## 4 Implementation of Joint Plans
### Joint Signal Design
 ^{T}\), and \(s_{p}[k]\) is the pseudo-random sequence encoding the information carried by the \(kth\) subcarrier in the \(pth\) sub-band. \(\mathbf{s}_{j}(t)=\mathbf{V}^{H} \begin{bmatrix}\left(\mathbf{J}_{1}\right)^{T },\left(\mathbf{J}_{2}\right)^{T},\cdots,\left(\mathbf{J}_{N_{Rb}}\right)^{T} \end{bmatrix} \) is the designed jamming signal, such that \(\mathbf{J}_{p}\) is the frequency-domain representation of the jamming signal in the \(pth\) sub-band. The radar signal intercepted by the integrated device being jammed is given by the expression:  
\[J(t)=e^{j\pi k_{r}t^{2}},-\frac{T_{p}}{2}\leq t\leq\frac{T_{p}}{2} \tag{8}\]  
The integrated device induces jamming effects on the intercepted radar signal through frequency shifting and retransmission, resulting in the transmitted jamming signal \(J_{1}(t)=e^{j\pi(k_{r}t^{2}+2 gt)}\), such that \(\xi\) represents the frequency shift amount. Consequently, the result of fixed frequency-shift jamming \(J_{1}(t)\) after undergoing time-domain matched filtering can be expressed as:  
\[ \begin{array}{ll}s_{out}(t)&=J_{1}(t)\otimes J^{*}(-t)\\ &=\int\limits_{-\infty}^{\infty}J_{1}(\tau)\cdot J^{*}(t-\tau)d\tau\\ &=rect\Big{(}1-\frac{|t|}{T_{p}}\Big{)}\sin c\Big{[}\pi k_{r}T_{p}\Big{(}t+ \frac{\xi}{k_{r}}\Big{)}\Big{(}1-\frac{|t|}{T_{p}}\Big{)}\Big{]}\cdot e^{j \pi gt}\end{array}  \tag{9}\]  
where \(\otimes\) represents convolution, \(*\) denotes the conjugate operation, and \(rect(\cdot)\) is a rectangular window function. Based on Equation (9), the maximum peak in the output after signal-matched filtering occurs at \(-\xi/k_{r}\), and this peak is generated by the offset \(\xi/k_{r}\) from the true signal peak moment. When there is a change in the frequency modulation slope (\(k_{r}\)) between radar pulses, false targets exhibit range jumps, preventing them from accumulating at the same distance. If the intercepted radar signal is delayed by time \(\tau\) and then truncated, the result can be expressed as:  
\[J(t-n\tau)=e^{j\pi k_{r}t^{2}}\cdot e^{-j2n\pi k_{r}\tau}\cdot e^{j\pi k_{r} (n\tau)^{2}} \tag{10}\]where \(n\) is a positive integer, \(n\geq 2\). From Equation (10), the second term (\(e^{-\jmath 2n\pi k_{r}\tau t}\)) can form the frequency shift term of the jamming signal by letting \(-nk_{r}\tau=\overline{\xi}\). Moreover, as long as the values of \(n\) and \(\tau\) are fixed, the ratio of \(\overline{\xi}\) to \(k_{r}\) remains constant, i.e., \(\overline{\xi}/k_{r}=-n\tau\). This allows \(\overline{\xi}\) to vary with the change in \(k_{r}\), enabling false targets to accumulate at the same distance. At this point, as long as the term \(e^{\jmath\pi k_{r}(n\tau)^{2}}\) is eliminated, the remaining terms, \(J(t-n\tau)\), can be used as the jamming signal for transmission. Note that Equation (10) can be transformed as follows:  
\[ \begin{split}& J(t-n\tau)\cdot\left(J^{*}(t-\tau)\right)^{n^{2}} \\ &=e^{\jmath\pi k_{r}t^{2}}\cdot e^{-\jmath 2n\pi k_{r}\tau t}\cdot e^{\jmath\pi k_{r}(n\tau)^{2}}\cdot\left(e^{-\jmath \pi k_{r}t^{2}}\cdot e^{\jmath 2\pi k_{r}\tau t}\cdot e^{-\jmath\pi k_{r}\tau^{2}} \right)^{n^{2}}\\ &=e^{-\jmath\pi k_{r}t^{2}(n^{2}-1)}\cdot e^{\jmath 2n\pi(n-1)\pi k_{r}t}\\ &=\left(J^{*}(t)\right)^{n^{2}-1}\cdot e^{\jmath 2n\pi(n-1)\pi k_{r}t} \end{split}  \tag{11}\]  
Multiplying both sides of Equation (11) by \(\left(J(t)\right)^{n^{2}}\) yields the original time-domain expression of the jamming signal:  
\[s_{I}^{\prime}(t)=J(t-n\tau)\cdot\left(J^{*}(t-\tau)\right)^{n^{2}}\cdot\left( J(t)\right)^{n^{2}}=J(t)\cdot e^{\jmath 2n(n-1)\pi k_{r}t} \tag{12}\]  
where \(-T_{p}/2+n\tau\leq t\leq T_{p}/2\). When comparing the conventional frequency-shifted retransmitted jamming signal \(J(t)\) with the expression of \(s_{I}(t)\) in Equation (12), the frequency shift amount of the generated jamming signal is \(\xi_{I}=n(n-1)\tau k_{r}\). The ratio of this frequency shift to the frequency modulation slope \(k_{r}\) depends on the values of \(n\) and \(\tau\).