## 2 Design Methodology for Integrated Signal  
### OFDM Chirp Signal  
To improve bandwidth efficiency, extending the Nyquist criterion from single-carrier to multi-carrier communication leads to the generation of overlapping orthogonal subcarrier signals in the frequency domain, known as OFDM [20]. Since the signals between different subcarriers of OFDM satisfy orthogonality, its odd and even carriers can be utilized to modulate radar and communication signals, respectively [19; 21; 22]. Figure 1 shows the frequency domain subcarrier assignments (left) and the corresponding time-frequency plots (right) for each orthogonal waveform [22]. Figure 2 represents a schematic diagram of the integrated radar and communication signal waveform, where x_com is the communication message modulated by \(16-QAM\) and \(x\_rad\) is the chirp information sampled in the frequency domain. Let \(X_{l}[2K+1]\) denote the \((2k+1)\)-_th_ radar transmission symbol sent on the \(l\)-_th_ subcarrier and \(X_{l}[2K]\) represent the \(2k\)-_th_ communication transmission symbol sent on the \(l\)-_th_ subcarrier, where \(l=0,1,2,\cdots,\infty,k=0,1,2,\cdots,N-1\). The transmission time of \(N\) symbols is expanded to the duration of a single symbol, i.e., \(T_{sym}=NT_{s}\), the integrated signal on the \(k\)-_th_ subcarrier is given by the following:  
\[\Psi_{l,k}(t)=\left\{ \begin{array}{l}e^{l2\pi f_{l}(t-T_{sym})},0<t\leqslant T _{sym}\\ 0,\hskip 56.905512 pti\textit{else}\end{array} \right. \tag{1}\]  
The time-continuous passband radar and communication signals are denoted, respectively [19; 20], as follows:  
\[\left\{ \begin{array}{l}x_{rad}(t)=Re\bigg{\{}\frac{1}{T_{sym}}\sum\limits_{ l=0}^{\infty}\bigg{\{}\sum\limits_{k=0}^{N-1}X_{l}[2k+1]\Psi_{l,2k+1}(t)\bigg{\}} \bigg{\}}\\ x_{com}(t)=Re\bigg{\{}\frac{1}{T_{sym}}\sum\limits_{l=0}^{\infty}\bigg{\{} \sum\limits_{k=0}^{N-1}X_{l}[2k]\Psi_{l,2k}(t)\bigg{\}}\bigg{\}}\end{array} \right. \tag{2}\]  
Figure 1: The subcarrier distribution diagram of the chirp signal and the time–frequency plot of the modulated signal in the time domain. Figure 2: Schematic diagram of OFDM-Chirp integrated signal. The baseband signals for radar and communication are denoted, respectively, as follows:  
\[\left\{ \begin{array}{l}x_{rad}(t)=\sum\limits_{l=0}^{\infty}\sum\limits_{k=0}^{N -1}X_{l}[2k+1]e^{j2\pi t/T_{sym}(t-IT_{sym})}\\ \\ x_{com}(t)=\sum\limits_{l=0}^{\infty}\sum\limits_{k=0}^{N-1}X_{l}[2k]e^{j2\pi t /T_{sym}(t-IT_{sym})}\end{array} \right. \tag{3}\]  
Sampling the integration signal \(x(t)=x_{rad}(t)+x_{com}(t)\) at \(t=IT_{sym}+nT_{s}\), \(T_{s}=T_{sym}/N\), \(f_{k}=K/T_{sym}\) yields the corresponding discrete-time integration symbols:  
\[x_{l}[n]=\sum\limits_{k=0}^{N-1}X_{l}[k]e^{j2\pi kt/N},n=0,1,\cdots,N-1 \tag{4}\]  
Since the multipath channel generates inter-symbol interference (ISI) to this integrated symbol, we insert a guard interval between two consecutive symbols. In this paper, we utilize the CP to achieve cyclic expansion of the integrated symbol. For the second subcarrier signal with a delay of \(t_{0}+T_{s}\), the following equation is satisfied:  
\[\frac{1}{T_{sub}}\int_{0}^{T_{sub}}e^{j2\pi f_{k}(t-t_{0})}e^{-j2\pi f_{k}(t-t _{0}-T_{s})}dt=0,k\
eq i \tag{5}\]  
where \(T_{sub}=NT_{s}=1/\Delta f\), orthogonality between subcarriers can be maintained when the length of the CP is greater than the maximum delay of the multipath channel. Additionally, since each subcarrier signal on each symbol is time-limited, the integrated signal will generate out-of-band radiation, leading to adjacent channel interference (ACI). To minimize out-of-band radiation, it is necessary to allocate guard bandwidth on the outermost subcarriers, i.e., using Hanning windows and VC. Under the condition that the transmit samples and channel samples satisfy the circular convolution, the integrated signal can be obtained at the receiver as follows [20]:  
\[