# Interference Alignment for Cognitive Radio Communications and Networks: A Survey
## 2 Interference Alignment  
### Principles  
IA is a precoding transmission technique for interference channels and networks that linearly encodes signals over multiple signaling dimensions e.g., time slots, frequency blocks or multiple antennas. By coding over multiple signaling dimensions, transmissions are designed to align the interfering signals observed at individual receivers into a low dimensional subspace. Consequently, IA maximizes multiplexing gain, i.e., the amount of non-interfering symbols that can be concurrently transmitted over the interference channel [24]. Realising the channel's maximum multiplexing gain, otherwise known as DoF, indicates that the sum rates provided by IA can approach sum capacity at high SNR. Initial research on IA indicated that a system's ability to find such IA precoders is directly related to the number of signal dimensions it can encode [20]. The more multiple signaling dimensions that are accessible for precoding, the more flexibility a communication system has in aligning interference [14]. To illustrate the general IA fundamentals, we consider the system model in Figure 1 where real-valued signals are coded over three dimensions and transmitted over real-valued channels. Individual receivers observe all three interference signals, each of which are denoted as a vector in the real three dimensions space. All three signal dimensions at the receiver are occupied by the three interference signals and the signals do not accidently align. IA users are permitted to cooperatively precode their communications such that individual three interference signals are completely contained in a two dimensional space [24]. IA consequently leaves one dimension to enable users decode \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Acronym** & **Definition** & **Acronym** & **Definition** \\ \hline AS & antenna selection & OFDM & orthogonal frequency-division multiplexing \\ BER & bit error rate & OIA & opportunistic interference alignment \\ BS & base stations & OSO & opportunistic spatial orthogonalisation \\ CP & cyclic-prefix & OTD & optimal transceiver design \\ CR & cognitive radios & PA & power allocation \\ CSI & channel state information & PU & primary users \\ CSS & cooperative spectrum sensing & Rx & receiver \\ DoF & degrees of freedom & SA & space-alignment \\ DOIA & distributed OIA & SD & spatial directions \\ DPC & dirty paper coding & SIA & selective IA \\ DSA & dynamic spectrum access & SIC & successive IC \\ EE & energy efficiency & SIMO & single-input multiple-output \\ FAP & femtocell access points & SINR & signal-to-interference-plus-noise \\ FBS & femtocell base stations & SNR & signal-to-noise ratio \\ FC & fusion centre & SR & spatial reuse \\ FSA & fixed spectrum access & SS & spectrum sensing \\ GA & grouping algorithm & ST-WF & space–time water-filling \\ GLRT & generalized likelihood ratio test & SU & secondary users \\ IA & interference alignment & SVD & singular value decomposition \\ IC & interference channel & SWF & spatial water-filling \\ LIF & leakage of interference & TBF & threshold-based beamforming \\ MBS & macrocell base station & TDD & time division duplex \\ MEB & maximum eigenmode & TDMA & time-division multiple-access \\ MIMO & multiple-input multiple-output & TMA & transmission-mode adaptation \\ MU & macrocell & TO & transmit opportunities \\ MUE & macrocell users equipment & Tx & transmitter \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of Important Acronyms. symbols free from interference by projecting the received signal onto the subspace orthogonal to the interference subspace. While there are many IA schemes that take much more sophisticated forms, the origins of the idea lie in elementary linear algebra. The illustration in Figure 1 deciphers into an equally intuitive mathematical representation. Considering a system of linear equations:  
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