## 5 System Constraints
### Design of Frequency Modulation Slope \(k_{r}\)  
According to Equation (8), the finely tuned signal with frequency modulation slope can be expressed as:  
\[\widetilde{J}(t)=rect(\frac{t}{T_{p}})e^{j\pi(k_{r}+\Delta k_{r})t^{2}} \tag{34}\]where \(\Delta k_{r}\) is the frequency modulation slope mismatch value. The frequency-domain expression of \(\widetilde{J}(t)\) can be derived from the method of stationary phase as:  
\[\widetilde{J}(f)\approx\frac{1}{\sqrt{k_{r}+\Delta k_{r}}}\text{rect}\Bigg{(} \frac{f}{\left((k_{r}+\Delta k_{r})T_{p}\right)}\Bigg{)}\cdot e^{\frac{i\pi}{4} }\cdot e^{-j\pi\frac{f^{2}}{(k_{r}+\Delta k_{r})}} \tag{35}\]  
The frequency-domain matched filtering of Equation (35) can be expressed as:  
\[\widetilde{J}_{\text{out}}(f)=\widetilde{J}(f)H(f) \tag{36}\]  
where \(H(f)\) is the frequency-domain representation of the matched filtering reference signal:  
\[H(f)\approx\frac{1}{\sqrt{k_{r}}}\text{rect}\bigg{(}\frac{f}{k_{r}T_{p}} \bigg{)}\cdot e^{-\frac{i\pi}{4}}\cdot e^{j\pi\frac{f^{2}}{k_{r}}} \tag{37}\]  
Substituting Equations (35) and (37) into Equation (36), the frequency-domain output of the matched filter is given by:  
\[\widetilde{J}_{\text{out}}(f)\approx\frac{1}{\sqrt{k_{r}(k_{r}+\Delta k_{r})} }\text{rect}\bigg{(}\frac{f}{k_{r}T_{p}}\bigg{)}\cdot\text{rect}\bigg{(} \frac{f}{(k_{r}+\Delta k_{r})T_{p}}\bigg{)}\cdot e^{(j\pi f^{2}\cdot\frac{ \Delta k_{r}}{k_{r}(k_{r}+\Delta k_{r})})} \tag{38}\]  
According to Equation (38), the positive and negative signs will yield different effects on \(\widetilde{J}_{\text{out}}(f)\). When \(\Delta k_{r}>0\), conducting the inverse Fourier transform on \(\widetilde{J}_{\text{out}}(f)\) produces the time-domain expression after matched filtering:  
\[\widetilde{J}_{\text{out}}(t)=\frac{1}{\sqrt{\Delta k_{r}}}\text{rect}\bigg{(} \frac{t(k_{r}+\Delta k_{r})}{\Delta k_{r}\cdot T_{p}}\bigg{)}\cdot e^{j\frac{ \pi}{4}}\cdot e^{-j\pi k_{r}t^{2}\frac{(k_{r}+\Delta k_{r})}{\Delta k_{r}}} \tag{39}\]  
When \(\Delta k_{r}<0\), Equation (39) becomes:  
\[\widetilde{J}_{\text{out}}(t)=\frac{1}{\sqrt{|\Delta k_{r}|}}\text{rect} \bigg{(}\frac{k_{r}t}{|\Delta k_{r}|\cdot T_{p}}\bigg{)}\cdot e^{j\frac{\pi} {4}}\cdot e^{-j\pi k_{r}t^{2}\frac{(k_{r}+\Delta k_{r})}{\Delta k_{r}}} \tag{40}\]  
From the above derivation, it can be concluded that the pulse compression output of frequency modulation slope mismatch jamming is no longer in the form of a \(\sin c\) function, but approximates a rectangle with a width of:  
\[\chi(\Delta k_{r})=\frac{|\Delta k_{r}|\cdot T_{p}}{\text{max}\{k_{r}+\Delta k _{r},k_{r}\}} \tag{41}\]  
In Equation (41), the width of the output rectangle is related to the frequency modulation mismatch slope. Selecting two mismatched frequency modulation slopes \(\Delta k_{r1}=-0.0400k\), and \(\Delta k_{r2}=0.0417k_{r}\), corresponding to \(k_{r1}=0.9600k_{r}\) and \(k_{r2}=1.0417k_{r}\), respectively, the pulse compression results are illustrated in Figure 6. The figure reveals that a mismatch in the frequency modulation slope leads to a sharp decline in the amplitude of the pulse compression result, accompanied by a significant widening of the time width. The analysis of peak output and time delay expansion for the above chirp rates is shown in Figure 7. In the figure, the black dashed line represents the normalized peak value and effective time width of the pulse compression output of the chirp rate mismatched signal when the bandwidth changes with a time width of 50 \(\upmu\)s. When the bandwidth is 10 MHz and the time width is 50 \(\upmu\)s, the signal is in a matched state. Note that, under these conditions, the peak value of the pulse compression is highest, and the effective time width is minimal, forming an effective main lobe peak. From a jamming perspective, the selected jamming pattern involves the generation of multiple false targets, which requires the chirp jamming signal to be in phase with the radar signal.