# A Physical-Layer Watermarking Scheme Based on 5G NR
## 3 System Performance Simulation
### System Performance in the Gaussian Channel
#### 3.1.2 System Performance with Carrier Frequency Offset  
In the previous section, the system's performance under ideal carrier synchronization was analyzed. However, in practical applications, it is very difficult to achieve this condition. When the frequency stability of a mobile phone is 0.1 ppm and the carrier frequency is 2.6 GHz, the possible carrier frequency offset is 260 Hz, and the 5G NR system is sensitive to frequency offsets. Therefore, this section takes the frequency offset of 90 Hz as an example to analyze the performance of the watermarking system. When the frequency offset is 90 Hz, the symbol error rate of the user's signal is shown in Figure 13. The symbol error rates of user data in Scheme II and Scheme III are still very close, and lower than that in Scheme I, which is also due to the symmetry of the watermark. However, the symbol error rates in the three schemes are much higher than the theoretical symbol error rate of 16 QAM, which reflects the sensitivity of the 5G NR system to frequency offsets. When the frequency estimation is incorrect, the symbol error rate of the user data will increase significantly. Figure 11: Success rates of user data transmission and watermark authentication under ideal synchronization in the Gaussian channel. Figure 12: Success rates of watermark-bit transmission and watermark authentication under ideal synchronization in the Gaussian channel. The transmission bit error rates of watermark bits are plotted, and the results are shown in Figure 14. Figure 14 shows the bit error rates of watermark source bits in three schemes with a frequency offset, and also shows the bit error rate of watermark source bits in Scheme III with ideal carrier synchronization. Simulation results show that the bit error rates of watermark source bits in Scheme II and Scheme III can keep the same level as the ideal synchronization, and are far lower than that in Scheme I, since the bit error rate of watermark bits is significantly reduced by channel coding. Figure 15 shows the success rates of user-data transmission and watermark authentication in three schemes with a frequency offset, and also gives the success rate of user-data transmission and watermark authentication in Scheme III under ideal synchronization for comparison. In Figure 15, the success rate of watermark authentication in Scheme I is lower than that of user-data transmission, because the watermark signal in Scheme I is not encoded, so the bit error rate of the watermark signal is high, which affects the success rate of watermark authentication. The success rates of user data in Scheme II and Scheme III are very similar, and both are lower than that in ideal synchronization, because the 5G NR system is very sensitive to frequency offsets. However, it is worth noting that the success rate of watermark authentication in Scheme III is higher than that in Scheme II, which can  
Figure 14: Bit error rate of the watermark bits with a frequency offset in the Gaussian channel. Figure 13: Symbol error rate of user data with a frequency offset in the Gaussian channel. be similar to the success rate of watermark authentication under ideal synchronization. This is because in Scheme III pseudo-user-bits are used instead of user bits as the input for the hash operation. When the received signal has a frequency offset, the error rate of user data will increase significantly, but its modulus will not change. Therefore, when pseudo-user-bits are used as the input of hash operation, even if the user symbols are determined incorrectly, the pseudo-user-bits mapped by user's signals are still correct, and authentication can still be completed. Figure 15 can clearly show the advantages of Scheme III. When there is a frequency offset, the success rate of watermark authentication is kept at the same level as that under ideal synchronization, which means Scheme III has good anti-frequency-offset performance.