## 4 Relative Jammer Screening Algorithm Based on Mixed-Strategy Game
### Multidimensional Utility Function Construction  
The factors considered for interference optimisation include three factors: time, power, and frequency. The algorithm needs to map the lost time, lost power, and Doppler shift into the utility function. The utility function is a measure of the overall interference effect or the strength of the strategy. By observing the utility function values, it is possible to understand more intuitively the impact of each primary jammer on the overall interference effect after the game. As shown in Equation (3), in order to fuse the parameters of the different domains into the utility function, the time loss \(\tau_{ip,j_{R}}\), power loss \(W_{ip,j_{R}}\), and Doppler shift loss \(\Delta f_{ip,j_{R}}\) are normalised for the \(i_{P}\)th primary jammer, using the \(i_{R}\)th relative jammer for forwarding. \[\left\{ \begin{array}{l}\tau_{ip,j_{R}}^{\prime}=\frac{\left|\tau_{ip,j_{R}}- \tau_{ip~{}min}\right|}{\left|\tau_{ip~{}max}-\tau_{ip~{}min}\right|}\\ W_{ip,j_{R}}^{\prime}=\frac{W_{ip,j_{R}}-W_{ip~{}min}}{\left|W_{ip~{}max}-W_{ip~ {}min}\right|}\\ \Delta f_{ip,j_{R}}^{\prime}=\frac{\left|\Delta f_{ip,j_{R}}-\Delta f_{ip~{} min}\right|}{\left|\Delta f_{ip~{}max}-\Delta f_{ip~{}min}\right|}\end{array} \right. \tag{3}\]  
where \(\tau_{ip,j_{R}}^{\prime}\) is the normalised forwarding delay, \(\tau_{ip~{}min}\) and \(\tau_{ip~{}max}\) are the minimum forwarding delay and maximum forwarding delay, respectively, for a primary jammer using a relative jammer for spoofing. \(W_{ip,j_{R}}^{\prime}\) is the normalised power loss, \(W_{ip~{}min}\) and \(W_{ip~{}max}\) are the minimum and maximum power loss, respectively, for a primary jammer using a relative jammer for deception. \(\Delta f_{ip,j_{R}}^{\prime}\) is the normalised frequency shift, and \(\Delta f_{ip~{}min}\) and \(\Delta f_{ip~{}max}\) are the minimum and maximum frequency shifts, respectively, for a primary jammer using a relative jammer for spoofing. As shown in Equation (4), the utility function brought about by the ipth primary jammer, using the ieth relative jammer for signal forwarding, is \(P_{ip,j_{R}}\). \[P_{ip,j_{R}}(x_{k}^{ip})=\left(A\cdot\tau_{ip,j_{R}}^{\prime}+B\cdot W_{ip,j_{ R}}^{\prime}+C\cdot\Delta f_{ip,j_{R}}^{\prime}+\frac{D}{\sum_{i_{P}=0}^{N_{p}}x_{k }^{ip}}\right)*x_{i_{R}}^{ip} \tag{4}\]  
where A, B, and C are the time loss, power loss, and frequency loss weighting factors, respectively. \(\left(\text{A}\cdot\tau_{ip,j_{R}}^{\prime}+\text{B}\cdot\text{W}_{ip,j_{R}}^{ \prime}+\text{C}\cdot\Delta f_{ip,j_{R}}^{\prime}\right)*x_{\text{lg}}^{\text{ ipr}}\) is the expenditures resulting from the decisions taken regarding the time, power, and frequency. D denotes the fixed efficacy gain from each relative jammer. If D is small, it results in a negative gain and the search for available relative jammers is not possible. The primary jammers will share equally the gain from a relative jammer, in relation to the overall decision. \(\sum_{i=0}^{N_{p}}x_{\text{lg}}^{\text{ipr}}\) is the number of times a relative jammer has been selected at a given moment. The selection of a particular relative jammer by too many primary jammers can lead to a reduction in \(\frac{D}{\sum_{i_{P}=0}^{N_{p}}x_{i_{R}}^{ip}}\). The total utility function \(P_{ip}\) for each primary interference source is \(P_{ip,j_{R}}\) summed over \(i_{R}\). In scenarios where the real-time requirements of jamming are quite high, the system considers increasing the weight A of the forwarding delay appropriately to increase the impact of time on the jamming effect. The value of A is also related to factors such as propagation path differences, jammer processing delays, and time synchronisation accuracy. In the case of low ambient signal-to-noise ratio, the power factor B is appropriately increased to improve the interference effectiveness. We focus on the effect of the Doppler shift on the specific gravity C, when the target to be jammed is moving rapidly relative to the LEO and the jammer.