# Remote Radio Frequency Sensing Based on 5G New Radio Positioning Reference Signals
## 3 Investigated Radar Scenario and Propagation Channel Model  
The scenario investigated in this study is presented in Figure 6 and is the same as the one provided in 3 GPP TS 38.101-4 Annex B.3 [30], which is the high-speed train scenario. A train is inherently an elongated target for tracking, which negatively affects the quality of detection. In this paper, this effect is simplified by considering it as a point target, i.e., assuming that it is moving exactly in the direction to/from (toward/against) the base station (gNb). We consider only the monostatic radar, which means that gNb uses transmitted signal \(x_{TX}(t)\) and received signal \(y_{RX}(t)\) to identify reflections of \(x_{TX}(t)\), which are present in \(y_{RX}(t)\). The simulation scenario consists of one gNb station that is mounted at \(D_{min}\) meters above ground and one target that is \(D_{s}\) meters from it in a straight line and ideally moving toward or against it with constant velocity \(vm/s\). Therefore, the distance \(D\) can be expressed as follows:  
\[D=\sqrt{D_{s}^{2}+D_{min}^{2}}. \tag{1}\]  
The two-way propagation time within the distance \(D\) can be expressed as follows:  
\[\tau=\frac{2D}{c}, \tag{2}\]  
Figure 5: Different PRS configurations with two PRS resources. The first PRS resource is marked with the red color, and the second PRS resource is marked with the blue color.  
Figure 6: Study scenario with one target.  
where \(c\) denotes the speed of light. As per 3 GPP TS 38.101-4 Anne B.3 [30], the Doppler frequency shift is dependent on \(\cos(\theta)\), which can vary in time because the target moves toward/against the base station and can be expressed as follows:  
\[\cos(\theta(t))=\frac{(D_{s}-vt)^{2}}{\sqrt{(D_{s}-vt)^{2}+D_{min}^{2}}}. \tag{3}\]  
The maximum Doppler shift is obtained when the target and gNb are positioned at the same height, and it is equal to the following:  
\[f_{max}^{(D)}=\frac{2v}{\lambda},\quad\lambda=\frac{c}{f_{c}}, \tag{4}\]  
where \(f_{c}\) is a carrier frequency of the waveform. Thus, the Doppler shift in a specific time \(t\) is given as follows:  
\[f^{(D)}(t)=f_{max}^{(D)}\cos(\theta(t)). \tag{5}\]  
Additionally, the delay of the transmitted signal \(x_{TX}(t)\) by \(\tau\) (2) causes a linear phase shift of the signal Fourier spectrum \(X(f)\):  
\[x_{TX}(t-\tau)\leftrightarrow X(f)e^{-|2\pi f\tau}. \tag{6}\]  
Finally, after using (5), a single reflected signal \(y_{p}(t)\) from the path \(p\) with complex-value reflection gain \(g_{p}\) (the strength of the reflection), Doppler frequency shift \(f_{p}^{(D)}\), and delay \(\tau_{p}\) is defined in a specific time \(t\) as follows:  
\[y_{p}(t)=g_{p}x_{TX}(t-\tau_{p})e^{|2\pi f_{p}^{(D)}(t)t}. \tag{7}\]  
After the following substitution:  
\[h_{p}(t)=g_{p}e^{|2\pi f_{p}^{(D)}(t)t}, \tag{8}\]  
(7) can be rewritten as:  
\[y_{p}(t)=h_{p}(t)x_{TX}(t-\tau_{p}), \tag{9}\]  
The signal \(y_{RX}(t)\) in a radar receiver is, in general, a sum of reflections of the following form (9):  
\[y_{RX}(t)=\sum_{p=1}^{p}y_{p}(t). \tag{10}\]  
The weight \(h_{p}(t)\) of the complex value channel, associated with the path \(p\)-th, has magnitude \(|g_{p}|\) and oscillates with drifting frequency \(f_{p}^{(D)}(t)\) (5), which has to be estimated in a radar sub-section of the receiver. In the investigated HST scenario, we assume only two reflections: one strong reflection \(p=1\) from the motionless object, treated as a disturbing clutter (for simplicity, it is a transmitted signal itself), and one weak reflection from the moving train:  
\[y_{RX}(t) =y_{1}(t)+y_{2}(t)\] \[=x_{TX}(t)+h_{2}(t)x_{TX}(t-\tau_{2})\] \[=x_{TX}(t)+g_{2}e^{|2\pi f_{2}^{(D)}(t)t}x_{TX}(t-\tau_{2}). \tag{11}\]