# Passive Location for 5G OFDM Radiation Sources Based on Virtual Synthetic Aperture
## 3 The Signal Model and Principle of Virtual Synthetic Aperture Radiation Source Location Method
### Phase Compensation  
The premise of the above conclusion is that the influence of \(C_{m}\) is ignored; however, \(C_{m}\) is an unknown variable whose phase will affect the location result. To solve the problem, we use the reference signals (or pilot subcarriers) for channel estimation to obtain the information carried on the modulation symbols \(c_{mlk}\). The reference signals (or pilot subcarriers) are specific signals in OFDM systems for channel estimation. Technical specifications for 5G NR include several different types of reference signals, which are configured and transmitted in different ways for different purposes of the receiving device. The main application of DM-RS (demodulation reference signal) in NR is to estimate the channel coefficients of physical channel coherence detection, and it can be argued that the transmitter precoding is transparent to the receiver and treated as part of the whole channel. To reduce the interpolation error and implementation complexity under the condition of a stationary channel, it is recommended to use uniform DM-RS assignment in both frequency and time domains. Since DM-RS does not transmit user data, it needs to be allocated at an appropriate density to maximize throughput. DM-RS adopts the front-loading mode, and the data is loaded after it. After estimating the channel based on the front-loaded DM-RS, the receiver can conduct coherent demodulation in the data region. As the signal model in this paper, the time delay is considered constant in the same slot. Therefore, the channel property is time-invariant in the same slot, and different types of DM-RS can be unified into the model shown in Figure 3. The modulation symbols of the reference signal are discrete in the frequency domain and the interval is \(\Delta l\). After DFT, the reference signal in a slot is processed as  
\[P(k)=d_{k}\exp\{-j2\pi f_{k}\tau\}+W_{p}(k)=d_{k}H_{p}(k)+W_{p}(k) \tag{16}\]  
where \(d_{k}\) denotes the modulation symbols of DM-RS, which are considered known to the receiver. \(H_{p}(k)\) is the channel property at the frequency \(f_{k}\). Therefore, the channel can be estimated at the reference signal as  
\[\hat{H}_{p}(k)=\frac{P(k)}{d_{k}}=H_{p}(k)+\frac{W_{p}(k)}{d_{k}}(d_{k}\
eq 0) \tag{17}\]  
Since \(H_{p}(k)=\exp\{-j2\pi f_{k}\tau\}\) is a single-frequency signal of \(f_{k}\) and the interval is uniform, we adopt the transform domain interpolation method to estimate the channel property without the reference signal. The basic idea of transform domain interpolation is that adding zero in the time domain is equivalent to interpolating in the frequency domain. In addition, in the time domain, the energy of the channel is concentrated at a certain delay of the channel, while the noise exists in the whole time domain, so the noise can be removed if only the maximum point is retained. Then the channel estimation on all subcarriers in the frequency domain can be obtained by performing DFT with zero-padding in the time domain. \[H_{p}(k,t_{m})=\exp\{-j2\pi f_{k}\tau(t_{m})\} \tag{18}\]  
According to the estimation of the channel, we can obtain \(C_{m}\) in the same slot. \[\hat{C}_{m}=\frac{m(t_{m})}{H_{p}(0,t_{m})} \tag{19}\]  
Thus, the phase of \(C_{m}\) can be compensated by:  
\[ \begin{array}{rl}m^{\prime}(t_{m})&=\hat{C}_{m}^{*}\cdot m(t_{m})\\ &=\Big{|}C_{m}\Big{|}2\exp\Big{\{}-j\frac{2\pi f_{0}R(t_{m})}{c}\Big{\}}\end{array}  \tag{20}\]  
where \(\hat{C}_{m}^{*}\) is the conjugate of \(\hat{C}_{m}\). In conclusion, the principle of the algorithm is shown in Figure 4. First, the received baseband signal is preprocessed to remove the CP and the preprocessed signal is rearranged into range and azimuth domain. Then FFT is performed along the range domain to extract the signal at a particular frequency. Next, to compensate for the phase of the data symbol, we use the least square method and transform domain interpolation method to estimate the channel based on the known DM-RS and use the channel characteristics for denoising.