# 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
### The Principle of the Proposed Localization Algorithm  
According to the virtual synthetic aperture technology in SAR, the data utilization mode is the \"go-stop-go\" mode. Take a period of the received data with duration \(T_{p}\) at an interval \(T\) and rearrange the data into two domains. \(T_{s}=L_{s}/v\) is the synthetic aperture time, where \(L_{s}=\theta R_{0}=kcR_{0}/(f_{0}D)\), \(D\) is the antenna aperture and \(k\) is the beam width factor. Using \(t_{m}=mt\) represents the azimuth time of the \(m\)th pulse. Thus, the slant distance (instantaneous range) at \(t_{m}\) is \(R(t_{m})\):  
\[R(t_{m})=\sqrt{R_{0}^{2}+\left(vt_{m}-X_{0}\right)^{2}}\approx R_{0}+\frac{ \left(vt_{m}-X_{0}\right)^{2}}{2R_{0}} \tag{7}\]  
Thus, the received signal is expressed as follows:  
\[s_{r}(t_{r},t_{m})=s(t_{r}-\tau(t_{m}))=s(t_{r}-\frac{R(t_{m})}{c}) \tag{8}\]  
Convert the signal down to the baseband and discretize it into a two-dimensional matrix \(\mathbf{H}\) with a size of \(M\times N\), where \(M\) is the number of slots and \(N\) is the sampling points of one OFDM symbol. The \(m\)th row of the matrix is  
\[\mathbf{H}(m,n)=\sum_{l=0}^{L-1}\sum_{k=0}^{K-1}c_{mlk}\exp\left\{j\frac{2\pi kn}{ N}\right\}\exp\left\{-j\frac{2\pi f_{k}R(t_{m})}{c}\right\} \tag{9}\]  
Discrete Fourier transform (DFT) is performed along the row of the matrix \(\mathbf{H}\), and the result is  
\[\mathbf{H}_{\mathbf{r}}(m,k)=\sum_{l=0}^{L-1}c_{mlk}\exp\left\{-j\frac{2\pi f_{k}R(t_{ m})}{c}\right\} \tag{10}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter \(\mathbf{\mu}\)** & **0** & **1** & **2** & **3** & **4** \\ \hline Subcarrier spacing (kHz) & 15 & 30 & 60 & 120 & 240 \\ Slot duration \(T_{slot}\) (\(\upmu\)s) & 1000 & 500 & 250 & 125 & 62.5 \\ OFDM symbol duration (without CP) \(T_{u}\) (\(\upmu\)s) & 66.67 & 33.33 & 16.67 & 8.33 & 4.17 \\ OFDM symbol duration & 71.35 & 35.68 & 17.84 & 8.92 & 4.46 \\ \(T_{s}\) (\(\upmu\)s) & 4.69 & 2.34 & 1.17 & 0.57 & 0.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The parameters of 5G frame structure. Then, we take a particular frequency \(k=0\), the signal is a LFM signal in the azimuth domain and can be written as  
\[ \begin{split}\mathbf{m}(t_{m})&=\mathbf{H}_{r}(m)=C_{m}\exp \Bigl{\{}-j\frac{2\pi f_{0}R(t_{m})}{c}\Bigr{\}}\\ &=C_{m}\exp\Bigl{\{}-j\frac{2\pi f_{0}R_{0}}{c}\Bigr{\}}\exp \biggl{\{}-j\frac{\pi f_{0}(\pi t_{m}-X_{0})^{2}}{cR_{0}}\Bigr{\}}\end{split}  \tag{11}\]  
where \(C_{m}=\sum\limits_{l=0}^{L-1}c_{ml0}\). The Doppler centroid frequency and Doppler chip rate are shown as follows, which contains the location information of the radiation source. \[ \begin{split}& f_{dc}=\frac{\varphi X_{0}f_{0}}{cR_{0}}\\ & f_{dr}=-\frac{\varphi f_{0}}{cR_{0}}\end{split}  \tag{12}\]  
According to the theory of matched filtering, construct a series of matched filters \(\mathbf{h}(t_{m},K_{i})\) with different chirp rates \(K_{i}\). \[\mathbf{h}(t_{m},K_{i})=\exp\Bigl{\{}j\pi K_{i}f_{m}^{2}\Bigr{\}} \tag{13}\]  
where \(K_{i}=f_{0}v^{2}/(cR_{i})\) is the chirp rate with the range \(R_{i}\), and the interval is \(\Delta R\). Take the signal \(m(t_{m})\) through the series of matched filters, the location of the radiation source can be obtained when the output is the maximum. The output of the matched filters is:  
\[ \begin{split}\mathbf{w}(t_{m})&=\mathbf{m}(t_{m})\otimes \mathbf{h}(t_{m},K_{i})\\ &=\text{IFFT}\{\text{FFT}\{m(t_{m})\}\cdot\text{FFT}\{h(t_{m},K_ {i})\}\}\\ &=\text{C}^{\prime}{}_{m}\text{IFFT}\Bigl{\{}\exp\Bigl{\{}-j\pi( \frac{f_{m}^{2}}{f_{dr}}+\frac{f_{m}^{2}}{K_{i}})\bigr{\}}\Bigr{\}}\Bigr{\}} \otimes\delta(t_{m}-\frac{X_{0}}{v})\end{split}  \tag{14}\]  
where \(\text{C}^{\prime}{}_{m}=C_{m}\exp\Bigl{\{}-j\frac{2\pi f_{0}R_{0}}{c}\Bigr{\}}\). When \(K_{i}=-f_{dr}\), the output is the maximum. Suppose that the peak coordinate is \((\hat{K},\hat{t}_{p})\). Thus, the corresponding range and azimuth location is the estimation of the radiation source location:  
\[ \begin{split}\hat{R}_{0}&=\frac{f_{0}v^{2}}{cR}\\ \hat{X}_{0}&=\
u t_{p}\end{split}