# Secure Beamforming in 5G-Based Cognitive Radio Network
## 3 System Model
### Proposed Technique in Cascaded Transmission of MIMO-NOMA Network
This will align the interference channel matrix in a cell for a different dimension. Then \(\mathrm{v}^{I,A}=Q^{A}\mathrm{v}^{I,A}\) where \(\mathrm{v}^{I,A}\in\mathbb{C}^{\mathrm{S}\mathrm{x}\mathrm{1}}\) is a proposed vector for the \(\mathrm{l}^{\mathrm{th}}\) cluster in a cell A. The BS will perform a precoding in an aligned interference channel matrix. The precoding must be in NxM dimensions for the better generation of a non-trivial solutions. So, there are no limitations in the number of antennas that is used in the BS. At the same time, the number of clusters is restricted by the antenna in the BS is also eliminated. The Equation (6) is derived by applying an alignment matrix \(Q^{A}\). The area of an aligned channel matrix is 2N(L\(-\)1) \(\times\) A, which is derived from the Equation (7). The 'S' value is nominated to get a non-trivial solution of an Equation (7). The alignment matrix \(Q^{A}\) is constituted as \(Q_{2}^{A}Q_{1}^{A}\) where \(Q_{2}^{A}\) is initialized to align the interference channel matrix in a neighboring cell. Then, \(Q_{1}^{A}\) is a proposed transmitting vector to overcome the data leakage in the neighboring cell. The proposed transmitting vector \(Q_{1}^{A}\) for the neighboring cell will satisfy the following term as:  
\[\left[G_{\mathrm{pu}}^{\mathrm{1,A}}\;G_{\mathrm{su}}^{\mathrm{1,A}}\;\ldots \;G_{\mathrm{pu}}^{\mathrm{-1,A}}\;G_{\mathrm{su}}^{\mathrm{-1,A}}\;G_{\mathrm{ pu}}^{\mathrm{1+1,A}}\;G_{\mathrm{su}}^{\mathrm{1+1,A}}\;\ldots\;G_{\mathrm{pu}}^{ \mathrm{L,A}}\;G_{\mathrm{su}}^{\mathrm{L,A}}\right]^{\mathrm{G}}\!\mathrm{v}^{ \mathrm{1,A}}=0 \tag{6}\]  
\[\left[G_{\mathrm{pu}}^{\mathrm{1,A}}Q^{\mathrm{A}}\;G_{\mathrm{su}}^{\mathrm{1,A }}Q^{\mathrm{A}}\;\ldots\;G_{\mathrm{pu}}^{\mathrm{-1,A}}Q^{\mathrm{A}}\;G_{ \mathrm{su}}^{\mathrm{-1,A}}Q^{\mathrm{A}}\;G_{\mathrm{pu}}^{\mathrm{1+1,A}}Q^{ \mathrm{A}}\;G_{\mathrm{su}}^{\mathrm{1+1,A}}Q^{\mathrm{A}}\;\ldots\;G_{\mathrm{ pu}}^{\mathrm{L,A}}Q^{\mathrm{A}}\;G_{\mathrm{su}}^{\mathrm{L,A}}Q^{\mathrm{A}}\right]^{ \mathrm{G}}\!\mathrm{v}^{\mathrm{l,A}}=0 \tag{7}\]  
\[\left[H_{\mathrm{pu}}^{\mathrm{1,B}}Q_{2}^{\mathrm{A}}\;H_{\mathrm{pu}}^{ \mathrm{2,B}}Q_{2}^{\mathrm{A}}\;\ldots\;H_{\mathrm{pu}}^{\mathrm{L,B}}Q_{2}^{ \mathrm{A}}\right]^{\mathrm{T}}\!\mathrm{Q}_{1}^{\mathrm{A}}=0 \tag{8}\]  
where, \(H_{\mathrm{pu}}^{\mathrm{L,B}}Q^{\mathrm{A}}\mathrm{v}^{\mathrm{l,A}}\in\mathrm{ span}\left(H_{\mathrm{pu}}^{\mathrm{L,B}}\right)\!\mathrm{V}\;Q^{\mathrm{A}}\mathrm{v}^{ \mathrm{l,A}}\)  
A non-trivial solution of \(Q_{1}^{A}\in\mathbb{C}^{\mathrm{P}\mathrm{S}\mathrm{P}\mathrm{S}}\), \(Q_{2}^{A}\in\mathbb{C}^{\mathrm{M}\mathrm{P}}\) is aligned with the interference channel matrix in a neighboring cell as \(\mathrm{P}>\mathrm{NL}\) for the Equation (8). The solution is given as [24]:  
\[\mathrm{X}_{1}^{\mathrm{A}}=\mathrm{span}\left\{\overline{H}^{\mathrm{G}} \!\left(\overline{H}\overline{H}^{\mathrm{G}}+\beta I_{\mathrm{d}}\right)^{ \mathrm{-1}}\!\overline{H}\right\} \tag{9}\]  
where \(\overline{H}\stackrel{{\mbox{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{ \tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny}}}}}}}}}}}}}}}}{\rm{=}} \left[H_{\mathrm{pu}}^{\mathrm{1,B}}Q_{2}^{\mathrm{A}}\;H_{\mathrm{pu}}^{ \mathrm{2,B}}Q_{2}^{\mathrm{A}}\;\ldots\;H_{\mathrm{pu}}^{\mathrm{L,B}}Q_{2}^{ \mathrm{A}}\right]^{\mathrm{T}}\). The condition of M\(>\)N is unrelated for a non-trivial solution of \(Q_{1}^{A}\) in an Equation (9). The regularization term of \(\beta\) contains a normalized mean square error (NMSE) method of an optimization that allows the balance between the noise covariance and the transmit power. Let \(I_{d}\) be the identity matrix of NxM dimensions. Let \(V^{I,A}\) represent the determination of a non-trivial solutions for the Equation (7).