# Precoder and Decoder Co-Designs for Radar and Communication Spectrum Sharing +
## 3 Max-SINR Joint Precoder-Decoder Design
### Reciprocity and Feasibility of IA  
In a radio link, the roles of the receiving and transmitting antennas are functionally interchanged, while the instantaneous transfer characteristics of the radio channel remain unchanged. This channel reciprocity can be exploited in IA design. The reciprocity of IA [62] comes from the identity IA constraints between the original interference channel and the reciprocal interference channel, where the original precoder and decoder are considered to be a reciprocal decoder and precoder, respectively. In other words, IA constraints are still feasible after the signal direction is reversed. Using reciprocity, the IA constraints of (7) can be written as follows:  
\[\mathbf{0}=\overline{\mathbf{Q}}_{[i]}^{H}\overline{\mathbf{H}}_{[ji]} \overline{\mathbf{P}}_{[i]},\overline{\widetilde{\mathbf{H}}}_{[ii]}= \overline{\mathbf{Q}}_{[i]}^{H}\overline{\mathbf{H}}_{[ii]}\overline{\mathbf{ P}}_{[i]},\forall i\
eq j;\forall i,j\in\mathcal{A}\cup\mathcal{B} \tag{8}\]  
where \(\overline{\mathbf{P}}\) and \(\overline{\mathbf{Q}}\) denote the precoder and decoder on the reciprocal channel, \(\overline{\mathbf{Q}}_{[i]}=\mathbf{P}_{[i]},\overline{\mathbf{P}}_{[i]}= \mathbf{Q}_{[i]}\), \(\forall i\in\mathcal{A}\cup\mathcal{B}\). Furthermore, \(\mathbf{0}_{d_{[i]}\times d_{[j]},[i]}\) denotes the interference nullspace from the \(i\)th receiver to the \(j\)th transmitter on the reciprocal channel. \(\overline{\widetilde{\mathbf{H}}}_{[ii]}\) denotes the channel matrix for the \(i\)th user itself on the reciprocal channel. The reciprocity of IA does not change the user-desired DoFs of each user while projecting an undesired signal into the nullspace. However, it plays an important role in the following distributed iteration algorithm.  
As with the communication signal, the DoFs of a radar subsystem need to be properly chosen to realize IA. The radar subsystem also works better with more DoFs available to achieve a more flexible waveform design and better diversity gain. In this paper, we assume that the number of desired DoFs is predetermined. The DoFs of radar need to satisfy the feasibility condition of IA, which is written as follows:  
\[\left\{\begin{array}{rl}&d_{[R]}+d_{[i]}\leq N_{sc},\forall i\in\mathcal{A} \cup\mathcal{B}\\ &2d_{[R]}(N_{sc}-d_{[R]})-\sum_{i\in\mathcal{A}_{sub}}d_{[R]}d_{[i]}\geq 0, \forall\mathcal{A}_{sub}\subset\mathcal{A}\cup\mathcal{B}.\end{array}\right. \tag{9}\]  
This condition can easily be formulated based on Theorem 2 in [64]. Note that (9) is a necessary condition. IA requires finding feasible signal strategies while the channel matrix is fixed. However, the feasibility is established in a reverse way, where the strategy is fixed and the study channel matrix is set for which strategy is feasible. The space of strategies can be represented by the product of Grassmannians (7, [64]). If IA is feasible, then the dimension of projection between the strategy space and channel matrix must be non-negative [64]. Furthermore, if \(d_{[i]}=d_{[R]}=d\), i.e., the desired DoFs of the communication and radar nodes are identical and equal to \(d\), the IA is feasible if and only if  
\[d\leq\frac{2N_{sc}}{K+1}. \tag{10}\]  
This condition degenerates into a necessary condition when \(d_{[i]}=d_{[R]}=d=1,K\geq 3\), where (Theorem 1, [65])  
\[K\leq 2N_{sc}-2. \tag{11}\]  
The general feasibility condition of a proper IA network with multiple streams for each user remains an open problem.