# Precoder and Decoder Co-Designs for Radar and Communication Spectrum Sharing +
## 4 Simulation Examples  
In this section, we present simulation results to demonstrate the performance of the proposed max-SINR joint precoder-decoder design algorithm. The simulation is done using MATLAB 2019b and a desktop with an i7-10700k processor. We consider a four-user interference channel where three communication users and one radar user colocated with one of the communication users. Each node employs a multicarrier signal model with \(N_{sc}=8\) subcarriers. Assume that communication users desire one DoF and radar desires three DoFs, which is the highest practical achievable DoFs for the radar subsystem under (9). We also assume that zero mean Gaussian noise with variance \(\sigma_{W}^{2}=1\) is present at each receiver. The power of the payload signal is \(\sigma_{S}^{2}=1\) for each communication user. Consider that coding matrix \(\mathbf{C}_{[i]}\mathbf{C}_{[i]}^{H}=\mathbf{I}\) is an identity matrix. The far-field point target is at azimuth angle \(\theta=0^{\circ}\). The total number of samples is 500. The initial estimates of the precoder and decoder are obtained from independent row vectors. One of these three communication users is not interfered with by radar. Increasing the transmit power at each transmitter will also increase the interference observed by the other receivers. The channel matrices are generated randomly according to the block fading channel. We evaluate the system performance by the sum of all the SINRs in the coexistence scenario. The performances of the communication and radar subsystems clearly depend heavily on the SINR values at the receivers. Radar performance in the detection task is studied using a ROC curve, which evaluates the performance of a radar detector by plotting the probability of detection (Pd) versus the probability of false alarm (Pfa) for a given conditions. We compare the performance of the proposed method with the switched small singular value space projection (SSVSP) in [18]. SSSVSP is an extension of nullspace-based precoder design in which the nullspace has been expanded to include the subspace spanned by singular vectors. These singular vectors correspond to small singular values that are selected based on a threshold value. However, it still faces the drawback that it cannot handle mutual interference between radar and communication systems. In the following simulations, we compare the proposed algorithm with SSSVSP when interference from radar is projected into communication users' switched small singular value space. In Figure 3, we compare the total SINR values from all radio receivers, including communication and radar. For fairness in the comparison of DoFs, the number of antennas in SSSVSP is selected to be eight. Indeed, the multicarrier signal can be mathematically regarded as a special case of a MIMO system in which the channel matrix is diagonal. For the proposed method, the SINR increases as a function of SNR, whereas the other approaches experience more interference, and consequently their SINR values decrease. However, the SSSVSP and original signal (without precoder or decoder) method decrease with it. This result may be due to the projection of the radar signal to the nullspace of all communication receivers while ignoring the interference experienced at the radar receiver and interferences among the communication nodes. Moreover, if the signal powers increase at the transmitters, it will increase the interference power at the other receivers. The proposed method approaches the DoF upper bound as the SNR increases. The SINR of the proposed design increases as a function of SNR because interferences are almost completely eliminated. We also observe a good total SINR improvement in a medium-SNR regime. In Figure 4, Neyman-Pearson detectors with different false alarm constraints are compared to the proposed design and SSSVSP method using ROC curves of the radar subsystem. Different SNR levels are considered. All radio transmitters are assumed to be active such that every receiver suffers from interference from all other transmitters.