# Cognitive Interference Alignment Schemes for IoT Oriented Heterogeneous Two-Tier Networks
## 3 Cognitive Interference Alignment Schemes Design
### Partial Cognitive IA
}^{d_{i}} \overset{\leftarrow}{\mathbf{\mathrm{H}}}_{k,i}\overset{\leftarrow}{ \mathbf{\mathrm{H}}}_{k,i}\overset{\leftarrow}{\mathbf{\mathrm{v}}}_{i,j}\overset{ \leftarrow}{\mathbf{\mathrm{v}}}_{i,j}\overset{\leftarrow}{\mathbf{\mathrm{v}}}_{i,j} \overset{\leftarrow}{\mathbf{\mathrm{H}}}_{k,i}+\sigma_{n}^{2}\mathbf{\mathrm{I}}_{ N_{T,k}}.\] Similarly, the optimal receiving vector for reciprocal channel is given by: \[\overset{\leftarrow}{\mathbf{u}}_{k,s}^{*} \overset{\leftarrow}{\mathbf{\mathrm{u}}}_{k,s} \overset{\leftarrow}{\mathbf{\mathrm{u}}}_{k,s}\overset{\leftarrow}{ \mathbf{\mathrm{O}}}_{k,s}\frac{\overset{\leftarrow-1-}{\mathbf{\mathrm{u}}}_{k,s} \overset{\leftarrow}{\mathbf{\mathrm{u}}}_{k,s}}{\|\mathbf{\mathrm{O}}_{k,s}\bar{ \mathbf{\mathrm{H}}}_{k,k}\overset{\leftarrow}{\mathbf{\mathrm{v}}}_{k,s}\|}.\] Reverse the communication channel and update all the decoder as \(\overline{\mathbf{\mathrm{v}}}_{k,s}=\overset{\leftarrow}{\mathbf{u}}_{k,s}^{*}\), for \(\forall k\in[1,K],\ \forall s\in[1,d_{k}]\). \[\mathbf{\mathrm{If}}\max\biggl{\{}\|\overline{\mathbf{\mathrm{V}}}_{k}-\mathbf{\mathrm{U}}_ {k}\|,\|\overset{\leftarrow}{\mathbf{\mathrm{U}}}_{k}-\overline{\mathbf{\mathrm{V}}}_ {k}\|\biggr{\}}\geq\varepsilon_{k}\text{, i.e., the momentum greater than a given threshold}\] Jump to Step 2. ```  
**Algorithm 2** Partial cognitive IA (P-CIA) (\(\mathbf{\mathrm{H}}_{k,k}\), \(\mathbf{\mathrm{H}}_{k,j}\), \(\mathbf{\mathrm{G}}_{p,k},\mathbf{\mathrm{g}}_{k,p}\), \(P_{k}\), \(\sigma_{n}\))  
The proposed partial cognitive IA also exploits the channel propagation duality and operates in an iterative manner. We identify \(\mathbf{\mathrm{U}}_{k}\) and \(\overset{\leftarrow}{\mathbf{\mathrm{U}}}_{k}\) to be the decoders at the \(k\)-th receiver in the original and reciprocal channels, respectively. Instead of eliminating the interference completely, we consider both the noise and interference power for each IoT node, aiming to maximize the transmission rate of the data streams directly. More specifically, the optimization problem for the \(s\)-th data stream of the IoT node, \(k\), in the original channel can be expressed as:\[ \begin{split}\mathbf{u}_{k,s}^{*}=\text{argmax}\ \log_{2}\left|1+\frac{p_{k}}{d_{k}} \cdot\frac{\mathbf{u}_{k,\tilde{\mathbf{H}}_{k,\tilde{\mathbf{H}}_{k,\tilde{ \mathbf{F}}_{k}}}}\mathbf{\tilde{\mathbf{H}}}_{k,\tilde{\mathbf{F}}_{k}}^{H}\mathbf{ \tilde{\mathbf{H}}}_{k,\tilde{\mathbf{F}}_{k}}^{H}\mathbf{\tilde{\mathbf{H}}}_{k, \tilde{\mathbf{F}}_{k}}^{H}\mathbf{\tilde{\mathbf{H}}}_{k,\tilde{\mathbf{F}}_{k}}^ {H}}{\mathbf{u}_{k,\tilde{\mathbf{O}}_{k}}\mathbf{u}_{k,s}^{H}}\right|,\\ s.t.\mathbf{u}_{k,s}^{H}\mathbf{u}_{k,s}=1,\quad for\ \ \forall k\in[1,K],\ \forall s \in[1,d_{k}],\end{split}  \tag{25}\]  
where the vectors, \(\mathbf{u}_{k,s}\) and \(\widetilde{\mathbf{v}}_{k,s}\), are the \(s\)-th column of the coding matrices, \(\mathbf{U}_{k}\) and \(\widetilde{\mathbf{V}}_{k}\), respectively. In (25), the interference and noise covariance matrix, \(\mathbf{O}_{k,s}\), is represented as:  
\[ \begin{split}\mathbf{O}_{k,s}=\sum_{i=1}^{K}\frac{P_{i}}{d_{i}} \sum_{\begin{subarray}{c}j=1\\ j\
eq s,for\ i=k\end{subarray}}^{d_{i}}\widehat{\mathbf{H}}_{k,\widetilde{ \mathbf{v}}_{i,j}}\widetilde{\mathbf{v}}_{i,j}^{H}\widetilde{\mathbf{H}}_{k,i}^{H}+P_ {p}\mathbf{g}_{k,p}\mathbf{g}_{k,p}^{H}+\sigma_{n}^{2}\mathbf{I}_{N_{R,k}}.\end{split}  \tag{26}\]  
It is known that, for the given interference and noise covariance, \(\mathbf{O}_{k,I}\), and the precoding vector, \(\widetilde{\mathbf{v}}_{k,s}\), the optimal decoding vector to problem (25) is given by [34]:  
\[\mathbf{u}_{k,s}^{*}=\frac{\mathbf{O}_{k,s}^{-1}\widetilde{\mathbf{H}}_{k,\tilde{ \mathbf{k}}}\widetilde{\mathbf{v}}_{k,s}}{\|\mathbf{O}_{k,s}^{-1}\widetilde{\mathbf{ H}}_{k,\tilde{\mathbf{k}}}\widetilde{\mathbf{v}}_{k,s}\|}. \tag{27}\]  
For the reciprocal channel, the partial cognitive IA algorithm operates in a similar way.