# Joint Transmit Waveform and Receive Mismatched Filter Design to Suppress Range Sidelobe
## 4 Numerical Simulations
In Figure 4, we compared the five highest peaks obtained after pulse compression processing, and the results indicate varying levels of signal loss caused by different template envelopes. The observed phenomenon arises from the different strategies used by traditional sidelobe suppression methods and the V-shaped filter in handling sidelobes. Traditional methods typically reduce sidelobes by smoothing the signal envelope, but they often have limited effectiveness in suppressing the first sidelobe, especially in the presence  
Figure 3: The different reference template sidelobe shapes. Figure 2: (**a**): With \(M=2\) and \(h=0.5\), the correlation between SNR and BER across various frequency pulse formulations. (**b**): With \(M=2\) and \(h=0.5\), the correlation between SNR and BER for differing partial response lengths. of strong clutter. Strong clutter can cause the first sidelobe to be prominent, making it difficult for conventional methods to effectively suppress it. In contrast, the V-shaped filter focuses on reducing sidelobes near strong clutter. By optimizing the frequency response for specific frequency bands, it significantly reduces the sidelobes close to strong clutter. However, this comes at the cost of increasing distant sidelobes, which can lead to more interference in other regions. While the V-shaped filter effectively detects weak targets near strong clutter, its unbalanced sidelobe suppression inevitably results in a loss of pulse compression gain, reflected in the 0.86 dB decrease in compression gain. Figure 5a-c shows the results of employing the initial algorithm with an explicit solution, based on the Barker code, P3 code, and P4 code, respectively. As the main sidelobe level over a specified region (MSLSR) decreases, the SNR loss increases. The outcomes of the iterative algorithm, grounded in the LFM, NLFM, and NLFM-CPM waveforms, are illustrated in Figure 5d-f, respectively. Owing to minor variations in the iterative depth, the curves exhibited in (d)-(f) are less smooth relative to those of the phase code, but the overall trends persist unchanged. We selected all sidelobes outside the 3 dB mainlobe width as the range for calculating the MSLSR, with \(R_{x}=60\) and \(R_{y}=70\). The minimum MSLSR is established at \(-80\) dB, and the MSLSR attainable for mitigating all sidelobes using the NLFM-CPM waveform is approximately \(-80\) dB. The observed phenomenon results from the balance between sidelobe suppression and SNR loss across different waveform structures and algorithm optimization strategies. Phase-coded waveforms like Barker, P3, and P4 exhibit strong sidelobe suppression but lead to higher SNR losses, whereas frequency-modulated waveforms such as LFM, NLFM, and NLFM-CPM demonstrate greater sidelobe suppression under iterative algorithms, albeit with slightly reduced curve smoothness. Ultimately, the NLFM-CPM waveform achieves sidelobe suppression down to \(-80\) dB outside the 3 dB mainlobe width while maintaining relatively low SNR loss. Figure 4: Different filter envelopes for the normalized peak loss obtained by the EWMF method. Figure 6 shows the normalized amplitude of the first sidelobe's peak and trough values as a function of the cavity depth of the reference template. The red curve represents the normalized peak value of the first sidelobe, which remains relatively stable around \(-40\) dB across different cavity depths until it increases sharply near 70 dB. This stability suggests that the peak of the first sidelobe is not significantly affected by the cavity depth in the lower range but begins to rise as the depth approaches 70 dB, indicating a weakening in sidelobe suppression effectiveness at this depth. The blue curve represents the normalized trough value of the mainlobe, which initially decreases as the cavity depth increases, reaching its lowest point around 40 dB, and then gradually rises as the cavity depth continues to increase.