# Airborne Radar Staggered PRF Coherent Processing Method for Down-Looking Target Detection
## 3 The Flow Design of Multilayer Detection and Blind Zone Optimization  
### Multilayer Detection and Coherent Processing of Pulse Train Signals  
Considering the possible existence of Doppler ambiguity for the target, let us assume that the Doppler ambiguity number of the \(n\)th signal group is \(\Gamma_{n}\) and the residual Doppler frequency is \(\tilde{f}_{d,n}\), which satisfies the following:  
\[f_{d,t}=F_{n}f_{r,n}+\tilde{f}_{d,n} \tag{15}\]  
where \(f_{r,n}\) is the \(n\)th PRF. Let \(U_{n}(f)=FFT\{u_{n}(t)\}\), then the fast-time frequency-domain signal with Doppler ambiguity can be expressed as:  
\[S_{k,n}(f,t_{n,k})= \sigma_{t}U_{n}(f)\exp\left(-j\frac{4\pi fR_{t,0}}{c}\right)\exp \left(j\frac{4\pi(R_{t,0}+v_{t}\Delta t_{n})}{c}\right) \tag{16}\] \[\exp(j2\pi F_{n}f_{r,n}t_{n,k})\exp\left(j2\pi(f+f_{c})\frac{ \tilde{f}_{d,n}}{f_{c}}t_{n,k}\right)\]  
Performing the Keystone transform, let \(\tau_{n,k}=\frac{(f+f_{c})}{f_{c}}t_{n,k}\). \[S_{k,n}(f,\tau_{n,k})= \sigma_{t}U_{n}(f)\exp\left(-j\frac{4\pi fR_{t,0}}{c}\right)\exp \left(j\frac{4\pi(R_{t,0}+v_{t}\Delta t_{n})}{c}\right)\] (17) \[\exp\left(j2\pi F_{n}f_{r,n}\frac{f_{c}}{(f+f_{c})}\tau_{n,k}  
Due to the varying slow-time sampling intervals, inconsistent Doppler intervals, and phase inconsistency caused by spectral leakage, direct Fourier transform application and coherent fusion processing of multiple groups are not feasible. To address this, DFT processing is employed for slow-time sampling moments, and a uniform Doppler sampling interval is established to ensure phase consistency. Assuming the Doppler spectrum interval during DFT processing is \(f_{l}\), and the DFT length is \(2L\), the corresponding Doppler spectrum vector is:  
\[\begin{array}{l}{\bf{f}}_{l}=[f_{d_{r}-L},f_{d_{r}-L+1},\cdots,f_{d,L-1}]\\ \quad\quad\quad\quad\quad  
N groups of PRF, the minimum number of PRF required for target detection is \(N_{1}\) groups. The first detection threshold \(T_{1}\) is set to \(\frac{13\,\mathrm{dB}}{N_{1}}\) of the noise power, and Doppler ambiguity resolution and coherent processing are applied to the detected targets. The second detection threshold \(T_{2}\) is set at 13 dB of the noise power for the coherent result, enabling determination of the target and estimation of its parameters. Assuming that the detection results of each group of RD cell \([t,f_{r}]\) are \(D(n)\), then:  
\[D(n)=\left\{ \begin{array}{ll}1,\ \text{When}\ \big{|}s_{k,n,F \subset}(t,\tilde{f}_{r,n})\big{|}>T_{1}\\ 0,\ \text{When}\ \big{|}s_{k,n,F\subset}(t,\tilde{f}_{r,n})\big{|}\leq T_{1} \end{array} \right. \tag{25}\]  
Equation (24) is hereby revised as follows:  
\[s_{b,n}\big{(}\hat{t},f_{r}\big{)} =\sum_{i=1}^{N}s_{k,n,F\subset}\big{(}t,\tilde{f}_{r,n}\big{)}*D(n )*W_{n}(f_{r}) \tag{26}\] \[=N_{1}\sigma\sin c\bigg{(}B\bigg{(}\hat{t}-\frac{2R_{t,0}}{c} \bigg{)}\bigg{)}\sin c\big{(}T_{r}\big{(}f_{r}-\tilde{f}_{r,n}\big{)}\big{)} \exp\bigg{(}-j4\pi\frac{R_{t,0}}{\lambda}\bigg{)}\]  
where \(N_{1}=\sum D(n)\) represents the number of range-Doppler cells corresponding to the PRF groups that have detected the target. Perform a secondary threshold test on the \(N_{1}\) group coherent results to determine if it is a target. \[\left\{ \begin{array}{ll}\big{|}s_{b,n}(\hat{t},f_{r})\big{|}>T_{2},\ \text{Target}\\ \big{|}s_{b,n}(\hat{t},f_{r})\big{|}\leq T_{2},\ \text{Noise}\end{array} \right. \tag{27}\]  
Based on the analysis above, when there is clutter that obstructs the detection of the target and the first-level threshold detection can detect \(N_{1}\) times, an \(N_{1}\) times SNR gain can be achieved. The conventional detection logic is based on the \(3/5\) or \(5/8\) rule, where \(N=5\), then \(N_{1}=3\), or \(N=8\), then \(N_{1}=5\). If the \(3/5\) rule is used, the correct target can be detected at least three times, and after multiple PRF coherent accumulation, a minimum of three times SNR gain can be obtained. The first detection threshold can be set to 8 dB times the noise power based on the coherent SNR gain.