## 3 Joint System Implementation
### Joint Signal Design
Because of the fixed frequency-shift jamming \(\mathbf{J}_{1}(t)\) after time-domain-matched filtering, the result can be calculated using Equation (8):  
\[ \begin{array}{ll}\mathbf{s}_{jout}&=\mathbf{J}_{1}(t)\otimes\mathbf{s}^{*}( -t)\\ &=\text{rect}\Big{(}1-\frac{|t|}{T_{r}}\Big{)}\sin c\Big{[}\pi k_{r}T_{r}\Big{(} t+\frac{\xi}{k_{r}}\Big{)}\Big{(}1-\frac{|t|}{T_{r}}\Big{)}\Big{]}\cdot e^{j\pi \xi t}\end{array}  \tag{8}\]  
where \(*\) is the conjugate operator, \(\otimes\) denotes the convolution operator, and \(\text{rect}(\cdot)\) is a window function. The maximum peak output after matched filtering of the jamming signal is at \(-\frac{\xi}{k_{r}}\), which is shifted by \(\frac{\xi}{k_{r}}\) from the moment of the real signal spike. When the FM slope \(k_{r}\) changes between the radar pulses, the false target will have a distance jump. If we want the false target to accumulate at the same distance, the shift (\(\xi\)) must be changed with the change in FM slope (\(k_{r}\)). Therefore, the \(\exp\Big{\{}j\pi k_{r}(n\tau)^{2}\Big{\}}\) term in Equation (7) needs to be further eliminated so that the first two terms in Equation (7) take the following form:  
\[ \begin{array}{ll}\mathbf{J}(t-n\tau)\cdot\left(\mathbf{J}^{*}(t-\tau) \right)^{n^{2}}\\ &=e^{j\pi k_{r}t^{2}}\cdot e^{-j2\pi nk_{r}\tau t}\cdot e^{j\pi k_{r}(n\tau)^{ 2}}\cdot\left(\mathbf{J}^{*}(t-\tau)\right)^{n^{2}}\\ &=\left(\mathbf{J}^{*}(t)\right)^{n^{2}-1}\cdot e^{j2\pi m(n-1)\tau k_{r}t} \end{array}  \tag{9}\]  
Multiply both sides of Equation (9) by \(\left(\mathbf{J}(t)\right)^{n^{2}}\) to obtain the jamming signal \(\mathbf{s}_{2}(t)\):  
\[\mathbf{s}_{2}(t)=\mathbf{J}(t)\cdot e^{j2\pi n(n-1)\tau k_{r}t} \tag{10}\]  
where \(-T_{r}/2+n\tau\leq t\leq T_{r}/2\). From Equation (10), the ratio of the frequency shift (\(\xi=n(n-1)\tau k_{r}\)) to the modulation frequency (\(k_{r}\)) depends on the values of \(n\) and \(\tau\). The ratio of the shift (\(\xi\)) to the FM frequency (\(k_{r}\)) is a constant when \(n\) and \(\tau\) take certain values in Equation (10). The expressions for the communication signal and the radar jamming signal can be derived from the signal generation matrix:  
\[\mathbf{s}_{com}(t)=\mathbf{s}_{1}(t)\cdot p\Bigg{(}\frac{t}{\frac{\tau}{2}} \Bigg{)}+\mathbf{s}_{1}(t-T/2)\cdot p\Bigg{(}\frac{t-T/2}{\frac{\tau}{2}} \Bigg{)} \tag{11}\]  
\[\mathbf{s}_{jam}(t)=\mathbf{s}_{2}(t)\cdot p\Bigg{(}\frac{t}{\frac{\tau}{2}} \Bigg{)}-\mathbf{s}_{2}(t-T/2)\cdot p\Bigg{(}\frac{t-T/2}{\frac{\tau}{2}} \Bigg{)} \tag{12}\]  
where \(t\) is the time width of one symbol of the joint signal, and \(p(\cdot)\) is the PHYDYAS modulation prototype filter pulse function, received at the receiver end using the demodulation analysis pulse \(p^{H}(t)\) and transposed conjugately with the prototype pulse. The FBMC system modulates the information on different carrier frequencies through this prototype filter. Transforming Equations (11) and (12) to the frequency domain, the frequency domain expression of the two quadrature signals can be obtained as  
\[X_{\text{com}}(2k)=\mathcal{F}\{s_{1}(n)\} \tag{13}\]  
\[X_{\text{jam}}(2k+1)=\mathcal{F}\bigg{\{}s_{2}(n)\cdot\exp\biggl{(}-j\frac{2 \pi}{N}(n-1)\biggr{)}\bigg{\}} \tag{14}\]where \(s_{1}(n)\) is the sampling sequence that denotes the discrete time samples of a complex signal \(\mathbf{s}_{1}(t)\) with the subcarrier length (\(N\)); \(s_{2}(n)\) is the sampling sequence of \(\mathbf{s}_{2}(t)\); \(\mathcal{F}\) represents the discrete Fourier transform operator. The subcarrier interval \(\Delta f\) is \(F_{S}/N\), and \(F_{S}\) is the sampling rate used for the discrete Fourier transform (DFT). Adding the two signals gives a sign of the joint signal, and Figure 4 shows the frequency domain form of the joint signal. The multiplexing of the signal bands is achieved by dividing the available frequency band into several sub-bands, transmitting one signal in each sub-band, modulating the information to be transmitted to each subcarrier, and transmitting it simultaneously.