# Precoder and Decoder Co-Designs for Radar and Communication Spectrum Sharing +
## 3 Max-SINR Joint Precoder-Decoder Design
### Distributed Max-SINR Precoder-Decoder Design  
Based on IA theory, (6a)-(6c) could then be formulated as an interference minimization problem as follows:  
\[ \begin{split}\min_{\mathbf{P}_{[i]}\mathbf{Q}_{[i]}}\mathbf{Q}_{[i]}^ {H}\mathbf{H}_{[ij]}\mathbf{P}_{[j]}\\ s.t.\quad\text{rank}(\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]}\mathbf{P} _{[i]})=d_{[i]}.\end{split}  \tag{12}\]  
However, the interference minimization criteria are not optimal when taking noise and the radar channel matrix into account. Without loss of generality, (6a)-(6c) could be formulated as a SINR optimization problem as follows:  
\[\max_{\mathbf{P}_{[i]}\mathbf{Q}_{[i]}}\mathbf{Q}_{[i]}\frac{ \mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]}\mathbf{A}_{[i]}\mathbf{A}_{[i]}^{H} \mathbf{H}_{[ii]}^{H}\mathbf{Q}_{[i]}}{\mathbf{Q}_{[i]}^{H}(\sum_{j=1}^{Z_{[ij]} }\mathbf{H}_{[ij]}\mathbf{A}_{[j]}\mathbf{A}_{[j]}^{H}\mathbf{H}_{[ii]}^{H}+ \mathbf{W}_{[i]}\mathbf{W}_{[i]}^{H}\mathbf{Q}_{[i]})\mathbf{Q}_{[i]}} \tag{13a}\] \[s.t.\quad\text{rank}(\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]}\mathbf{ P}_{[i]})=d_{[i]}. \tag{13b}\]  
Here, noise power is described by \(\mathbb{E}(\mathbf{W}_{[i]}\mathbf{W}_{[i]}^{H})=\sigma_{W}^{2}\mathbf{I}_{N_{ \text{\tiny{${\text{${\text{${\text{${\text{${\text{$\text{$\text{$}$}$}$}$}$}$}}}} $}}}}\times N_{\text{\tiny{${\text{\[s.t.\quad\text{rank}(\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]}\mathbf{ P}_{[i]})=d_{[i]}. \tag{13b}\]0{\text{\[s.t.\quad\text{rank}(\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]}\mathbf{ P}_{[i]})=d_{[i]}. \tag{13b}\]1{\text{\[s.t.\quad\text{rank}(\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]}\mathbf{ P}_{[i]})=d_{[i]}. \tag{13b}\]2 \[s.t.\quad\text{rank}(\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]}\mathbf{ P}_{[i]})=d_{[i]}. \tag{13b}\]3\text{\[s.t.\quad\text{rank}(\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]}\mathbf{ P}_{[i]})=d_{[i]}. \tag{13b}\]4\text{\[s.t.\quad\text{rank}(\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]}\mathbf{ P}_{[i]})=d_{[i]}. \tag{13b}\]5\text{\[s.t.\quad\text{rank}(\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]}\mathbf{ P}_{[i]})=d_{[i]}. \tag{13b}\]6}\[s.t.\quad\text{rank}(\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]}\mathbf{ P}_{[i]})=d_{[i]}. \tag{13b}\]7}\[s.t.\quad\text{rank}(\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]}\mathbf{ P}_{[i]})=d_{[i]}. \tag{13b}\]8}\[s.t.\quad\text{rank}(\mathbf{Q}_{[i]}^{H}\mathbf{H}_{[ii]}\mathbf{ P}_{[i]})=d_{[i]}. \tag{13b}\]9{\text{${\text{$0{\text{{\text{${\text{$1{\text{${\text{$2 ${\text{$3\text{${\text{$4\text{${\text{$5}}${\text{$6}${\text{$7}${\text{$8}}}\), and \(Z_{[i]}\) is the number of interfering sources for the \(i\)th receiver. Projecting interference into the designed nullspace with fixed SNR implies maximizing the SINR as (13a). Let us recall (1); the signal power emitted from the \(i\)th transmitter may be written as  
${\text{$9  
where \(\mathcal{V}_{d_{[i]}}(\mathbf{A})\) denotes the eigenvectors corresponding to the \(d_{[i]}\) smallest eigenvalues of \(\mathbf{A}\). The objective function in (13) is not convex. However, we could find a solution by employing channel reciprocity and a distributed iterative algorithm. This algorithm is guaranteed to converge, but it may not necessarily find the global optimum [62]. By taking all radio nodes into account, we can write an algorithm for solving this optimization problem in (13). See Algorithm 1 for detailed steps. The reciprocal interference channel is still 1A feasible by choosing the original precoder as a decoder of the reciprocal interference channel and the original decoder as a precoder, respectively. The algorithm finds \(\mathbf{Q}_{[i]}\) and \(\mathbf{P}_{[i]}\) iteratively in two stages.