## 4 Target Model and Frequency Offset Extraction Method
### Target Detection Processing Method
#### 4.2.1 Channel Estimation  
The frequency domain least squares (LSs) method is a commonly used channel estimation method in OFDM systems. For OFDM systems that rely on a pilot symbol, the transmitted pilot symbol vector is \(\mathbf{X}_{P}\) and the received pilot symbol vector is \(\mathbf{Y}_{p}=\mathbf{X}_{p}\mathbf{H}_{p}+\mathbf{N}_{p}\), where \(\mathbf{N}_{p}\) is the noise vector. The objective function of the method can be expressed as follows:  
\[J\big{(}\hat{\mathbf{H}}_{P}\big{)}=\big{\|}\mathbf{X}_{P}\hat{\mathbf{H}}_{P}- \mathbf{Y}_{p}\big{\|}^{2} \tag{6}\]  
To obtain the minimum value of \(J\big{(}\hat{\mathbf{H}}_{P}\big{)}\), set its partial derivative to 0, i.e., \(\partial J\big{(}\hat{\mathbf{H}}_{P}\big{)}/\partial\hat{\mathbf{H}}_{P}=0\), and the estimated channel response value is [21].  
\[\hat{\mathbf{H}}_{P}=\frac{\mathbf{Y}_{P}}{\mathbf{X}_{P}} \tag{7}\]  
The flow chart of target frequency offset estimation based on 5G signals is shown in Figure 8 [22].  
Figure 6: Bilateral rotating target model. Target 1 and Target 2 are two targets with rotation radii \(r_{1}\) and \(r_{2}\), respectively. Two targets are connected by a pole, and they will rotate at the same angular velocity.  
Figure 7: (**a**) Equal-radius bilateral target Doppler frequency offset curve; (**b**) unequal-radius bilateral target Doppler frequency offset curve.  
The transmission resource grid of 5G signals can be represented as \(G_{TX}\), where \(G_{TX}\) is a matrix of size \(N\times M\), and \(M\) is the number of OFDM modulation symbols contained in the signal. The baseband modulation signal can be represented as follows:  
\[x(t)=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}G_{TX}(n,m)\exp(j2\pi f_{n}t)rect\bigg{(} \frac{t-T/2-mT}{T}\bigg{)} \tag{8}\]  
where \(T\) is period which includes the basic symbol length and the cyclic prefix length, \(f_{n}=n\Delta f\) is the frequency of the n-th subcarrier in the baseband signal, and \(rect(\cdot)\) is the rectangular window function.  
Assuming the time delay of the target echo signal is \(\tau\) and the Doppler frequency shift is \(f_{d}\), the received signal can be represented as follows:  
\[y(t)=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}G_{RX}(n,m)\exp(j2\pi f_{d}t)rect\bigg{(} \frac{t-T/2-mT-\tau}{T}\bigg{)} \tag{9}\]  
\[G_{RX}(n,m)=A_{y}(n,m)G_{TX}(n,m)\exp[j2\pi(-n\Delta f\tau+f_{d}mT)] \tag{10}\]  
where \(G_{TX}(n,m)\) is the transmission resource grid of the signal, \(G_{RX}(n,m)\) is the resource grid obtained from the received signal, and \(A_{y}(n,m)\) is the amplitude.  
According to the LS method, the obtained channel response can be expressed as follows:  
\[H(n,m)=\frac{G_{RX}(n,m)}{G_{TX}(n,m)}=A_{y}(n,m)\exp(-j2\pi n\Delta f\tau+j2 \pi f_{d}mT) \tag{11}\]  
In this paper, CSI-RS is applied to obtain channel responses, and the Equation (11) can be represented as follows:  
\[H(n,m)=A_{y}(n,m)\exp(-j2\pi n\Delta f_{CSI}\tau+j2\pi f_{d}mT_{CSI}) \tag{12}\]  
where \(T_{CSI}\) is the period of the CSI-R, and \(\Delta f_{CSI}\) is the interval between adjacent subcarriers of the CSI-RS. The dimension of the channel response matrix is \(N_{1}\times M_{1}\), where \(N_{1}\) is the number of subcarriers occupied by the CSI-RS in the frequency domain, and \(M_{1}\) is the number of OFDM symbols occupied in the time domain.