## 5 Results and Discussions
Taking the period from 0 to \(\pi\) as an example, when \(\Delta\varphi\) < \(\pi/4\), the BER exhibits minimal changes and remains around 0.5. This occurs because the single-frequency interference threshold is greater than the given \(-26.5\) dB, and the BER is primarily determined by the single-frequency interference. When \(\pi/4\) < \(\Delta\varphi\) < \(\pi/2\), the BER decreases rapidly with increasing phase deviation as the threshold is crossed, and the influence of the single-frequency interference component on the system gradually diminishes. When \(\pi/2\) < \(\Delta\varphi\) < \(3\pi/4\), the BER increases with the increase in the single-frequency interference component. When \(\Delta\varphi\) > \(3\pi/4\), the actual SIR is lower than the threshold SIR, and the BER stabilizes at approximately 0.5. It is clear from the above analysis that adjusting the signal phase can enhance the system's ability to resist interference in practical communication systems. Figure 5 illustrates the BER of the system when the single-frequency interference deviates from the signal carrier frequency. A m-sequence with a period of 15 is employed, with an \(E_{b}/N_{0}\) value set at 8.4 dB and a SIR of \(-14.5\) dB. The theoretical BER curve is established using the BER formula derived from the SGA assumption. In Figure 5, we can observe the BER curve for an interference frequency offset equivalent to the bit bandwidth. The change pattern of the theoretical curve is consistent with Figure 4, although there is a notable phase difference compared to the simulation curve. This discrepancy arises because the x-axis of the theoretical curve represents the composite phase deviation \(\beta_{k}\) in  
Figure 3: System BER in the presence of single-frequency interference that has the same frequency as the signal carrier. Equation (32), while the x-axis of the simulation curve represents the actual phase deviation \(\Delta\varphi\), with the disparity between them being the frequency offset phase deviation \(\alpha_{k}\). Next, the system's BER performance is simulated and experimentally verified as a function of the interference frequency offset. Figure 6 illustrates the theoretical BER of the long-code system compared to the simulated and experimental BERs of the short-code system with a code length of 15. In this scenario, the \(E_{b}\)/\(N_{0}\) is 8.4 dB, the single-frequency interference SIR is \(-10\) dB, and the bit bandwidth is 5 Hz. The results indicate that when the interference frequency offset is an integer multiple of the bit bandwidth, the theoretical and simulated BERs of the short-code system align closely, with the experimental BER slightly higher. The overall trend of change for the three BERs remains consistent, which confirms the accuracy of the proposed BER calculation formula for the short-code system. However, when the interference frequency offset is not an integer multiple of the bit bandwidth, the simulated BER of the short-code system exhibits periodic oscillation as the frequency offset increases. The period of oscillation is approximately twice the bit bandwidth, consistent with the minimum sinusoidal fluctuation period of the interference sequence as defined in Equation (9). This finding validates the correctness of the theoretical derivation presented in Section 2. Furthermore, comparing the BER curves of the long-code system and the short-code system, it is evident that the BER of the long-code system decreases gradually with increasing interference frequency offset. The impact on system performance is most significant when the interference is directed at the carrier frequency of the signal. On the other hand, the BER of the short-code system experiences drastic fluctuations with small changes in the interference frequency offset. At zero frequency offset, the BER of the short-code system is  
Figure 4: BER of the system in the presence of single-frequency interference at the same frequency as the signal carrier, and with additive white Gaussian noise.