## 3 The Processing Method for Integrated Radar-Communication Signals
### Symbol Time Offset and Carrier Frequency Offset  
In this integrated system, the starting point of the integrated signal front end and the FFT window can be inconsistent due to the symbol timing offset (STO). We perform a fast Fourier transform (FFT) on the received samples \(\{x_{l}[n+\delta]\}_{n=0}^{N-1}\) in the time domain, where \(\delta\) is the number of samples. We obtain the received signal in the frequency domain as follows:  
\[Y_{l}=\frac{1}{N}\sum\limits_{n=0}^{N-1}x_{l}[n+\delta]e^{-\beta\tau mk/N}=X_{ l}[k]e^{2\pi k\delta/N} \tag{14}\]  
When there is a phase offset in the received signal, it causes the constellation of the signal to rotate around the origin. To ensure accurate sampling of the transmitted signal, we need to consider symbol timing synchronization. Due to the overlap between the CP and the corresponding data, as Figure 5 shows, when the CP falls within the sliding window, the similarity between the two sampling blocks within the window reaches its maximum. We use the correlation between the two sampling blocks in \(W1\) and \(W2\) and then estimate the STO by maximizing the log-likelihood function. Let \(\rho=SNR/(SNR+1)\)  
Figure 4: BER variation with different multiplicative reduction factors \(\alpha\) under NMSA algorithm. Figure 5: STO estimation technique using double sliding windows. and \(L\) represent the actual number of samples averaged within the window. Then, STO is estimated as follows:  
\[\hat{\delta}_{\mathrm{ML}}=\underset{\delta}{\mathrm{argmax}}\{|\gamma[\delta]|- \rho\phi[\delta]\} \tag{15}\]  
\[\gamma[m]=\sum_{n=m}^{m+L-1}y_{l}[n]y_{t}{{}^{*}}[n+N] \tag{16}\]  
The \(\gamma(\delta)\) and \(\phi(\delta)\) in (15) are shown in (16). Inevitably, carrier signal distortion occurs at the receiving end of the signal, and we also consider the carrier frequency offset (CFO) caused by the Doppler shift \(f_{d}\). Let \(f_{offset}=f_{c}-f_{c^{\prime}}\), where \(f_{c}\) and \(f_{c^{\prime}}\) represent the carrier frequencies of the transmitter and receiver, respectively. The Doppler shift \(f_{d}\) is determined by both the carrier frequency \(f_{c}\) and the speed of the mobile terminal \(v\), i.e., \(f_{d}=v\cdot f_{c}/c\). Define the normalized CFO as the ratio of the CFO to the subcarrier spacing: \(\varepsilon=f_{offset}/\Delta f\). For normalized CFO, it can be decomposed into two parts, i.e., \(\varepsilon=\varepsilon_{i}+\varepsilon_{f}\), where \(\varepsilon_{i}\) represents the integer frequency offset (IFO), and \(\varepsilon_{f}\) represents the fractional frequency offset (FFO). When there is a CFO of size \(\varepsilon\) between the transmitter and the receiver, the received signal is represented as follows:  
\[y_{l}[n]=\frac{1}{N}\sum_{k=0}^{N-1}H[K]X_{L}[K+\delta]e^{2\pi(k+\varepsilon) n/N}+z_{l}[n] \tag{17}\]  
We transform the two symbols \(y_{l}[n]\) and \(y_{l+D}[n]\) into a frequency domain signal \(\{Y_{l}[k]\}_{K=0}^{N-1}\) and \(\{Y_{l+D}[k]\}_{K=0}^{N-1}\) by FFT, which is used to extract the guide frequency [20]. Then, the CFO of the guide frequency estimation is used to compensate for the time-domain received signal. The specific scheme is shown in Figure 6.  
We obtain the IFO and FFO estimates, respectively:  
\[\hat{\varepsilon}_{i}=\frac{1}{2\pi^{T}\cdot_{\mathrm{amb}}}\max_{ \varepsilon}\left\{\left|\sum_{j=0}^{L-1}Y_{l+D}[p[j],\varepsilon]Y_{l}{{}^{* }}[p[j],\varepsilon]X_{l+D}{{}^{*}}[p[j]]X_{l}{{}^{*}}[p[j]]\right|\right\} \tag{18}\] \[\hat{\varepsilon}_{f}=\frac{1}{2\pi^{T}\cdot_{\mathrm{amb}}}\max \left\{\sum_{j=0}^{L-1}Y_{l+D}[p[j],\varepsilon_{i}]Y_{l}{{}^{*}}[p[j], \varepsilon_{i}]X_{l+D}{{}^{*}}[p[j]]X_{l}{{}^{*}}[p[j]]\right\}\]  
Figure 7 shows the performance of CP-based STO estimation using correlation-based maximum and interpolation minimum techniques. The solid line represents the method based on maximum correlation, while the dashed line represents the method based on minimum difference.