# Joint Radar and Communications Waveform Design Based on Complementary Sequence Sets
## 4 Results
### Communications Performance  
In this subsection, the BER and communications rate are applied to evaluate the communications performance of the JRC waveforms.  
The variations in the BER with the bit energy-to-noise power spectral density ratios (which is denoted as \(E_{b}/N_{0}\) in Figure 9) of the JRC waveforms with the BPSK and 16-PSK modulation schemes are shown in Figure 9a,b, respectively. Moreover, the BERs of the traditional BPSK waveform and the traditional 16-PSK waveform are also illustrated in Figure 9. The OPP-16-PSK in Figure 9b denotes the OPP waveforms with 16-PSK modulation. In Figure 9a, we can see that both the designed JRC-CWG-BPSK and DR-JRC-CWG-BPSK have the same BER as the traditional BPSK waveforms. To make sure the BER is lower than \(10^{-5}\), the \(E_{b}/N_{0}\) should be greater than 9.6 dB for the JRC-CWG-BPSK and DR-JRC-CWG-BPSK, greater than 11.1 dB for the OPP-BPSK waveform, and greater than 12.4 dB for the LFM-BPSK waveform. For the JRC waveforms with 16-PSK modulation in Figure 9b, the designed JRC-CWG-16-PSK and DR-JRC-CWG-16-PSK have the same BER performance as the traditional 16-PSK waveforms. To ensure that the BER is lower than \(10^{-5}\), the \(E_{b}/N_{0}\) should be greater than 17.5 dB for the designed JRC-CWG-16 PSK and DR-JRC-CWG-16 PSK, and greater than 23.4 dB for the OPP-16-PSK. Obviously, compared with the other two JRC waveforms, the designed JRC-CWG and DR-JRC-CWG have the best communications performances in terms of BER for the same modulation scheme.  
The communications rates of the JRC-CWG and the DR-JRC-CWG are given in (7). It can be seen from (7) that the communications rate of the JRC-CWG or the DR-JRC-CWG will be increased when the number of WCIs in each JRC-CWG or DR-JRC-CWG \(M_{c}\) is  
Figure 8: Output SNRs of the JRC waveforms at the radar receiver with different Doppler frequencies.  
Figure 9: **(a)** BERs of the JRC waveforms with BPSK modulation; **(b)** BERs of the JRC waveforms with 16-PSK modulation.  
increased. Let \(M\) be a fixed value of 16, the PRI \(T\) be 25 \(\mu\)s, and the length of each waveform \(N\) be 64; the relationship between the communications rate and the PSLR of the designed JRC-CWG-16-PSK and the DR-JRC-CWG-16-PSK is shown in Figure 10. According to (7), the communications rate of the designed JRC-CWG-16-PSK and DR-JRC-CWG-16-PSK is \(R_{b}=0.64M_{c}\) Mbps. It can be seen that for the designed JRC-CWG-16-PSK and DR-JRC-CWG-16-PSK, the PSLR increases as the communications rate or \(M_{c}\) increases. Specifically, when the communications rate is 0.64 Mbps, i.e., \(M_{c}=1\), the PSLRs of the designed JRC-CWG-16-PSK and the DR-JRC-CWG-16-PSK are \(-165.3\) dB and \(-118.8\) dB, respectively. When the communications rate is increased to 9.6 Mbps, i.e., \(M_{c}=15\), the PSLRs of the designed JRC-CWG-16-PSK and the DR-JRC-CWG-16-PSK are increased to \(-33.0\) dB and \(-30.7\) dB, respectively. Since \(M=M_{r}+M_{c}\), the number of WTOs \(M_{r}\) in each JRC-CWG or DR-JRC-CWG will decrease when \(M_{c}\) increases with \(M\) being fixed. Since only the WTOs are optimized to improve the sidelobe performance of the designed JRC waveforms in the optimization problems (9) and (25), the degrees of freedom of the optimization problems will decrease when \(M_{r}\) decreases, which will lead to a worse PSLR performance. It can be observed that there is a trade-off between the communications rate and the PSLR of the designed JRC-CWG-16-PSK and DR-JRC-CWG-16-PSK. In practice, when designing the JRC-CWG or the DR-JRC-CWG, the trade-off curve in Figure 10 can be used to satisfy the requirements of the PSLR and the communications rate.  
The maximum communications rates and the corresponding PSLRs of the JRC waveforms are listed in Table 3. The simulation parameters used in Table 3 are the same as those in Figure 10. Although the OPP-16-PSK has a slightly higher communications rate than the designed JRC-CWG-16-PSK and DR-JRC-CWG-16-PSK, it has much worse performance in terms of PSLR and BER according to Figure 9b.