# Passive Detection of Low-Altitude Signal Sources Using an Improved Cross-Correlation Algorithm
## 2 System Model  
The communication signals between a controller and a low-altitude source can be used for passive detection. In general, they consist of several fixed carrier signals. The system model includes a signal source \(u\), a controller \(c\), and a passive receiver Rx, as shown in Figure 1. The dashed lines indicate that the receiver collects the signals passively, while the solid line represents the active communications between the source and controller. A line-of-sight (LOS) channel is assumed between the source and receiver, and a non-line-of-sight (NLOS) channel between the controller and receiver [2]. A UAV is considered in this paper as a typical low-altitude signal source. A UAV communicates with the corresponding controller in the frequency range 2.4 GHz to 2.5 GHz using orthogonal frequency division multiplexing (OFDM) and frequency hopping (FH). The Phantom 4 Pro UAV from DJ-Innovations was employed in the experiments. One experiment was carried out outdoors with the Rx placed on a bridge in the Jimo District, Qingdao, Shandong Province, China, with the UAV moving away from the Rx at distances of 500 m, 1000 m, 1500 m, 2000 m, 2500 m, and 3000 m, as shown in Figure 2a. The UAV hovered at a height of 100 m in the outdoor test. Another experiment was conducted indoors in a research laboratory with the Rx on a table 1.1 m above ground and the UAV on another table at distances of 5 m and 10 m from the Rx, as shown in Figure 2b. The antenna gain is 24 dBi, the beam width is 10\({}^{\circ}\), and the frequency range is 2.3 GHz to 2.7 GHz. During the experiments, the antenna elevation angle was varied from 0\({}^{\circ}\) to 12\({}^{\circ}\) and the azimuth angle from 0\({}^{\circ}\) to 180\({}^{\circ}\). The same parameters were used in the experiments with multiple UAVs. The detection environment contains static and nonstatic clutter as well as Gaussian noise. The discrete frequency domain signal received from the antenna is \(X(\omega)\) with length \(N=5120\). The corresponding time domain signal can be obtained using an inverse discrete Fourier transform (IDFT), which gives  
\[x(n)=\frac{1}{N}\sum_{\omega=0}^{N-1}X(\omega)e^{\frac{\omega\pi n\omega}{N}},n =0,1,\ldots,N-1 \tag{1}\]  
This includes the UAV signal, nonstatic clutter, additive white Gaussian noise (AWGN), and static clutter such as Bluetooth and WiFi signals. The autocorrelation of this signal is  
\[r\big{(}n,n^{\prime}\big{)}=x^{*}\big{(}n-n^{\prime}/2\big{)}x\big{(}n+n^{ \prime}/2\big{)} \tag{2}\]  
Figure 1: The system model which includes a signal source \(u\), a controller \(c\), and a passive receiver Rx. The dashed lines indicate that the receiver collects the signals passively, while the solid line represents the active communications between the source and controller. Figure 2: The test environments (**a**) outdoors and (**b**) indoors. where \(n^{\prime}\) is the time delay and \(\left(\cdot\right)^{*}\) denotes conjugate. In order to observe the time and frequency domain features of the UAV signal simultaneously [26], the Wigner-Ville distribution (WVD) of \(x(n)\) can be expressed as  
\[y_{n\omega}=\sum_{n^{\prime}=-(N-n-1)/2}^{(N-n-1)/2}r\big{(}n,n^{\prime}\big{)} e^{\frac{-2\pi m^{\prime}\omega}{N}} \tag{3}\]  
where \(n\) denotes time, \(\omega\) denotes frequency, and \(y_{n\omega}\) denotes the power at frequency \(\omega\) and time \(n\). The passive receiver rotates from \(0^{\circ}\) to \(180^{\circ}\) with a rotation speed of \(22.5^{\circ}/\)s, so the antenna rotates \(180^{\circ}\) in 8 s. The time sampling interval is \(t_{a}\) = 0.02 s, the frequency range is 2.4 GHz to 2.5 GHz, and the frequency sampling interval is \(f_{w}\) = 19.53 kHz. The discrete time-frequency values are then \(y_{jk},j=1,\dots,8/t_{a},k=1,\dots,(2.5-2.4)\times 10^{9}/f_{w}\). \(y_{jk}\) is the power at frequency \(kf_{w}+2.4\times 10^{9}\) and time \(jt_{a}\). These values in an ideal environment are shown in Figure 2(a) with the UAV at a relative angle of \(97.88^{\circ}\) and a distance of 1.5 km from the Rx.