# Cognitive Interference Alignment Schemes for IoT Oriented Heterogeneous Two-Tier Networks
## 2 System Model
Hence, the precoding and decoding matrices for each IoT node should be designed to satisfy the following conditions simultaneously [24]:  
\[\mathbf{U}_{k}^{H}\mathbf{H}_{k,i}\mathbf{V}_{i}=\mathbf{0}_{d_{k}\times d_{i} }\quad k\
eq i \tag{2}\]  
\[\text{rank}\Big{\{}\mathbf{U}_{k}^{H}\mathbf{H}_{k,k}\mathbf{V}_{k}\Big{\}}=d _{k},\quad for\quad k=1,2,\cdots K. \tag{3}\]  
Condition (2) guarantees that the interference signals of unintended transmitters can be projected into the orthogonal subspace of the \(k\)-th receiver by \(\mathbf{U}_{k}\) and \(\mathbf{V}_{i}\). Condition (3) preserves the required interference-free degree of freedom (DoF) for the desired signal of each IoT node. In the Rayleigh fading channel where the entries are independently distributed with no special structure, Condition (3) would be satisfied naturally when Condition (2) is satisfied [25]. If the feasibility condition is held for the tiered network, the received signal of the IoT node \(k\) can be given as:  
\[\mathbf{y}_{k}=\mathbf{U}_{k}^{H}\mathbf{H}_{k,k}\mathbf{V}_{k}\mathbf{x}_{k}+\mathbf{ U}_{k}^{H}\mathbf{n}_{k}. \tag{4}\]  
Therefore, the transmission data rate of the \(k\)-th IoT node can be expressed as:  
\[\mathcal{R}_{k}=\text{log}_{2}\Bigg{|}\mathbf{L}_{k}+\frac{p_{k}}{d_{k}\sigma _{n}^{2}}\cdot\frac{\mathbf{U}_{k}^{H}\mathbf{H}_{k,k}\mathbf{V}_{k}\mathbf{V} _{k}^{K}\mathbf{H}_{k,k}^{H}\mathbf{U}_{k}}{\mathbf{U}_{k}^{H}\Big{(}\mathbf{ I}_{N_{Rx}}+\mathbf{Q}_{k}\Big{)}\mathbf{U}_{k}}\Bigg{|}, \tag{5}\]  
where \(\mathbf{Q}_{k}\) is the co-tier interference covariance. Note that if IA is fully achieved, the interference covariance, \(\mathbf{Q}_{k}\), would be a zero matrix and the sum rate of IoT nodes can achieve its maximum by transmit power water-filling [26]:  
\[\mathbf{Q}_{k}=\sum_{i=1,j\
eq k}^{K}\frac{P_{i}}{d_{i}\sigma_{n}^{2}}\mathbf{ H}_{k,i}\mathbf{V}_{i}\mathbf{V}_{i}^{H}\mathbf{H}_{k,i}^{H} \tag{6}\]  
However, when the macrocell is active, since the MU will not sacrifice its own transmission rate to satisfy the IA conditions of IoT nodes, the sum rate of the entire network would be severely decreased. The received signals of both the macrocell and the femtocell would be disrupted by the cross-tier interference as:  
\[\mathbf{y}_{p}=\overbrace{\mathbf{U}_{p}^{H}\mathbf{H}_{p}\mathbf{x}_{p}}^{\text{ desired signal}}+\overbrace{\sum_{k=1}^{K}\mathbf{U}_{p}^{H}\mathbf{G}_{p,k}\mathbf{V}_{k} \mathbf{x}_{k}}^{\text{cognitive interference}}+\overbrace{\mathbf{U}_{p}^{H}\mathbf{n}_{p}}^{ \text{noise}}, \tag{7}\]  
\[\mathbf{y}_{k}=\overbrace{\mathbf{U}_{k}^{H}\mathbf{H}_{k,k}\mathbf{V}_{k}\mathbf{x}_{ k}}^{\text{desired signal}}+\overbrace{\sum_{i=1,i\
eq k}^{K}\mathbf{U}_{k}^{H}\mathbf{H}_{k,i} \mathbf{V}_{i}\mathbf{x}_{i}}^{\text{co-tier interference}}+\overbrace{ \mathbf{U}_{k}^{H}\mathbf{n}_{k}}^{\text{noise}}, \tag{8}\]where \(\mathbf{h}_{p}\) is the channel coefficient vector between the MU and MBS. \(\mathbf{G}_{p,k}\) and \(\mathbf{g}_{k,p}\) are the interference matrix/vector between the two tiers. If the transmission power constraint of the MU is \(P_{p}\), even with the co-tier interference being eliminated by full-IA, the data rate of the \(k\)-th IoT node is degraded as:  
\[R^{\prime}_{k}=\log_{2}\left|\mathbf{I}_{d_{k}}+\frac{P_{k}}{d_{k}}\cdot\frac {\mathbf{U}_{k}^{H}\mathbf{H}_{k,k}\mathbf{V}_{k}\mathbf{V}_{k}^{K}\mathbf{H} _{k,k}^{H}\mathbf{U}_{k}}{\mathbf{U}_{k}^{H}\left(\sigma_{n}^{2}\mathbf{I}_{N _{k,k}}+P_{p}\mathbf{g}_{k,p}\mathbf{g}_{k,p}^{H}\right)\mathbf{U}_{k}}\right|. \tag{9}\]  
The average received SINR of each data stream is also decreased by \(10\lg\left(1+P_{p}/\sigma_{n}^{2}\right)\) dB [27]. As can be seen, the performance of the full-IA algorithm will be severely degraded by the cross-tier interference induced by cognitive spectrum sharing. Hence, the precoding schemes should be redesigned as cognitive empowered IA to mitigate the cross-tier interference and achieve a better network sum rate.