. \(\mathcal{CN}(\mathbf{0},\mathbf{C})\) denotes the complex white Gaussian distribution of random vector with zero mean vector \(\mathbf{0}\) and the covariance matrix \(\mathbf{C}\). \(\mathbb{C}^{N}\) denotes the \(N\) dimensional complex vector space. Finally, \(\mathbb{E}_{n}[x]\) takes the expectation of \(x\) with respect to \(n\).