## 4 Implementation of Joint Plans
### Joint Signal Processing Flow
Following the joint signal generation principle outlined in Section 4.2, the UFMC chirp spectrum is obtained by up-sampling the spectrum of UFMC, resulting in the change in the number of subcarriers in each sub-band from \(c\) to \(2c\). \(\mathbf{H}\) refers to the frequency response of the multipath channel, and \(\mathbf{X}\) is the frequency-domain representation of the transmitted joint signal, determined by Equation (17). \(\mathbf{Z}\) is the frequency-domain form of additive Gaussian white noise. The matrix \(\mathbf{Q}\) can be expressed as:  
\[\mathbf{y}(t)=\mathbf{h}\mathbf{s}(t)+\mathbf{z}(t) \tag{18}\]1  
where \(\mathbf{V}\) is the FFT transformation matrix, the conjugate transpose of \(\mathbf{V}\) is denoted as \(\mathbf{V}^{H},\mathbf{F}\) is the FIR matched filter at the receiving end, and the conjugate transpose of \(\mathbf{F}\) is denoted as \(\mathbf{F}\). After the communication system is designed, the value of \(\mathbf{Q}\) can be precisely determined. Therefore, in Equation (20), when noise influence is not considered, it is only necessary to obtain the value of \(\mathbf{H}\) to demodulate the communication information. Due to the up-sampling process in UFMC chirp, which shortens the original subcarrier spacing by half, two discrete frequency components are generated at fixed intervals. The two frequency components at fixed intervals have a strong correlation with the frequency selectivity of the channel. It can be approximated that these two channels are the same:  
\[\mathbf{y}(t)=\mathbf{h}\mathbf{s}(t)+\mathbf{z}(t) \tag{18}\]2  
where \(m=2k,k\in\{0,1,\cdots,c-1\}\). As per Equation (17), the frequency-domain information structure of the joint signal sent is intertwined with both communication information and jamming information. Therefore, Equation (26) can be decomposed into odd and even terms:  
\[\mathbf{y}(t)=\mathbf{h}\mathbf{s}(t)+\mathbf{z}(t) \tag{18}\]3  
\[\mathbf{y}(t)=\mathbf{h}\mathbf{s}(t)+\mathbf{z}(t) \tag{18}\]4  
where \(\mathbf{X}_{p}^{even}\) is the communication information and \(\mathbf{X}_{p}^{odd}\) is the jamming signal resulting from frequency-shifted retransmission, with its value given by Equation (14). Since the frequency spectrum structure of the LFM signal is fixed, \(\mathbf{X}_{p}^{old}\) can be considered as a pilot signal suitable for channel estimation. The received part of \(\mathbf{X}_{p}^{old}\) can be expressed as:  
\[\mathbf{y}(t)=\mathbf{h}\mathbf{s}(t)+\mathbf{z}(t) \tag{18}\]5  
where \(\mathbf{H}_{p}^{even}\) is the corresponding channel frequency response, \(\mathbf{Q}_{p}^{even}\) is a known term derived from \(\mathbf{Q}\), and \(\mathbf{Z}_{p}^{even}\) is Gaussian white noise. Channel estimation can be obtained through the least squares (LS) method, resulting in an estimated channel frequency response:  
\[\mathbf{y}(t)=\mathbf{h}\mathbf{s}(t)+\mathbf{z}(t) \tag{18}\]6  
Based on the result from Equation (28), the corresponding channel frequency response for the transmission of communication information is obtained as:  
\[\mathbf{y}(t)=\mathbf{h}\mathbf{s}(t)+\mathbf{z}(t) \tag{18}\]7  
By reversing Equation (33), the overall multipath channel frequency response for the received signal is obtained as \(\mathbf{\hat{H}}\). The channel can be equalized to recover the communication information and achieve the communication target. At the radar receiver, this signal is considered as the reflected radar signal originating from the target but with superimposed jamming signals. Consequently, standard radar processing procedures are applied, employing moving target detection (MTD) technology to assess target motion. From the perspective of jamming, maintaining consistency in the jamming signal within each pulse repetition period is essential to obtain coherent processing gain. In addition, during the radar power detection process, the detection threshold should be controlled to maximize detection power.