# Cognitive Interference Alignment Schemes for IoT Oriented Heterogeneous Two-Tier Networks
## 3 Cognitive Interference Alignment Schemes Design
### Cognitive Interference Elimination  
According to the cognitive radio based paradigm [5], the cognitive network should first protect the primary user from the cognitive interference caused by spectrum sharing, rather than selfishly design its precoding scheme. In this subsection, we consider the proposal of a linear precoder design for the femtocell to nullify the cognitive interference of (7) in a totally autonomous way. By looking at (7), the cognitive interference constraint should be satisfied as:  
\[\sum_{k=1}^{K}\mathbf{U}_{p}^{H}\mathbf{G}_{p,k}\mathbf{V}_{k}\mathbf{x}_{k}= \mathbf{0}. \tag{10}\]  
Note that one possible way to obtain the channel state information (CSI), for instance the cognitive interference channel, \(\mathbf{G}_{p,k}\), has been proposed in [28]. In such a method, the duality of TDD signals and the sounding reference in the LTE/LTE-A frame is exploited to estimate the channel. The data transmitted by the macrocell is unknown to the femtocell, which disqualifies the interference cancellation algorithms, such as interweave IA [29] or dirty paper coding. Furthermore, since the macrocell does not change its optimal decoding design, techniques that rely on cooperative coding between the network tiers are not implementable [17]. Self-configuring and self-optimizing precoding procedures should be designed for the femtocell. Assuming that, for the \(k\)-th IoT node, the precoder is divided to \(\mathbf{V}_{k}=\mathbf{W}_{k}\widetilde{\mathbf{V}}_{k}\), then (10) can be rewritten as:  
\[\mathbf{G}_{p}\mathbf{W}\widetilde{\mathbf{V}}=\mathbf{0}, \tag{11}\]  
where \(\mathbf{G}_{p}\) is the aggregated interference channel as:  
\[\mathbf{G}_{p}=\left[\mathbf{G}_{p,1},\mathbf{G}_{p,2},\cdots,\mathbf{G}_{p,K} \right]\in\mathbb{C}^{N_{K,p}\times\sum_{k=1}^{K}N_{T,k}}. \tag{12}\]The aggregated precoder, \(\mathbf{W}\), in (11) is the direct sum [30] of the total \(K\) precoders:  
\[\mathbf{W}=\mathop{\oplus}\limits_{k=1}^{K}\mathbf{W}_{k}, \tag{13}\]  
where \(\mathbf{W}_{k}\) is the linear precoder that aims to fully eliminate the cognitive interference to the macrocell. It is straight-forward to see when the interference constraint (10) is always satisfied. Note that the macrocell does not need to cooperate or share any transmit information to create the aggregated precoder, \(\mathbf{W}\), which results in a lower backhaul transmission:  
\[\mathbf{G}_{p,k}\mathbf{W}_{k}=\mathbf{0},\quad\ for\ \ k=1,2,\cdots,K, \tag{14}\]  
Now, we focus on the \(k\)-th IoT node to devise \(\mathbf{W}_{k}\). By looking into (14), we can see that the precoder must lie within the kernel of \(\mathbf{G}_{p,k}\). Here, we exploit the LQ decomposition to achieve a kernel space with lower computational complexity. Let \(\mathbf{G}_{p,k}=\mathbf{L}_{k}\mathbf{Q}_{k}\) be the LQ decomposition of the interference channel matrix, where \(\mathbf{L}_{k}\in\mathbb{C}^{N_{R_{p}}\times N_{T,k}}\) is a lower triangular matrix and \(\mathbf{Q}_{k}\in\mathbb{C}^{N_{T,k}\times N_{T,k}}\) is a unitary matrix. The matrix, \(\mathbf{Q}_{k}\), can be given in columns as \(\left[\boldsymbol{q}_{1},\boldsymbol{q}_{2},\cdots,\boldsymbol{q}_{N_{T,k}}\right]\), and the last \(N=N_{T,k}-N_{R,p}\) orthonormal vectors of \(\mathbf{Q}_{k}\) span the kernel space, \(\text{ker}\left(\mathbf{G}_{p,k}\right)\). Therefore, the linear precoder, \(\mathbf{W}_{k}\), at the \(k\)-th IoT node can be defined as:  
\[\mathbf{W}_{k}\triangleq\left[\boldsymbol{q}_{N+1},\boldsymbol{q}_{N+2}, \cdots,\boldsymbol{q}_{N_{T,k}}\right], \tag{15}\]  
This fulfills the cognitive interference constraint (10). Note that, if the IoT nodes can obtain the channel vector, \(h_{p}\), of the macrocell with the help of X2 interface [31], \(\mathbf{W}_{k}\) can be designed directly as the orthonormal basis of \(\boldsymbol{h}_{p}\), which may preserve more interference-free DoF for the network. After eliminating the cognitive interference, we focus on the received signal of the MBS in the macrocell.