# Interference Alignment for Cognitive Radio Communications and Networks: A Survey
## 3 Interference Alignment in Cognitive Radio: First Paradigm
### System Model For Paradigm 1  
Consider a \(K\)-user MIMO interference channel with a single PU link and \(K\) SUs as shown in Figure 2. Each Tx/Rx pair are fitted out with \(N\) and \(M\) antennas, respectively. It is assumed that the interference from the PU-Tx does not affect the SUs receivers. From the point of view of IA, a \(d_{i}\) dimensional transmit signal vector \(d_{i}\in\mathbb{C}^{d_{i}\times 1}\) obtained at receiver \(j\) is given as:  
\[y_{i}=H_{ii}V_{i}x_{i}+\sum_{l=1,j\
eq k}^{K}H_{ij}V_{j}x_{j}+z_{i},\;i=1,\ldots K, \tag{2}\]  
where \(H_{ij}\in\mathbb{C}^{M\times N}\) denotes the channel between \(j\)th Tx to \(i\)th Rx, \(V_{i}\in\mathbb{C}^{N\times d_{i}}\) is the precoding matrix with columns comprising of \(d_{i}\) suitably selected linearly independent beamforming vectors and \(z_{i}\in\mathbb{C}^{M\times 1}\) denotes the receiver thermal noise, modelled as complex additive white Gaussian noise vector, i.e., \(z_{i}\sim\mathcal{CN}(0,1)\). The PU transmission is a point-to-point communication without considering the secondary links. It is also assumed that the primary channel matrix \(H_{pp}\) is perfectly known at PU-Tx and PU-Rx.  
In designing IA in single-tier networks (without PU links) over MIMO block fading channel, the precoding matrices \(\{V_{i}\}_{i=1}^{K}\)\(k=1\) and interference receiving matrices \(\{U_{i}\}_{i=1}^{K}\) satisfy the following conditions [13]:  
\[\begin{array}{c}U_{i}^{H}H_{ij}V_{j}=0,\;\;\forall_{j}\
eq i\\ rank\big{(}U_{i}^{H}H_{ii}V_{i}\big{)}=d_{i},\;\forall i\in\{1,2,\ldots,K\}. \end{array} \tag{3}\]  
Condition (3) make certain no interference arises from other SU links at the output of the \(k\)th SU-Rx, and guarantees that the wanted signal space at the \(k\)th receiver achieves \(dk\) DoF when \(H_{ii}\) is full rank. The wanted message for \(k\)th SU can be decoded by projecting onto the orthogonal complement of \(U_{k}\) and zero forcing the interference. Notwithstanding, it is an ongoing open problem for general interference channels to determine closed-form precoding matrices and received interference subspaces. For three-user interference channel, the closed-form solution of \(V_{i}\) for any \(d_{i}\) has been found in refs. [7; 12]. For the two-tier CR network, the goal of IA is to choose precoder matrices \(\{V_{i}\}_{k=1}^{K}\) and interference receiving matrices \(\{U_{i}\}_{k=1}^{K}\) such that each SU-Rx can decode its own signal by forcing interfering SUs to share a reduced-dimensional subspace while keeping an allowable interference level to the PU within the specified limit. Firstly, transmitter \(i\) adjusts \(V_{i}\) to make sure the most of its induced  
\begin{table}
\begin{tabular}{c c} \hline \hline
**References** & **Applications** \\ \hline
[56; 57; 58; 59; 60] & Cognitive Radio Networks \\
[30; 41; 47; 61] & \(K\) User Interference Channel \\
[27; 30; 31; 46; 47] & \(K\) User \(M\times N\) MIMO Interference Channels \\
[62; 63] & 5G Cellular Wireless Networks \\
[64; 65; 66] & Cooperative Interference Networks \\
[66; 67] & Multihop Interference Networks \\
[68; 69] & Ad hoc Networks \\
[70; 71] & Physical Layer Security \\
[72; 73] & Satellite Networks \\
[74; 75] & D2D Networks \\
[50; 76] & IoT Networks \\
[34; 35; 38; 51] & Heterogeneous Networks \\ \hline \hline \end{tabular}
\end{table}
Table 3: Applications of IA.  
interference at other receiver falls into the subspaces \(\{U_{i}\}_{k=1}^{K}\), and keep the interference to the PU below a pre specified level. Secondly, each SU-Rx chooses a subspace \(U_{i}\) to guarantee most of interference falls into the interference subspace when transmit precoding \(V_{i}\) is fixed. This system model serves as the foundation on which various IA algorithms have been applied towards implementing the solutions under this paradigm, as will be discussed in the preceding section of this overview.