# Joint Transmit Waveform and Receive Mismatched Filter Design to Suppress Range Sidelobe
## 4 Numerical Simulations  
In this chapter, we validate the effectiveness of the EWMF algorithm through simulation experiments that assess the impact of communication parameters and radar detection performance on the system. We set the pulse duration and bandwidth of the transmitted signal to 20 \(\upmu\)s and 10 MHz in the monostatic system, respectively. The initial waveform uses an NLFM-CPM waveform with binary modulation and an LRC pulse envelope. The maximum number of iterations is 50, with a stopping threshold \(\delta=1\times 10^{-4}\). The initial SINR is \(-45\) dB. The main lobe of strong clutter is at the 65 th range cell, and a weak target at the 55 th range cell is buried by the strong clutter sidelobe. In Figure 2a, the symbol error rate (SER) of different pulse envelopes (Lorentzian pulse, Rectangular pulse, GMSK pulse, Raised Cosine pulse) is compared across various signal-to-noise ratio (SNR) levels. The \"Bound \(M=2\), \(h=0.5\)\" line represents the theoretical lower bound for SER performance, given a 2-phase modulation and a modulation index of 0.5. From the curves in the figure, we can observe that the Raised Cosine pulse has the closest SER performance to the theoretical lower bound, exhibiting the best error rate performance. This is because the Raised Cosine envelope provides good spectral efficiency and noise resistance, which effectively lowers the error rate at a given SNR. Additionally, the smooth shape of the Raised Cosine envelope helps suppress sidelobes in the frequency domain, reducing spectral leakage, which is crucial for minimizing error rates. In contrast, the Rectangular pulse performs poorly at lower SNR levels. The Rectangular pulse has higher sidelobes in the frequency domain, leading to significant spectral leakage and inter-symbol interference, which introduces more errors at low SNR, resulting in a higher SER. In summary, the Raised Cosine pulse achieves the best error rate performance due to its smooth spectrum characteristics, while the Rectangular pulse has worse error rates due to its higher spectral sidelobes. This explains the performance differences in SER among the different pulse envelopes shown in the figure. Figure 3 shows the amplitude response characteristics of four different reference template sidelobe shapes, each with unique features suited for specific applications. The linear envelope remains stable across the filter length without significant amplitude changes, making it ideal for applications that require a uniform sidelobe level in the spectrum. It has a simple design and minimizes ripple effects, though it may be less effective in scenarios requiring precise sidelobe suppression. The concave envelope drops sharply in the middle and quickly recovers on both sides, making it suitable for applications that need strong suppression at the center frequency while allowing higher sidelobes at other frequencies; the sharp dip helps isolate or attenuate unwanted frequencies at the  
filter's center. The arc-shaped envelope shows a gradual, smooth decline, forming an arc curve and balancing sidelobe suppression with a smooth transition, making it suitable for applications requiring smoother transitions. It serves as a compromise between the linear and concave designs. The V-shaped envelope remains stable at both ends and drops sharply in the middle, creating a distinct V shape, which is useful for applications needing strong suppression at the center frequency and stability at the edge frequencies. It is similar to the concave envelope but with sharper transitions, helping to focus suppression in a narrow frequency band while maintaining stability at the edges. In summary, the linear envelope is for uniformity, the concave envelope for center frequency suppression, the arc-shaped envelope for smooth transitions, and the V-shaped envelope for focused center suppression and edge stability.