_Applied_  
_sciences_  
_Applied_  
_Optimal Multi-Antenna Transmission for the Cooperative Non-Orthogonal Multiple-Access System_  
**[PERSON] 1**, [PERSON] 2**\({}^{\ast}\)**\({}^{\copyright}\), [PERSON] 3**\({}^{\copyright}\) and [PERSON] 4**\({}^{\copyright}\)  
(MISO) case, where only the AP is equipped with multiple antennas, such clustering is not possible; thus, the AP beamforming serves single channel NOMA UTs only [12; 21; 22]. Only partial results of the optimal beamforming for the MISO-NOMA system are available in Reference [21]. In the cooperative NOMA system, where some UTs participate in helping the AP transmission [23; 24; 25; 26; 27] so that the performance of NOMA system is improved in diversity sense and in average transmit power sense.  
This article considers the beamforming in the cooperative NOMA system, where multiple antennas are adopted at the AP and at the helping UTs to further enhance the performance of NOMA system. To the author's best knowledge, the beamforming problem for the cooperative NOMA system has not been discussed much, while the design criteria and actual beamforming schemes are essential to efficiently utilize the spatial dimension into the benefit of transmission. Therefore, we formally present the optimal beamformer design criterion for the AP and the cooperative UTs in the cooperative NOMA system and develop two optimal beam fomers (BFs) schemes depending on the power constraint applied at the UTs. These schemes encompass the non-cooperative NOMA when no UT collaborates with the AP. First, we discuss the sufficiency of rank one beamforming scheme in achieving the optimal points so that the ranks of the beamformer matrices at the AP and at the cooperative UTs all need simple rank one structure. This leads to an optimal beamforming criterion with the sum of signal strength terms at all UTs to be equalized. The rank one structure and the derived criterion suggests that the semi-definite programming (SDP) is well suited for the given optimization problem; thus, we propose two beamforming schemes based on optimization with the rank one constraint. In the numerical experiments, these two beamforming schemes perform very closely regardless of the UT power constraint, while they show advantage both in diversity and in average power compared to the non-cooperative NOMA. In other words, they either need less power in achieving a certain outage level or achieve a better outage level with the same power consumption.  
Section 1 presents the system model of the proposed cooperative NOMA. Section 2 two beamforming schemes are presented after a discussion on the optimal condition of the cooperative NOMA. Section 3 presents relevant numerical results. Finally, Section 4 concludes this paper.  
Notations: The notations \(\mathbf{A}^{H}\) and \(\mathbf{A}^{\mathsf{T}}\) 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 \(\mathbf{A}_{i,j}\) 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 \(n\).