# Joint Transmit Waveform and Receive Mismatched Filter Design to Suppress Range Sidelobe
## 4 Numerical Simulations
This trend indicates that the trough value is more sensitive to changes in cavity depth, with the minimum value suggesting optimal sidelobe suppression effectiveness. The difference between the two curves--specifically the gap between the peak and trough values--indicates the effectiveness of suppression. The maximum separation between the red and blue curves occurs around 35 dB to 40 dB. In this range, the amplitude of the first sidelobe trough is minimized relative to its peak, achieving optimal suppression. Beyond this range, as the cavity depth increases, the difference between the two curves decreases, implying a reduction in suppression effectiveness. Thus, an optimal cavity depth for the reference template lies within 35 dB to 40 dB, where the balance between minimizing the trough and maintaining a stable peak provides effective sidelobe suppression. Figure 7 shows the impact of code lengths P = 30, P = 40, and P = 50 on the peak clutter sidelobe as SINR varies. The longer the code length or the higher the input SINR, the more likely the mainlobe of the target exceeds the level of the PSL of strong interference. Conversely, under the same PSL, a longer code length requires a lower input SINR. This implies that under identical experimental conditions, a longer code length is more adept at detecting weak targets near strong clutter in the presence of a reference template. Figure 5: LPG for attenuating all range sidelobes derived from the following baseband signals: (a) Barker code, (b) P3 code, (c) P4 code, (d) LFM, (e) NLFM, and (f) NLFM-CPM waveform. In CPM modulation, the Modulation Index \(h\) determines the magnitude of phase variation with each symbol, directly affecting the spectral efficiency and characteristics of the modulation waveform. A smaller \(h\) results in smoother phase transitions, improving bandwidth efficiency, while a larger \(h\) enhances signal distinguishability and improves noise resistance. Therefore, selecting an appropriate \(h\) is crucial for balancing bandwidth usage and bit error rate performance. Additionally, the value of \(h\) also directly impacts the size of LPG. Figure 8 indicates that an increase in the modulation index \(h\) generally leads to higher LPG. However, the curve shows fluctuations at certain points, reflecting a complex nonlinear relationship. This suggests that while the Modulation Index significantly impacts LPG, the relationship is not a simple linear decrease. This is because a larger \(h\) value leads to faster phase changes, which effectively suppress the sidelobe level of the signal, thereby reducing LPG. Figure 6: Curve of strong interference first sidelobe peak and trough values as a function of reference template cavity depth. Figure 7: With different lengths of codes, the PSL after mismatch filtering varies with the change in input SINR. In Figure 9, it illustrates the variation of the objective function (ISL) during the iterative optimization process for different waveforms: P3 code, LFM, and NLFM-CPM. It can be observed that the ISL values for all three waveforms significantly decrease as the number of iterations increases, indicating that the iterative optimization process effectively reduces the ISL. Among them, the NLFM-CPM curve decreases the fastest, demonstrating the quickest convergence rate. Its final ISL value is also the lowest, showcasing its superior sidelobe suppression performance. In contrast, LFM is second-best, while the P3 code has the highest final ISL value, indicating relatively poorer performance. At the initial state, the P3 code exhibits the highest ISL value, while NLFM-CPM has the lowest, suggesting that NLFM-CPM inherently possesses better characteristics. Moreover, during the first 20 iterations, the ISL values for NLFM-CPM and LFM decrease rapidly, whereas the P3 code shows a slower decline. After approximately 50 iterations, the reduction rates for all three waveforms gradually level off.