## 5 System Constraints
### Minimum Detectable Signal-to-Noise Ratio and Matched Gain Ratio  
To enhance detection performance in a multi-target environment, two widely employed CFAR detection methods are the average CFAR for cluttered environments and the ordered statistic CFAR designed for scenarios with multiple targets. In general, average CFAR performs well in backgrounds with uniform noise and clutter distribution. However, its performance may diminish significantly when multiple false targets are distributed on both sides of the target detection unit.  
Building on prior detection information and assuming knowledge of the false alarm probability and detection probability for ordered statistic CFAR, the signal-to-noise ratio (SNR) of the target echo pulse, after coherent accumulation and matched filtering, is derived in ordered statistic CFAR detection under the condition of no jamming:  
\[SNR=\frac{P_{t}G_{t}G_{r}\lambda^{2}\sigma}{(4\pi)^{3}R^{4}kT_{0}BFL} \tag{43}\]  
Figure 6: The pulse compression results in different frequency modulation slope ratios.  
Figure 7: Impact of effective pulse width on peak output of pulse compression for chirp rate mismatched signals: (**a**) peak pulse compression results under different frequency modulation slopes and bandwidth conditions; (**b**) pulse width results of pulse compression under different time widths and bandwidth conditions.  
The variables in the equation represent the following parameters: \(P_{t}\) is the radar transmission power, \(G_{t}\) is the radar transmit antenna gain, \(G_{r}\) is the radar receive antenna gain, \(\lambda\) is the signal wavelength, \(\sigma\) is the target scattering cross-section, \(R\) is the radial distance between the radar and the target, \(k\) is the Boltzmann constant, \(T_{0}\) is the effective noise temperature, \(B\) is the signal bandwidth, \(F\) is the receiver noise figure, and \(L\) accounts for losses such as atmospheric losses and losses in the radar feedline.  
The SNR for the joint radar jamming and communication integration system, assuming it is located in the direction of the transmitted jamming signal from the radar, can be expressed as:  
\[J_{SNR}=\alpha_{i}^{2}\frac{P_{jt}G_{j}^{2}\lambda^{2}\sigma}{(4\pi)^{2}R_{j} ^{2}kT_{0}BFL_{j}} \tag{44}\]  
where \(\alpha_{i}\) represents the loss ratio of the matching filter gain between the secondary frequency-shift jamming and the target echo, \(P_{jt}\) is the integrated signal transmission power of the joint system, \(G_{j}\) is the jamming antenna gain of the joint system, and \(L_{j}\) is the atmospheric loss and losses in the joint system. Based on Equation (12), the ratio of the jamming signal's frequency shift \(n(n-1)\tau k_{r}\) to the frequency slope \(k_{r}\) is dependent on \(n\) and \(\tau\). Therefore, \(\alpha_{i}\) can be expressed as:  
\[\alpha_{i}=1-\frac{n((n-i+1)!-(n-i)!)\tau}{T_{r}} \tag{45}\]  
In the CFAR detection of the jammed radar system, given the false alarm probability \(P_{fa}\), minimum detectable probability \(P_{dmin}\), and radar-related parameters, the minimum SNR \(J_{SNR}\) required for the joint system can be determined. Substituting the obtained \(J_{SNR}\) into Equation (44) yields the corresponding \(\alpha_{i}\) for the joint system.  
As the jammed radar approaches the protected target, the radial distance \(R\) between the radar and the target decreases. Figure 8 presents the joint system's minimum required jamming SNR \(J_{SNR\text{min}}\) in this scenario and the minimum matching gain ratio \(\alpha_{\text{imin}}\) for different positions. As shown in the figure, as \(R\) decreases, the values for \(J_{SNR\text{min}}\) and \(\alpha_{\text{imin}}\) both increase. This trend arises because as the distance decreases, the jammed radar receives stronger jamming signals, leading to an increase in SNR. The heightened SNR, in turn, results in an increase in the target echo-matched filter gain-loss ratio.