## 2 System Model  
Consider the cooperative NOMA system in Figure 1, where a multiple-antenna AP transmits messages to \(J\) NOMA UT. Among these \(J\) UTs, \(L\) (\(L\leq J\)) UTs are also equipped with multiple-antennas to cooperate with the AP transmission in half duplex mode. In theory, up to \(J-1\) UTs can join the cooperation to make \(J-1\) cooperation stages. However, more stages require more cooperation time slots (\(L+1\) time slots are required in total); thus, the performance gain, such as diversity from additional stages, comes at the cost of the increased spectral resource. Therefore, one or two (\(L=1\) or 2) stages can be considered to be a practical candidate stage number. The remaining \(J-L\) single antenna UTs only receive the cooperative NOMA transmissions during \(L+1\) time slots. The AP uses an \(M\times 1\) AP beamformer vector \(\mathbf{w}_{A}\), where \(M\) denotes the number of transmit antennas at the AP. The cooperative UTs are equipped with \(N\) antennas to receive and to transmit. The decode-and-forward (DF) protocol-based cooperative UTs apply \(N\times 1\) unit-norm receive beamforming vectors \(\mathbf{w}_{R,\ell}\), \(\ell=1,\ldots,L\) to the received signals to decode the messages from the AP. If the \(\ell\)-th cooperative UT decodes the messages, the signal is re-encoded before being beamformed by the \(N\times 1\) transmit vector \(\mathbf{w}_{T,\ell}\). The MIMO channel vector from the AP to the \(\ell\)-th cooperative UT is denoted by the \(N\times M\) matrix \(\mathbf{H}_{\ell}\) and the MISO channel from the AP to the \(j\)-th non-cooperative UT is denoted by the \(1\times M\) vector \(\mathbf{h}_{j}\). In addition, the MIMO channel from the \(\ell\)-th the cooperative UT to the \(m\)-th cooperative UT is denoted by the \(N\times N\) matrix \(\mathbf{G}_{m,\ell}\) and the MISO channel from the \(\ell\)-th cooperative UT to the \(j\)-th non-cooperative UT (\(j=L+1,\ldots,J\)) is denoted by the \(1\times N\) vector \(\mathbf{g}_{j,\ell}\). Note that all the elements of \(\mathbf{H}_{\ell},\mathbf{h}_{j},\mathbf{G}_{m,\ell}\), and \(\mathbf{g}_{j,\ell}\) are independent and identically distributed random variables with \(\mathcal{CN}(0,1)\). We assume that full channel state information (CSI) is available at the AP so that all the beamformer vector design is conducted at the AP. At the \(n\)-th time slot, the AP transmits a composite message toward \(J\) NOMA UTs. Recall that there are \(J-L\) non-cooperative UTs, where the received signals at the \(\ell\)-th cooperative UT and \(j\)-th non-cooperative UT are expressed as  
\[ \begin{array}{rl}\mathbf{y}_{\ell}(n)&=\mathbf{H}_{\ell}\mathbf{w}_{A}x(n)+ \mathbf{n}_{\ell}(n)\\ y_{j}(n)&=\mathbf{h}_{j}\mathbf{w}_{A}x(n)+n_{j}(n),\end{array}  \tag{1}\]  
where \(x(n)=\sum_{j=1}^{J}x_{j}(n)\) is the composite AP message symbol with \(\mathbb{E}\big{[}|x(n)|^{2}\big{]}=1\). The message \(x_{j}(n)\) is destined to the \(j\)-th UT with \(\mathbb{E}\big{[}|x_{j}(n)|^{2}\big{]}=P_{j}\) and \(\sum_{j=1}^{j=J}P_{j}=1\). The \(1\)-st cooperative UT applies the receive beamforming to the signal in (1) as \(\mathbf{w}_{R,1}^{H}\mathbf{y}_{1}(n)\), decodes \(J\) messages with the SIC principle from \(x_{1}(n)\) to \(x_{j}(n)\), re-encodes \(J\) message signals \(x(n)=\sum_{j=1}^{J}x_{j}(n)\), and transmits \(x(n)\) with beamforming as \(\mathbf{w}_{T,1}x(n)\) at the time slot \(n+1\). Note that the \(j\)-th cooperating UT of conventional cooperative NOMA systems [25; 26; 27] relays only the superposition of \(x_{j+1}(n),\ldots,x_{j}(n)\), while all the cooperating UTs in this paper repeat the superposition of \(x_{1}(n),\ldots,x_{J}(n)\). This fact helps each UT to decode easily with the SIC principle, and the beamformer design criteria in the sequel to be easily derived.