# Airborne Radar Staggered PRF Coherent Processing Method for Down-Looking Target Detection
## 2 Airborne Radar Clutter Model and Signal Model
### PRF Interleaved Transmission and Echo Model  
We propose a novel airborne radar mode that addresses the coherence issue in traditional multi-PRF systems. This mode involves interleaved transmission of multiple low PRF signal groups, each utilizing orthogonal discrete frequency coded (DFC) signals. The first pulse of each group is interleaved, followed by sequential emission of subsequent pulses based on each group's PRF. Echo processing involves matched filtering to filter out signals from other PRF groups, enabling coherent processing based on the pulse transmission interval. This approach enhances target detection performance in low SNR conditions by obtaining high SNR results and improving target detection performance. Figure 3: Schematic diagram of target detection using multiple PRFs. As shown in Figure 4, the PRF interleaved transmission model assumes the existence of \(N\) PRF signal groups, each using a different encoded DFC signal. It is also assumed that the radar alternates the transmission of N sets of DFC waveforms, with the start times of the signal transmissions being \([0,\Delta t_{1},\cdots,\Delta t_{N-1}]\). The PRI of the signals for different PRF groups are denoted as \(T=[T_{1},\cdots,T_{N}]\). The \(n\)th pulse group transmission of the radar can be expressed as:  
\[s_{n}(t)=u_{n}(t-T_{n})\exp(j2\pi f_{c}(t-T_{n})) \tag{9}\]  
where \(f_{c}\) is the center frequency of the radar system. In Equation (9), \(u_{n}(t)\) denotes the DFC waveform used by the \(n\)th PRF signal group, which can be denoted as:  
\[u_{n}(t)=\sum_{m=0}^{M-1}rect\bigg{(}\frac{t-mT_{sp}}{T_{sp}}\bigg{)}\exp(j2 \pi f_{m,n}t) \tag{10}\]  
where \(rect(\cdot)\) is a rectangular function, \(M\) is the number of frequency chips in a pulse, \(T_{sp}\) is the time width of the frequency chip, \(T_{p}=MT_{sp}\) is the entire pulse time width, \(f_{m,n}=a_{m,n}\cdot\Delta f\) is the frequency encoding sequence, \(\Delta f\) is the frequency interval of the frequency chips, and \(\Delta f=1/T_{sp}\). \(\mathbf{a_{n}}=[a_{1,n},\cdots,a_{m,n},\cdots,a_{M,n}]\) is the pulse frequency coding coefficient in the \(n\)th PRF signal group, which is randomly rearranged by \([0,1,\cdots,M-1]\). Assuming that \(K\) pulse signals are transmitted for each PRF signal group during a beam dwell time, the transmitted signal can be represented as:  
\[s(t)=\sum_{k=0}^{K-1}\sum_{n=0}^{N-1}rect\bigg{(}\frac{t-kT_{n}-\Delta t_{n}}{ T_{p}}\bigg{)}u_{n}(t-kT_{n}-\Delta t_{n})\exp(j2\pi f_{c}t) \tag{11}\]  
Let \(t_{n,k}=kT_{n}+\Delta t_{n}\), the echo signal of each PRF signal group can be represented as follows:  
\[s_{k}(t,t_{n,k})=s_{k,c}(t,t_{n,k})+s_{k,t}(t,t_{n,k}) \tag{12}\]  
where \(s_{k,c}(t,t_{n,k})\) represents clutter signal, while \(s_{k,t}(t,t_{n,k})\) represents target signal. Figure 4: Airborne radar model for transmitting orthogonal signals in an alternating manner using multiple PRFs. For ground echoes, the echo signals of each PRF group can be represented after frequency down-conversion as:  
\[s_{k,c}(t,t_{n,k})=\int_{r_{1}}^{r_{2}}\int_{\theta_{1}}^{\theta_{2}}\sigma_{r, \theta}u_{n}\bigg{(}t-\frac{2r}{c}\bigg{)}\exp(j\sigma_{r,\theta})\exp(j2\pi f_ {d,\theta}t_{n,k})d\theta dr \tag{13}\]  
where \(\sigma_{r,\theta}=G(r,\theta)\cdot\sigma_{c}(r,\theta)\) refers to the clutter unit echo intensity, \(\sigma_{r,\theta}\) corresponds to the phase of the clutter unit, and \(f_{d,\theta}\) denotes the Doppler of the clutter unit. For a target, assuming the initial distance is \(R_{t,0}\) and the velocity is \(v_{t}\), the echo of each PRF group can be represented as:  
\[s_{k,n}(t,t_{n,k})=\sigma_{t}u_{n}\bigg{(}t-\frac{2R(t_{n,k})}{c}\bigg{)}\exp \bigg{(}j\frac{4\pi(R_{t,0}+v_{t}\Delta t_{n})}{c}\bigg{)}\exp(j2\pi f_{d,t}t_ {n,k}) \tag{14}\]  
where \(\sigma_{t}\) denotes the intensity of the target echo, while \(R(t_{n,k})=R_{t,0}+v_{t}(\Delta t_{n}+kT_{n})\) represents the distance of the target at each pulse time.