# Interference Alignment for Cognitive Radio Communications and Networks: A Survey
## 4 Interference Alignment in Cognitive Radio: Second Paradigm
### System Model for Paradigm II  
The system model for this paradigm is a MIMO CR network that consists of a single PU link (PU-Tx and PU-Rx) and \(k\) SUs (\(SU_{1}\),..., \(SU_{k}\)) as shown in Figure 3. Every user is assumed to have \(M\) transmit and \(N\) receive antennas. The PU link is a point-to-point MIMO link, while the SU network can be a point-to-point or multi-user MIMO network. The following assumptions are made for the purpose of the system model as follows:  
* The PU and SUs operate in the same frequency band and all channels are Rayleigh flat-fading.
* The PU link is a single user MIMO channel, which is represented as a \(N_{pu}\times M_{pu}\) matrix, \(H_{pu}\) with channel gains \(h_{ij}\).
* CSI is perfectly known at both the transmitter and receiver, thus \(H_{pu}\) is also perfectly known.  
The capacity of this channel can thus be defined as:  
\[C=\max_{Q:Tr(Q)=P}log\big{|}I_{N}+H_{pu}QH_{pu}^{H}\big{|}, \tag{8}\]  
where \(Q\) is the \(M\times M\) input covariance matrix. The SU setup is assumed to be either a single user MIMO SU link or a multi-user MIMO channel. While the PU has reserved rights to the spectrum, the SU transmitters sense vital information about the PU to avoid causing interference at both the PU-Rx and other SUs. Each transmitter therefore transmits a sequence of Gaussian encoded symbols to its corresponding receiver by processing its symbols using a \(M_{i}\times d_{i}\) precoding matrix \(V_{i}\) to form the transmitted signal vector \(V_{i}x_{i}\). The assumption is that the SUs possess perfect knowledge of the entire channel transfer matrices. Although unrealistic, this condition provides an upper bound on the achievable rate of each SU. The IA condition states that the primary and secondary received signals are represented by [24]  
\[y_{i}=\mathrm{H}_{ii}V_{i}x_{i}+\sum_{j=1}^{K}\mathrm{H}_{ij}V_{j}x_{j}+z_{i}, \tag{9}\]  
where \(y_{i}\) denotes the \(N_{i}\times 1\) received signal vector at the \(j^{th}\) receiver; \(z_{i}\) denotes the \(N_{i}\times 1\) zero mean unit variance circularly symmetric AWGN noise vector at the \(j^{th}\) receiver; \(x_{i}\) denotes the \(M_{i}\times 1\) signal vector transmitted from the \(i^{th}\) transmitter; \(\mathrm{H}_{ij}\) is the \(N_{i}\times M_{i}\) matrix of the channel coefficients between the \(i^{th}\) transmitter and the \(j^{th}\) receiver; also, \(P_{i}=E\big{[}x_{i}\,x_{i}^{H}\big{]}\), where \(P_{i}\) is the transmit power of the \(i^{th}\) transmitter. It should be noted that \(i\) and \(j\) are used as a generalization denoting each transmitter and receiver pair.  
When CSI is known at the Tx and immediate variation is probable, the capacity gain distribution is evaluated by making use of the renowned water-filling technique where either the temporal or spatial domains have been presented to be optimal. There is also another optimization technique called maximum eigen beamforming that can be used in place of water-filling. This paradigm's operation is based on three main procedures that characterize the design of OIA.  
Firstly, the PU maximizes its own rate on its MIMO channel singular values to leave unused spatial dimensions. Then, the SUs perform linear precoding, which aligns the SUs transmission with such unused spatial directions, thereby ensuring orthogonality between their transmissions and enables the SUs to share the SUs spectrum without causing any interference to the PU's transmission. Finally, the SU implements an optimization procedure to ensure useful data rates for their transmission. The next sections will therefore review each of the water-filling techniques as well as the various IA techniques that have been exploited under each of the water-filling solutions.