# Cognitive Interference Alignment Schemes for IoT Oriented Heterogeneous Two-Tier Networks
## 3 Cognitive Interference Alignment Schemes Design
### Interference Nulling Based Cognitive IA  
With the cognitive interference in (7) being fully eliminated by the linear precoder, we next consider the communication in the femtocell and attempt to mitigate both the co-tier and cross-tier interference for the precoding design. After designing the precoder as \(\mathbf{V}_{k}=\mathbf{W}_{k}\widetilde{\mathbf{V}}_{k}\), the received signal in (8) can be simplified by introducing \(\widetilde{\mathbf{H}}_{k,l}=\mathbf{H}_{k,l}\mathbf{W}_{i}\) as:  
\[\boldsymbol{y}_{k}=\mathbf{U}_{k}^{H}\widetilde{\mathbf{H}}_{k,k}\widetilde {\mathbf{V}}_{k}\mathbf{x}_{k}+\sum_{i=1,l\
eq k}^{K}\mathbf{U}_{k}^{H} \widetilde{\mathbf{H}}_{k,l}\widetilde{\mathbf{V}}_{i}\boldsymbol{x}_{i}+ \mathbf{U}_{k}^{H}\boldsymbol{g}_{k,p}\boldsymbol{x}_{p}+\widetilde{ \boldsymbol{n}}_{k}. \tag{18}\]  
The linear precoder, \(\mathbf{W}_{k}\), is constructed from the unitary matrix, \(\mathbf{Q}_{k}\), hence, the entries of \(\widetilde{\mathbf{H}}_{k,l}\) remain the same statistic and can still be modeled as the Rayleigh fading channel. As we can only design the precoding and decoding matrices of the femtocell rather than changing the optimal transmission strategies of the macrocell, we consider mitigating the co-tier and cross-tier interferences with IA in an iterative way. By exploiting the channel propagation duality [32], the precoding and decoding matrices can be obtained via iterations between the interference channel and its reciprocal channel. To simplify the presentation, we summarize the interference nulling based cognitive IA in Algorithm 1. More specifically, in the original channel, the optimization problem to minimize the interference power imposed on the \(k\)-th IoT node can be given as:  
\[ \begin{split}&\mathbf{U}_{k}^{*}=\operatorname*{argmin}_{\mathbf{U}_ {k}}P_{l,k}=\operatorname*{argmintr}\big{\{}\mathbf{U}_{k}^{H}\mathbf{O}_{k} \mathbf{U}_{k}\big{\}}\\ & s.t.\quad\mathbf{U}_{k}^{H}\mathbf{U}_{k}=\mathbf{I}_{N_{R,k}}, \quad\text{ \emph{for} }\ k=1,2,\cdots,K,\end{split}  \tag{19}\]  
where the interference covariance matrix:  
\[\mathbf{O}_{k}=\sum_{i=1,i\
eq k}^{K}P_{i}\widetilde{\mathbf{H}}_{k,i} \widetilde{\mathbf{V}}_{i}\widetilde{\mathbf{V}}_{i}^{H}\widetilde{\mathbf{H} }_{k,i}^{H}+P_{p}\mathbf{g}_{k,p}\mathbf{g}_{k,p}^{H}. \tag{20}\]  
The interference channel, \(\mathbf{g}_{k,p}\), in (20) should be estimated at the FBS by the blind channel estimation [33], which can obtain the CSI with insufficient or even no pilot. On the other hand, for the reciprocal channel, the optimization problem over the signal of the \(k\)-th IoT node can be expressed as:  
\[ \begin{split}&\mathbf{U}_{k}^{*}=\operatorname*{argmin}_{ \widetilde{\mathbf{U}}_{k}}\left[\overset{\leftarrow}{\mathbf{U}_{k}}P_{l,k} =\operatorname*{tr}\left\{\overset{\leftarrow H}{\mathbf{U}_{k}}\overset{ \leftarrow}{\mathbf{O}_{k}}\overset{\leftarrow}{\mathbf{U}_{k}}\right\} \right]\\ & s.t.\quad\overset{\leftarrow H\leftarrow}{\mathbf{U}_{k}} \mathbf{U}_{k}=\mathbf{I}_{N_{R,k}},\quad\text{\emph{for} }\ k=1,2,\cdots,K,\end{split}  \tag{21}\]  
where the reciprocal interference covariance matrix:  
\[\overset{\leftarrow}{\mathbf{O}_{k}}=\sum_{i=1,i\
eq k}^{K}P_{i}\overset{ \leftarrow}{\mathbf{H}_{i,k}}\overset{\leftarrow H\leftarrow H}{\mathbf{V}_ {i}}\overset{\leftarrow H\leftarrow H}{\mathbf{H}_{i,k}}. \tag{22}\]  
Here, the corresponding variables in the reciprocal channel are denoted with a left arrow on top, i.e., \(\overset{\leftarrow}{\mathbf{U}_{k}}=\widetilde{\mathbf{V}}_{k}\overset{ \leftarrow}{\mathbf{V}_{k}}=\mathbf{U}_{k}\) and \(\overset{\leftarrow}{\mathbf{H}_{i,k}}=\widetilde{\mathbf{H}}_{k,i}^{H}\). It should be noted that the cognitive interference on the macrocell is not involved in the reciprocal channel, since it has been eliminated by the linear precoder, \(\mathbf{W}_{k}\). The iterations alternate between the original and reciprocal channels, and in each iteration the receivers optimize their decoding matrix, \(\mathbf{U}_{k}\) or \(\overset{\leftarrow}{\mathbf{U}_{k}}\).