# Joint Transmit Waveform and Receive Mismatched Filter Design to Suppress Range Sidelobe
## 2 Design of NLFM-CPM Integrated Waveform and Signal Processing Methods  
The selection of the waveform plays a crucial role in shaping the performance of the echo signal across both range and Doppler dimensions. Conversely, the communication modulation technique influences the transmission speed and resilience of the communication data. In radar-communication integrated systems, when range resolution is not the primary concern, LFM signals are typically used as the radar baseband signal, with communication modulation applied on top. After analyzing the ambiguity functions of different radar waveforms and communication modulation methods, we ultimately selected NLFM signals as the radar baseband signal and combined them with CPM modulation to achieve communication functionality. Compared to LFM, NLFM achieves lower range sidelobes by optimizing spectral characteristics, which helps integrate communication functionality with minimal impact on radar performance, enabling efficient resource sharing. The CPM modulation method can control the stability of communication transmission by adjusting multiple modulation parameters. The formulation for the NLFM-CPM transmitted signal model is presented as follows:  
\[s(t)=G\bigg{(}t-\frac{T_{p}}{2},T_{p}\bigg{)}\cdot\exp[j(\varphi(t)+\phi(t;I)+ \phi_{0})] \tag{1}\]  
\[G\big{(}t,T_{p}\big{)}= \begin{cases}1,|t|\leq\frac{T_{p}}{2}\\ 0,|t|>\frac{T_{p}}{2}\end{cases}  \tag{2}\]  
where \(G\big{(}t,T_{p}\big{)}\) is the envelope of the signal, \(T_{p}\) is the signal width, \(\varphi(t)\) is the phase function, and \(\phi(t,I)\) denotes communication continuous phase modulation signal. \(\phi(t,I)\) can be further written as  
\[\phi(t,I)=\sum_{n=0}^{N_{c}-1}\text{rect}\big{(}\big{(}t-nT_{p}\big{)}/T_{p} \big{)}\phi(t,I_{n}) \tag{3}\]  
where \(N_{c}\) is number of symbols, rect is the rectangular function, \(\phi(t,I_{n})\) is the phase value of the \(n\)-th symbol. \(\phi(t,I_{n})\) is given as  
\[\phi(t,I_{n})=\pi h\sum_{m=-\infty}^{n-L_{c}}I_{m}+2\pi h\sum_{m=n-L_{c}+1}^{n }I_{m}q\big{(}t-mT_{p}\big{)},nT_{p}\leq t<(n+1)T_{p} \tag{4}\]  
where \(L_{c}\) is the partial response length, \(I_{m}\) is the symbol sequence to be modulated, \(I_{m}\in\left\{\pm 1,\pm 3,\ldots,\pm(M-1)\right\}\), \(M\) is the modulation order, \(h\) is the modulation index, \(q(t)\) is obtained by integrating the impulse function \(g(t)\). \[q(t)= \begin{cases}0,&t<0\\ \int_{0}^{t}g(\tau)d\tau,&0\leq t\leq L_{c}T_{s}\\ 1/2,&t>L_{c}T_{s}\end{cases}  \tag{5}\]  
where the raised cosine (RC) pulse \(g(t)\) is  
\[g(t)=\frac{1}{2L_{c}T_{p}}\bigg{[}1-\cos\bigg{(}\frac{2\pi t}{L_{c}T_{p}} \bigg{)}\bigg{]},\quad 0\leq t\leq L_{c}T_{p} \tag{6}\]  
The RC pulse is molded using a raised cosine function, which ensures smooth transitions and minimal spectral dispersion. Additionally, the RC pulse demonstrates robustresilience against multipath fading and interference, rendering it ideal for high-speed data transmission with a low bit error rate and high transmission efficiency. When \(L_{c}=1\), CPM signals are classified as full response signals. When \(L_{c}>1\), CPM signals are categorized as partial response signals. The pulse repetition interval (PRI) for the transmitted signal is designated as \(T\), and it must adhere to  
\[T\geq 2T_{p} \tag{7}\]  
to avoid pulse overlap and ensure accurate signal processing [27]. The total number of samples per pulse is given by  
\[N=T\cdot f_{s} \tag{8}\]  
where \(f_{s}\) is the sampling frequency. The discrete form of \(s(t)\) is  
\[s(n)=s(t)\big{|}_{t=\frac{n}{f_{s}}},n=0,1,\ldots,N-1. \tag{9}\]  
Consider multiple scattering points, where the target is represented by \(b\) and the strong interference by \(k\). The distance from the radar to the target is represented by \(R_{b}\), and the distance from the radar to the intense interference is denoted as \(R_{k}\).