# Airborne Radar Staggered PRF Coherent Processing Method for Down-Looking Target Detection
## 2 Airborne Radar Clutter Model and Signal Model  
### Airborne Radar Clutter Model  
The main problem in airborne radar air-to-air target detection is the range-Doppler ambiguity. The primary factor influencing target range-Doppler ambiguity is the PRF. Pulse radar typically uses a fixed PRF, categorized as low pulse repetition frequency (LPRF), medium pulse repetition frequency (MPRF), or high pulse repetition frequency (HPRF). When an airborne radar operates in down-looking mode, ground clutter within a resolution cell consists of numerous randomly scattered objects with random amplitude and phase. Therefore, it is common to describe the amplitude characteristics of clutter using probability density functions, such as Gaussian, Rayleigh, and Rice distributions. Figure 1 illustrates the down-looking scene of an airborne radar, showing clutter echoes that resemble target echoes. Clutter power can be quantified by the clutter scattering cross section \(\sigma_{c}\). In a radar system, the clutter region's unit area can be represented as the equivalent scattering cross section per unit area, denoted as:  
\[\sigma_{c}=\sigma^{0}A_{c} \tag{1}\]  
where \(A_{c}\) refers to the unit area of clutter and \(\sigma^{0}\) denotes the clutter scattering coefficient, which follows the clutter statistical characteristics mentioned above. Figure 1: Airborne radar down-looking detection model. The Doppler spectrum of stationary ground clutter broadens in the Doppler dimension when observed by a down-looking airborne radar, due to varying radial velocities of different angles within the radar beam. The clutter's Doppler spectrum width is influenced by the aircraft speed, wavelength, and beamwidth. The Doppler frequency of ground scatter echoes depends on the aircraft velocity and the angle between the scatter point and the aircraft's velocity vector. It can be formulated as:  
\[f_{\eta}=\frac{2v}{\lambda}\cos\theta \tag{2}\]  
where \(v\) represents the velocity of the carrier aircraft and \(\theta\) denotes the angle between the clutter point-to-radar vector and the velocity vector of the radar, and \(\lambda\) is the wavelength of the radar system. In general, the clutter area within the main radar beam has a significant impact on target detection. We assume that the radar beamwidth is \(\theta_{bw}\) and the radar beam center slant angle is \(\theta_{c}\), and the radar beamwidth is relatively small. The Doppler bandwidth and the Doppler center of ground clutter can be approximated as:  
\[B_{d}\approx\frac{2v}{\lambda}\theta_{bw} \tag{3}\]  
\[f_{dc}=\frac{2v}{\lambda}\cos\theta_{c} \tag{4}\]  
Assuming the aircraft altitude is \(h\), the clutter distance from the leading edge to the trailing edge of the main beam can be denoted as:  
\[r_{1}=\frac{h}{\cos(\theta_{c}+\theta_{bw}/2)} \tag{5}\]  
\[r_{2}=\frac{h}{\cos(\theta_{c}-\theta_{bw}/2)} \tag{6}\]  
As radar typically transmits fixed PRF signals, target and ground clutter echoes will suffer from range ambiguity and velocity ambiguity in the received echoes, as shown in Figure 2. For a given PRF, the maximum unambiguous range of the radar can be denoted as \(R_{mm}\), and the maximum unambiguous velocity as \(v_{mm}\). Assuming the distance and Doppler of a certain scatter point are \(R\) and \(f_{d}\), respectively, the observable distance and Doppler of the radar can be expressed as:  
\[R_{v}=R-IR_{um} \tag{7}\]  
\[f_{d,v}=f_{d}-gPRF \tag{8}\]  
Figure 1:0 Schematic diagram of target and clutter in fixed PRF mode. where \(l\) represents the number of range ambiguities and \(g\) represents the number of Doppler ambiguities. To mitigate ambiguity in radar echo signals from targets and ground clutter, a single fixed PRF signal may not provide accurate target information. Hence, a conventional approach involves employing multiple signal groups with different medium PRFs (Multi-MPRF) for detection. By varying the PRF, which affects both the maximum unambiguous range and velocity, the ambiguous region of the clutter spectrum can be adjusted.