# Joint Radar and Communications Waveform Design Based on Complementary Sequence Sets
## 4 Results
### Performance of the DR-JRC-CWG  
In this subsection, the performance of DR-JRC-CWG will be shown. The AF defined in (18) and Delay-Doppler maps of the designed JRC-CWG-BPSK are shown in Figure 6(a1,a2). Let \(\mathbf{w}=\left[\bar{w}_{0},\bar{w}_{1},\ldots,\bar{w}_{Q}\right]^{T}\) collect the weights in (26), and the AFs and Delay-Doppler maps of the designed DR-JRC-CWG-BPSK and DR-JRC-CWG-16-PSK with different weights \(\mathbf{w}\) are shown in Figure 6(b1-d2). Furthermore, the AFs and Delay-Doppler maps of the LFM-BPSK waveform and OPP-BPSK waveform are shown in Figure 6(e1-f2). The simulation parameters used in Figure 6 are listed in Table 1.  
Figure 6: AFs and Delay-Doppler maps of the JRC waveforms. (**a1**) AF of the JRC-CWG-BPSK; (**a2**) Delay-Doppler map of the JRC-CWG-BPSK; (**b1**) AF of the DR-JRC-CWG-BPSK with \(\mathbf{w}=\mathbf{w}_{1}\); (**b2**) Delay-Doppler map of the DR-JRC-CWG-BPSK with \(\mathbf{w}=\mathbf{w}_{2}\); (**c2**) Delay-Doppler map of the DR-JRC-CWG-BPSK with \(\mathbf{w}=\mathbf{w}_{2}\); (**d1**) AF of the DR-JRC-CWG-16-PSK with \(\mathbf{w}=\mathbf{w}_{2}\); (**d2**) Delay-Doppler map of the DR-JRC-CWG-16-PSK with \(\mathbf{w}=\mathbf{w}_{2}\); (**e1**) AF of the OPP-BPSK waveform; (**e2**) Delay-Doppler map of the OPP-BPSK waveform; (**f1**) AF of the LFM-BPSK waveform; and (**f2**) Delay-Doppler map of the LFM-BPSK waveform. From Figure 6, we can see that all the AFs have a 'thumbtack' shape. It can be seen that there is a valley in the AFs of the designed JRC-CWG-BPSK, DR-JRC-CWG-BPSK, and DR-JRC-CWG-16-PSK around \(f_{d}=0\), while there is no valley in the AFs of the LFM-BPSK waveform and the OPP-BPSK waveform. This means that the designed JRC-CWG-BPSK, DR-JRC-CWG-BPSK, and DR-JRC-CWG-16-PSK have better PSLR performances around \(f_{d}=0\) than the other two JRC waveforms. Moreover, the valley indicates that the PSLRs of the JRC-CWG-BPSK, DR-JRC-CWG-BPSK, and DR-JRC-CWG-16-PSK increase as \(f_{d}\) increases, which is consistent with the analysis of (17). Compared with Figure 6(a2), the valley in Figure 6(b2,c2,d2) is much wider. The wider the valley is, the more slowly the PSLR will increase with the increase in \(f_{d}\). Therefore, compared with the designed JRC-CWG-BPSK, the devised DR-JRC-CWG-BPSK and DR-JRC-CWG-16-PSK have better Doppler resilience. Furthermore, compared with Figure 6(b2), the valley in Figure 6(c2) is wider, which indicates that the Doppler resilience of the designed DR-JRC-CWG-BPSK will be improved when the weight \(\widehat{w}_{1}\) in \(\mathbf{w}\) is increased. From Figure 6, we can see that compared with the LFM-BPSK waveform, OPP-BPSK waveform, and the JRC-CWG, the proposed DR-JRC-CWG has the best Doppler resilience. This indicates that the designed DR-JRC-CWG is more suitable to be employed by automotive JRC systems than other JRC waveforms when \(f_{d}\) cannot be obtained or ignored. In Figure 7, the relationship between the PSLR and Doppler frequency is shown. The simulation parameters in Figure 7 are the same as those in Figure 6. In accordance with the IEEE 802.11p standard, we assume that the carrier frequency of the JRC waveforms in Figure 7 is 5.9 GHz [46], i.e., the wavelength \(\lambda\) is 0.051 m. In Figure 7, the horizontal line for \(-\)35 dB is plotted. To make sure the PSLRs of the JRC waveforms are lower than \(-\)35 dB, the maximum Doppler frequencies and the corresponding radial velocities of the target in Figure 7 are listed in Table 2.