# Interference Alignment for Cognitive Radio Communications and Networks: A Survey
## 4 Interference Alignment in Cognitive Radio: Second Paradigm
### Water-Filling Techniques For IA in CR Networks
#### 4.2.1 Spatial Water Filling (SWF)  
The work done in ref. [111] has shown that when CSI is available at the transmitter, the capacity achieving distribution can be found by using the well-known water-filling algorithm. Thus given the input-output relationship of a point to point MIMO system as  
\[y\ =\ Hx\ +\ z \tag{10}\]  
where \(y\) is the \(N_{r}\ \times\ 1\) received symbol vector, \(H\) is the \(N_{r}\ \times\ N_{t}\) MIMO channel matrix, \(x\) is the \(N_{t}\ \times\ 1\) transmitted symbol vector, and \(z\) is the \(N_{r}\ \times\ 1\) additive white Gaussian noise vector with variance \(E\!\left[\!\left[\!\left[\!\left[\!\left.\!\left.\!\left.\!\cdot\right.\right. \right.\right.\right]\!\right]\ =\ \sigma^{2}I\), where \((\cdot)^{\dagger}\) denotes the operation of matrix complex conjugate transpose [112]. The MIMO channel \(H\) is modeled as \(H\ =\ \sqrt{s}H_{w}\), where \(H_{w}\) is an \(N_{r}\ \times\ N_{t}\) Rayleigh fast fading MIMO channel whose entries are i.i.d. complex Gaussian random variables, and \(s\) is a scalar log-normal random variable, that is, \(10\ \log_{10}s\ \sim\ \mathcal{N}\!\left(\!\left[\!\left.\!\left.\!\left. \!\!\left.\!\!\left.\!\!\left.\!\!\!\left.\!\!\!\left.\!\!\!\left. at Tx and Rx ends, capacity is gained by implementing a water-filling PA scheme over the spatial directions related with the singular values of its channel transfer matrix. Significantly, when the PU Txs maximize their transmission rates, power constraints generally lead the Txs to retain some spatial directions (SD) unused. The unused SD can consequently be reused by an alternate system operating in the same frequency band. Certainly, an opportunistic Tx can transmit its individual data to its respective Rx by processing its signal in such a manner that the interference originating from the PU link impairs only the unused SDs. Therefore, the SDs which are calculated analytically and presented to be adequately high, are beneficial for the SUs when the available spectral resources are fully exploited over a certain period in a certain geographical area. The process is as follows:  
The PU-Tx follows a water-filling power allocation by choosing its pre-coding and post-processing matrices in such a way that their channel transfer matrix is diagonalised. The PU allocates it's transmit power over an equivalent channel, which consists of parallel sub-channels with non-zero singular values, referred to as the used transmit dimensions. If \(m_{1}\in\{1,\ldots,\ M\}\) denotes the number of transmit dimensions used by PU, then the unused transmit directions can be computed. There exist a total number of dimensions \(N\) described in the range as:  
\[1\leq m_{1}\leq rank\Big{(}H_{11}^{11}H_{11}\Big{)}\leq N, \tag{13}\]  
where \(H_{11}\) is the SVD of the \(N_{pu}\times M_{pu}\) matrix. The PU equivalent channel can alternatively be denoted as the \(N_{1}\times M_{1}\) matrix whose main diagonal consists of \(m_{1}\) non-zero entries and \(N-m_{1}\) zero entries or \(N_{1}-m_{1}\) unused receive dimensions without any PU signal. It is these empty unused SDs (also known as transmit opportunities (TO)) that the SU will exploit to increase the DoF on the reporting channels. The opportunistic transmitter has to avoid interfering with the \(m_{1}\) dimensions used by the PU Tx, which is achieved by aligning the transmission from the SU using linear precoding, with the \(N_{1}-m_{1}\) unused receive dimensions of the PU link. This can be described as more or less being equivalent to the SU-Tx placing its transmitted signal in the null space of the PU-Rx by pre-multiplying its transmitted signal with a matrix of size \(M_{1}-m_{1}\times M_{1}\), which is also equivalent to a Tx with \(M_{1}-m_{1}\) antennas and no PU. Thus, the opportunistic link is said to satisfy the IA condition of Equation (3), if the opportunistic SUs transmission rate is equal to the PUs transmission rate.