# Interference Alignment for Cognitive Radio Communications and Networks: A Survey
## 3 Interference Alignment in Cognitive Radio: First Paradigm
### Techniques For IA in CR Networks
#### 3.2.2 Leakage of Interference Signals/Distributed Algorithm IA  
Distributed IA is centred on the local channel knowledge instead of global channel knowledge. The work done in ref. [78] assumes that IA among the SUs has already been done and thus only focuses on subspace selection. While this will simplify computational complexities, it could also place a limit on finding novel solutions. As a matter of fact, the \"half-the-cake\" solutions [20; 21] are actually sub-optimal because of the following reasons. Firstly, they present closed form expressions, which require global CSI, a requirement that can prove overwhelming in practice. Secondly, the feasibility of IA over limited number of dimensions is still very much an open problem [79]. Lastly, the more favourable performance of IA in the high SNR regimes is also not very feasible in practice. The work done in ref. [79] proposes distributed cognitive algorithms that utilizes only local CSI, minimizes each transmitters interference to unintended receivers and utilizes the principle of reciprocity [79]. Typically, some interference power remains in the received signal after applying receiver interference suppression matrices, otherwise known as leakage of interference (LIF). The distributed IA algorithm achieves IA by progressively reducing this leakage interference at each receiver. The total leakage interference due to undesired transmission at each Rx is given as:  
\[I_{i}=Tr\Big{[}U_{i}^{H}Q_{ij}U_{j}\Big{]}, \tag{6}\]  
where \(Q_{i}=\sum_{j=1,j\
eq k}^{d^{[i]}_{i}}\frac{p_{j}}{d_{j}}H_{ij}V_{j}V_{j}^{H \dagger}H_{ij}{}^{H\dagger}\) is the interference covariance matrix at receiver \(i\). For the reciprocal channel, if \(\overset{\leftarrow}{P}^{[k]}>0\) is the power constraint at a SU, then the total leakage interference can also be defined as:  
\[\overset{\leftarrow}{I}_{i}=Tr\Big{[}\overset{\leftarrow}{U_{i}}\overset{ \leftarrow}{Q_{ij}}\overset{\leftarrow}{U_{j}}\Big{]}, \tag{7}\]  
where \(\overset{\leftarrow}{Q_{i}}=\sum_{j=1,j\
eq k}^{d^{[i]}_{i}}\frac{p_{j}}{d_{j} }\overset{\leftarrow}{H_{ij}}V_{j}\overset{\leftarrow}{V_{j}}^{H\dagger}H_{ij }{}^{H\dagger}\) is the interference covariance matrix at receiver \(j\). The algorithm is modelled to alternate between both the original and reciprocal network and only the Rx's update their interference suppression filters to minimize leakage interference. The significance of this solution is about two things: Firstly, each SU-Tx is only required to learn the effective channel of their desired SU-Rx. This lessens the burden of global CSI of all available matrices. Secondly, these algorithms can be used for further analytical study of IA. For a two-tier CR network, the distributed IA algorithms can be applied with some enhanced algorithms, the goal being to concurrently reduce the leakage of interference signals while preserving interference to the PU-Rx below a satisfactory measure. Such an enhanced algorithm has been aptly described in the work in ref. [80], which introduces the concept of matrix distance so that at receiver \(i\), the distance between the subspace spanned by the interference signals \(H_{ij}V_{j},\ i\
eq 1\) and its interference receive subspace spanned by \(U_{i}\) is kept as close as possible. Unlike the work done in ref. [78], this work has no constraints on the number of users. Thus by assuming perfect local CSI, the distributed algorithm iteratively solves the optimization problem. Each SU-Tx updates its precoding matrix to minimize the total interference leakage from interference subspace to signal subspace as well as guarantee its interference to the PU below a certain level. The matrix distance, used as a measurement metric, can be defined as the distance between two orthonormal matrices A and B, such that the following expression holds \(\left\|A-\ BB^{H}A\right\|_{F}\)[80]. The work in ref. [56] is shown to improve sum rates of the CR network and employs a similar enhancement to ref. [80] to protect the PUs.