# Cognitive Interference Alignment Schemes for IoT Oriented Heterogeneous Two-Tier Networks
## 3 Cognitive Interference Alignment Schemes Design
### Partial Cognitive IA  
The interference nulling based IA algorithm achieves its optimum only with negligible noise. In this sub section, we consider improving the performance interference nulling based IA algorithm under low and intermediate SNR conditions and propose a partial cognitive IA scheme. The interference nulling based cognitive IA aims to perfectly align both co-tier and cross-tier interferences, providing a totally interference-free transmission for the femtocell users. However, the full-IA method considers just the interference nulling rather than capacity optimization, and achieves its optimum only in the networks with rather high SNR. When the transmit power is limited, the orthogonal precoder makes no attempt to suppress the noise or maximize the desired signal, which is generally a suboptimal solution with low or intermediate SNR values. The performance of the cognitive IA can be further enhanced by partially aligning the interference. Hence, we propose the partial cognitive IA to jointly consider the signal, interference, and noise power, and maximize the transmission rate directly. The steps of the partial cognitive IA scheme are summarized in Algorithm 2. ```
1.Input: \(\mathbf{H}_{k,k}\): Transmission channel; \(\mathbf{g}_{k,p}\): Cross-tier channel; \(\mathbf{\mathrm{H}}_{k,j}\): Co-tier channel; \(\sigma_{n}\): White noise power; \(\mathbf{\mathrm{G}}_{p,k}\): Cognitive interference channel; \(P_{k}\): Transmit power. Calculate LQ decomposition of \(\mathbf{\mathrm{G}}_{p,k}=\mathbf{\mathrm{L}}_{k}\mathbf{\mathrm{Q}}_{k},k=1,\ \cdots,\ K\). Construct the linear precoder \(\mathbf{\mathrm{W}}_{k}\) as the last N columns of the unitary matrix \(\mathbf{\mathrm{Q}}_{k}\) as (15). Let \(\overline{\mathbf{\mathrm{H}}}_{k,j}=\mathbf{\mathrm{H}}_{k,j}\mathbf{\mathrm{W}}_{i}\), initialize with arbitrary unitary matrix \(\overline{\mathbf{\mathrm{V}}}_{k}=\{\overline{\mathbf{\mathrm{v}}}_{k,1},\cdots, \overline{\mathbf{\mathrm{v}}}_{k,d_{k}}\}\), \(\mathbf{\mathrm{U}}_{k}=\{u_{k,1},\cdots,u_{k,d_{k}}\}\), whose columns are linearly independent unit vectors. Formulate the optimization problem for the \(s\)-th data stream of the \(k\)-th node as: \[\mathbf{u}_{k,s}^{*}=\mathrm{argmax}\ \log_{2}\left|1+\frac{p_{k}}{d_{k}} \right|\frac{u_{k,s}\overline{\mathbf{\mathrm{H}}}_{k,j}\overline{\mathbf{\mathrm{v}}}_ {k,j}\overline{\mathbf{\mathrm{v}}}_{k,j}^{H}\overline{\mathbf{\mathrm{H}}}_{k,j}^{H} \overline{\mathbf{\mathrm{H}}}_{k,j}^{H}\overline{\mathbf{\mathrm{H}}}_{k,j}^{H} \overline{\mathbf{\mathrm{H}}}_{k,j}^{H}\overline{\mathbf{\mathrm{H}}}_{k,j}^{H} \overline{\mathbf{\mathrm{H}}}_{k,j}^{H}}\right|,\] \[\left.s.t.\quad\mathbf{u}_{k,s}^{H}\mathbf{u}_{k,s}=1,\quad for\ \forall k\in[1,K],\ s\in[1,d_{k}],\right.\] where the interference plus noise covariance matrix \(\mathbf{\mathrm{O}}_{k,s}\) is: \[\mathbf{\mathrm{O}}_{k,s}=\sum_{i=1}^{K}\frac{p_{i}}{d_{i}} \sum_{ \begin{array}{c}j=1\\ j\
eq s,\end{array} }^{d_{i}} \overline{\mathbf{\mathrm{H}}}_{k,j}\overline{\mathbf{\mathrm{v}}}_{i,j} \overline{\mathbf{\mathrm{H}}}_{k,j}^{H}\overline{\mathbf{\mathrm{H}}}_{k,j}^{H}+P_{p }\mathbf{\mathrm{g}}_{k,p}\mathbf{\mathrm{g}}_{k,p}^{H}+\sigma_{n}^{2}\mathbf{\mathrm{I}}_{ N_{k,k}}.\]  Calculate the optimal receiving vector for the problem as: \[\mathbf{u}_{k,s}^{*}=\frac{\mathbf{\mathrm{O}}_{k,s}^{-1}\overline{\mathbf{\mathrm{H}}}_ {k,k}\overline{\mathbf{\mathrm{v}}}_{k,s}}{\|\mathbf{\mathrm{O}}_{k,s}^{-1}\overline{ \mathbf{\mathrm{H}}}_{k,k}\overline{\mathbf{\mathrm{H}}}_{k,k}\|}.\] In reciprocal channel, update all the precoder \(\overset{\leftarrow}{\overline{\mathbf{\mathrm{v}}}}_{k,s}=\mathbf{u}_{k,s}^{*}\) for \(\forall k\in[1,K]\) and \(\forall s\in[1,d_{k}]\). Compute reciprocal interference plus noise covariance matrix as: \[\overset{\leftarrow}{\mathbf{\mathrm{O}}}_{k,s}=\sum_{i=1}^{K}\frac{p_{i}}{d_{i}} \sum_{ \begin{array}{c}j=1\\ j\
eq s,\end{array}