## 3 Joint System Implementation
### Joint Signal Design  
OFDM chirp waveforms have been proposed in previous work [29; 30; 31]. Their basic principle is the separate modulation of the chirp signals onto different subcarrier groups. Since the OFDM signal subcarriers are orthogonal to each other, the signals on different subcarrier groups naturally satisfy orthogonality, so they can be separated at the transmitter and receiver ends, thus achieving multiple equivalents in the azimuthal direction. This allows the separation of the signals at the transmitter and receiver ends to form multiple equivalent phase centers in the azimuthal direction for high-resolution wide mapping band imaging. Figure 2: Comparison of three subcarrier prototype filters. In this paper, to achieve the jamming function, we modulate the FBMC signal with a chirp signal, i.e., FBMC chirp signal. We propose an FBMC chirp signal with a generation matrix:  
\[\mathbf{S}= \begin{bmatrix}\mathbf{s}(0,0)&\mathbf{s}(0,1)&\cdots&\mathbf{s}(0,M -1)\\ \mathbf{s}(1,0)&\mathbf{s}(1,1)&\cdots&\mathbf{s}(1,M-1)\\ \mathbf{s}(2,0)&\mathbf{s}(2,1)&\cdots&\mathbf{s}(2,M-1)\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{s}(M-1,0)&\mathbf{s}(M-1,1)&\cdots&\mathbf{s}(M-1,M-1)\end{bmatrix}  \tag{4}\]  
where \(\mathbf{S}\) is the signal generation matrix, \(M\) denotes the number of quadrature signals, \(\mathbf{s}(n,m)=\exp\left\{\dot{\gamma}\pi k_{r}t^{2}\right\}\times\exp\left\{ n\cdot\frac{m\cdot\dot{\gamma}\pi}{M}\right\}\), \(t\in[0,T_{r}]\), \(n\) denotes the row index of the matrix, \(m\) denotes the column index of the matrix, \(T_{r}\) is the pulse width of the LFM signal, and \(k_{r}\) is the modulation frequency. When \(M\) quadrature signals are generated, the pulse width of the signal will be expanded to \(M\) times the original one. When the same LFM signal is modulated on each set of subcarriers, the number of false targets in a single pulse period can be increased to achieve multi-false-target jamming. Taking the two orthogonal joint communication jamming signals as an example, the signal generation matrix is  
\[\mathbf{S}= \begin{bmatrix}\mathbf{s}_{1}&\mathbf{s}_{1}\\ \mathbf{s}_{2}\cdot e^{\frac{0.2\pi}{2}}&\mathbf{s}_{2}\cdot e^{\frac{1.2\pi }{2}}\end{bmatrix}  \tag{5}\]  
where \(\mathbf{s}_{1}\) is the communication signal to be modulated; \(\mathbf{s}_{1}(t)=\sum\limits_{i=1}^{N}c_{i}e^{i2\pi f_{i}t}\), where \(N\) represents the number of subcarriers, \(c_{i}\) represents the sub-band modulation weight, \(C=\{c_{i}\}\) is the pseudo-random sequence of information encoding, and \(f_{i}\) is the \(i\)th sub-band carrier frequency. \(\mathbf{s}_{2}\) is the designed multi-false-target jamming signal. Jamming designs for radar can be divided into three categories [32; 33; 34]: non-coherent signal (e.g., noise jamming), fully coherent signal (e.g., forwarding jamming), and semi-coherent signal (e.g., frequency shift jamming). In this paper, we achieve the multi-false-target jamming mode with semi-coherent jamming by frequency shifting. Assume the radar signal intercepted by the joint communication jamming system is a general chirp (also called LFM) signal \(\mathbf{J}(t)\),  
\[\mathbf{J}(t)=e^{j\pi k_{r}t^{2}},\ -T_{r}/2\leq t\leq T_{r}/2 \tag{6}\]  
where \(T_{r}\) is the pulse width and \(k_{r}\) is the modulation slope. The intercepted radar signal is frequency-shifted and forwarded by the joint communication jamming system, which can be expressed as \(\mathbf{J}_{1}(t)=e^{j\pi(k_{r}t^{2}+2 gt)}\), where \(\xi\) is the frequency shift of the jamming signal. Figure 3: Comparison of time and frequency representations between FBMC-OQAM and FBMC-QAM waveforms: (**a**) FBMC-OQAM; (**b**) FBMC-QAM. The intercepted radar signal \(\mathbf{J}(t)\) is truncated after a delay time \(\tau\), thus introducing the frequency-shifted term related to \(k_{r}\)  
\[\mathbf{J}(t-n\tau)=e^{j\pi k_{r}t^{2}}\cdot e^{-j2\pi nk_{r}\tau t}\cdot e^{j \pi k_{r}(n\tau)^{2}} \tag{7}\]  
where \(n\geq 2\) and \(n\) is a positive integer.