# Precoder and Decoder Co-Designs for Radar and Communication Spectrum Sharing +
## 3 Max-SINR Joint Precoder-Decoder Design
### Ideal IA Constraints  
The ideal IA constraints in a multicarrier radar-communication coexistence scenario can be written as follows [62]:  
\[\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ij]}\mathbf{P}_{[j]}=\mathbf{0}_ {d_{[ij]}\times d_{[ij]}}, \tag{6a}\] \[\text{rank}(\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]}\mathbf{P}_{[i]} )=d_{[ij]},\] (6b) \[\forall i\
eq j;\forall i,j\in\mathcal{A}\cup\mathcal{B}, \tag{6c}\]  
where \(\mathbf{0}_{d_{[ij]}\times d_{[ij]}}\) is a \(d_{[ij]}\times d_{[ij]}\) zero matrix, \(\mathbf{Q}_{[i]}\) is an \(N_{sc}\times N_{[ij]}\)-dimensional decoder matrix of the \(i\)th communication node, and recall that \(N_{[i]}=d_{[ij]}d_{[ij]}\) denotes the user-desired DoFs for the \(i\)th user. The channel matrix of the interference channel between the \(i\)th radio node and the \(j\)th radio node is \(\mathbf{H}_{[ij]}\), and the matrix of the signal channel at the \(i\)th user is \(\mathbf{H}_{[ii]}\). When \(i\in\mathcal{A}\), \(\mathbf{H}_{[ii]}\) is a target response matrix \(\mathbf{H}_{[ii]}(\theta)\), which is of interest to the radar subsystem.  
Constraint (6a) means that the precoder is designed to project the interference from the \(j\)th radio transmitter to the \(i\)th receiver's nullspace. The nullspace is designed by decoder \(\mathbf{Q}_{[i]}\) to align interference from all \(j\) transmitters, \(\forall i\
eq j;\forall j\in\mathcal{A}\cup\mathcal{B}\). Equation (6b) implies that the designed signal space should satisfy the desired DoFs. The ideal IA constraints design precoders \(\mathbf{P}_{[j]}\) to project interference such that it is aligned in the nullspace of \(\mathbf{Q}_{[i]}\) and design \(\mathbf{Q}_{[i]}\) to guarantee that the signal space and interference space exits.  
Constraint (6b) is automatically satisfied almost surely in the case where the channel matrix does not have a special structure [50], for example, in the MIMO case. However, (6b) must be taken into account due to the diagonal channel state matrix in the case of an OFDM waveform.  
As the obtained precoder and decoder solution is not unique, there may be some solutions that project both useful signals and interference signals into the nullspace [63]. For example,  
\[\mathbf{Q}=\begin{bmatrix}0\\ 1\end{bmatrix}\mathbf{P}=\begin{bmatrix}1\\ 0\end{bmatrix}\]  
will project any signal into the nullspace for any diagonal channel state matrix. Intuitively, it follows from (6a), and (6b) may be described as  
\[\mathbf{0}=\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ij]}\mathbf{P}_{[ij]}, \tag{7a}\] \[\tilde{\mathbf{H}}_{[ii]}=\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]} \mathbf{P}_{[ij]},\] (7b) \[\forall i\
eq j;\forall i,j\in\mathcal{A}\cup\mathcal{B} \tag{7c}\]where expression (7a) indicates that after the application of the precoder and decoder, an undesired signal from the \(j\)th transmitter will be projected into the nullspace that is designed by the \(i\)th decoder. The user-desired interference-free signal channel matrix \(\widetilde{\mathbf{H}}_{[i]}\) is a square rank \(d_{[j]}\) matrix. Recall that the radar channel matrix includes the target information that the radar is interested in when \(i\in\mathcal{A}\). This equivalent channel matrix offers an intuitive way to understand how IA-based precoders and decoders process radar signals.