# Interference Alignment for Cognitive Radio Communications and Networks: A Survey
## 4 Interference Alignment in Cognitive Radio: Second Paradigm
### Leakage Interference
#### 4.2.2 Space-Time Water-Filling (ST-WF)  
Another the capacity achieving water-filling technique is the ST-WF for MIMO channels [112]. The comparative analysis done in ref. [113] has shown that the ST-WF achieves higher capacity per antenna than SWF at low to moderate SNR regimes. It is also simpler to enumerate the solution for space-time water-filling because it eludes the cut-off value computation for each channel realization, but as a trade-off, its spectral efficiency gain is also set out to be associated with a higher channel outage probability. Similar to Equation (10), the problem of space-time water-filling can be formulated as:  
\[ \begin{split}\underset{Q}{\text{max}}E\big{[}log\big{|}I\,+\, \frac{1}{\sigma^{2}}HQH^{\dagger}\big{|}\big{]}\\ \underset{subject\,\,to\,\,tr(Q)\leq\,\overline{P},\end{split}  \tag{14}\]where \(\overline{P}\) is the average power constraint; \(H\) and \(Q\) have the same meaning as in Equation (11), making \(Q\) a function of \(H\). The expectation in \(E[tr(Q)]\) is carried over all MIMO channel realizations. This notation can be assumed as the symbol rate is considerable quicker than the MIMO channel variation. \(Q\) is evaluated from all symbols within one channel realization. Solving Equation (14) as a function of all eigenvalues gives the following relationship:  
\[E\Big{[}log\Big{|}I\,+\frac{1}{\sigma^{2}}HQH^{\dagger}\Big{|}\Big{]}=E\Bigg{[} \sum_{k=1}^{M}\log\Bigg{(}1+\frac{p\Big{(}\overline{\lambda}_{k}\Big{)} \overline{\lambda}_{k}}{\sigma^{2}}\Bigg{)}\Bigg{]}=ME\Bigg{[}\log\Bigg{(}1+ \frac{p(\lambda)\lambda}{\sigma^{2}}\Bigg{)}\Bigg{]}, \tag{15}\]  
where \(\overline{\lambda}_{k}\) is the \(kth\) unordered eigenvalue of \(H^{\dagger}H\), and \(p(\lambda)\) denotes the power adaption as a function of \(\lambda\). Therefore, Equation (15) can be rewritten as a function of all eigenvalues as follows:  
\[ \begin{array}{l}\underset{p(\lambda)}{\max}\,M\,\,\int\log\Big{(}1+\frac{p( \lambda)\lambda}{\sigma^{2}}\Big{)}f(\lambda)d\lambda\\ \text{\emph{subject to }}M\int p(\lambda)f(\lambda)d\lambda=\overline{P}\,\end{array}  \tag{16}\]  
where \(f(\lambda)\) is the empirical eigenvalue probability density function. Similar to Equation (12), the optimal power adaption can be formulated as:  
\[p(\lambda)=\Bigg{(}\Gamma_{0}{}^{(\sigma^{2},M)}-\frac{\sigma^{2}}{\lambda_{i }}\Bigg{)}^{+}, \tag{17}\]  
where \(\Gamma_{0}{}^{(\sigma^{2},M)}\) is found numerically to satisfy the average power constraint in Equation (17). The power adaptation is zero for the channel eigenvalue \(\lambda\) smaller than \(\sigma^{2}/\Gamma_{0}{}^{(\sigma^{2},M)}\) implying zero transmission in this eigenmode. As such, the technique in ref. [127] is based on a SVD parallel channel decomposition technique that makes use of this ST-WF algorithm for power allocation. Although, majority of the research done for IA in CR require that the SU post coding matrix be approximated with symbols sent by the PU, the goal of any CR network is for the SUs to have the minimum possible interaction with the PU and vice versa. This work proposes a solution in which the SUs would require no interaction with the PU system, by estimating the required CSI in a blind manner. This work proposes an OIA technique in which two IA constraints \(\text{Q}_{t}\) and \(\text{Q}_{r}\) ought to be adequate as opposed to the initial approach of ref. [24], after which the SUs transmissions are then maximised. Given that most of the work on IA assumes that the SUs have all the CSI regarding the PUs transmission, this work takes on the challenge of designing a blind CSI estimation scheme for \(\text{Q}_{t}\) and \(\text{Q}_{r}\), which contain information on the PU. Assuming that a primary user system utilizes a time division duplex (TDD) scheme and henceforth the possibility of channel reciprocity is considered between the onward (Tx to Rx) and the opposite (Rx to Tx) channels, then the PU-Rx will intermittently communicate data or feedback to the PU-Tx using the Rx to Tx channels channel in timeslots of symbols.