# Cognitive Interference Alignment Schemes for IoT Oriented Heterogeneous Two-Tier Networks
## 4 Performance Analyses of the Cognitive IA Schemes
### Feasibility Condition Analysis  
The feasibility condition is also a primary concern in IA based schemes. In this subsection, we analyze the feasibility condition of the proposed cognitive IA schemes and show that more antennas are required to empower a network with cognition. Despite being organized in different forms, the two cognitive IA schemes hold the same feasibility condition. Inspired by the [PERSON]'s theorem, a generic polynomial system is solvable if, and only if, the number of variables is not below the number of equations. Thus, we count the number of cognitive IA equations and that of the unsolved variables to derive the feasibility condition. We first focus on the co-tier interference constraint in (2) with the equivalent channel matrices, \(\widetilde{\mathbf{H}}_{k,i}\), \(k,i\in[1,K]\). The total number of Equations in (2) is counted as:  
\[\mathcal{N}_{e}^{IA}=\sum_{k,i\in[1,K],\ k\
eq i}d_{k}\times d_{i}. \tag{37}\]  
when counting the number of variables, we can see that not all the variables are mutually independent. After eliminating \(2d_{k}^{2}\) superfluous variables for the \(k\)-th IoT node, the total number of variables can be given as:  
\[\mathcal{N}_{v}^{IA}=\sum_{k=1}^{K}d_{k}\big{(}N_{T,k}+N_{R,k}-N_{R,p}-2d_{k} \big{)}. \tag{38}\]  
To simplify the calculations, we assume that the tiered network is a symmetrical one, i.e., \(N_{T,k}=N_{T}\), \(N_{R,k}=N_{R}\) and \(d_{k}=d\), for \(\forall k\in[1,K]\). Note that the assumption of network symmetry does not limit the generality of the analysis. For asymmetrical networks, we can also derive the feasibility condition with [PERSON]'s theorem by comparing the total number of equations and unsolved variables in the whole network. The unsolved variables in both the precoder and decoder can be determined by the IA polynomials, and are solvable if, and only if, the number of the equations, \(\mathcal{N}_{e}^{IA}\), does not exceed that of the variables, \(\mathcal{N}_{v}^{IA}\), as:  
\[d^{2}K(K-1)\leq dK\big{(}N_{T}+N_{R}-N_{R,p}-2d\big{)}, \tag{39}\]  
Which means the maximum number of data streams has the limit:  
\[d\leq\frac{N_{T}+N_{R}-N_{R,p}}{K+1}. \tag{40}\]  
When considering the cognitive interference, the following constraint should be satisfied as:  
\[\mathbf{U}_{k}^{H}\mathbf{g}_{k,p}=0\in\mathbb{C}^{d\times 1},\quad for\ \ k=1,\ 2,\ \cdots,\ K. \tag{41}\]Hence, the polynomial (41) is also involved to derive the feasibility condition of the proposed cognitive IA schemes. It can be seen that considering (41), there exists \(d\) independent equations for each IoT node, and the total number of the equations is \(\mathcal{N}_{e}^{CIA}=\sum_{k=1}^{K}d_{k}=Kd\). Similarly, we count the variables and remove the superfluous ones, hence, the total number of the variables is:  
\[\mathcal{N}_{v}^{CIA}=\sum_{k=1}^{K}d_{k}(N_{R,k}-d_{k})=Kd(N_{R}-d) \tag{42}\]  
Since the \(\mathcal{N}_{v}^{CIA}\) variables can be solved only by the \(\mathcal{N}_{e}^{CIA}\) equations, and the decoder also needs to cooperate with the precoder to align the co-tier interference, it should satisfy the condition, \(\mathcal{N}_{v}^{CIA}\geq\mathcal{N}_{e}^{CIA}+1\), i.e., \(N_{R}\geq d+1\). With the analysis above, we can obtain the total number of equations as \(\mathcal{N}_{e}=\mathcal{N}_{e}^{IA}+\mathcal{N}_{e}^{CIA}\), and the number of variables, \(\mathcal{N}_{v}^{IA}\), remains the same. The feasibility condition of the proposed cognitive IA schemes with \(d\) data streams for each IoT node can be represented as:  
\[\left\{ \begin{array}{l}dK+d+1\leq N_{T}+N_{R}-N_{R,p},\\ d\leq\min\{N_{R}-1,\ N_{T}-N_{R,p}\}.\end{array} \right. \tag{43}\]  
Although the proposed cognitive IA schemes can achieve the same transmission rate as the full-IA homogeneous network, we should note that, in the femtocell, more antennas at the transmitters and receivers are required to deal with the interference introduced by the heterogeneous structure, as well as to make the cognitive IA schemes feasible.