# Airborne Radar Staggered PRF Coherent Processing Method for Down-Looking Target Detection
## 3 The Flow Design of Multilayer Detection and Blind Zone Optimization
### Optimization of Clutter Blind Zone  
Changing the PRF of a radar system can alter the ambiguous distribution characteristics of a clutter spectrum. As shown in Figure 6, when using different PRFs, the periodic extension of the Doppler spectrum will change and the clutter spectrum periodic extensions of different PRFs will overlap. When N groups of PRFs are used and the minimum PRF required for target detection is \(N_{1}\) groups, a blind zone will be considered when the overlap is greater than \(N_{1}\).  
Assuming the radar operates within the power range of \([0,R]\) and the velocity range of \([-v,v]\), for ease of calculation, assuming there are \(M_{r}\) range cells and \(M_{v}\) velocity cells, the blind zone of the \(k\)-th PRF can be expressed as:  
\[B_{k}(r,v)=\left\{\begin{array}{ll}0,\tilde{B}_{k}(r,v)\
otin\tilde{B}_{k}\\ 1,\tilde{B}_{k}(r,v)\in\tilde{B}_{k}\end{array}\right. \tag{28}\]  
where \(\tilde{B}_{k}(r,v)\) represents the blind zone of the clutter Doppler spectrum under the \(k\)-th PRF, and \(\tilde{B}_{k}(r,v)\) represents the Range-Doppler cell with ambiguity at unit \((r,v)\) under the \(k\)-th PRF.  
Therefore, after the fusion of multiple PRFs, the cells greater than \(N_{1}\) are regarded as blind zone cells. The resulting fused blind zone map is represented as:  
\[B(r,v)=\left\{\begin{array}{ll}0,\sum\limits_{k=1}^{N}B_{k}(r,v)\leq N_{1} \\ 1,\sum\limits_{k=1}^{N}B_{k}(r,v)>N_{1}\end{array}\right. \tag{29}\]  
Figure 6: Cyclical extension of clutter spectrum schematic diagram  
The proportion of clutter blind zone within the radar detection range can be expressed as:  
\[\omega=\frac{\sum\limits_{r=1}^{M_{r}}\sum\limits_{v=-M_{v}/2}^{M_{v}/2}B(r,v)}{M _{r}M_{v}} \tag{30}\]  
Assuming an optimized PRF sequence is given as \(\mathbf{fp}=[PRF_{1},PRF_{2},\cdots,PRF_{N}]\), where \(PRF_{min}\) and \(PRF_{max}\) represent the minimum and maximum PRF values that the radar can operate at, respectively, the optimization problem of the radar blind zone map can be formulated as:  
\[\begin{array}{ll}\min&w\\ s.t.&\min\{\mathbf{fp}\}\geq PRF_{min}\\ &\max\{\mathbf{fp}\}\leq PRF_{max}\end{array} \tag{31}\]  
The genetic algorithm can be employed to optimize the PRF sets. The flow of the algorithm is described in Algorithm 1.  
```
Input: the initial PRF group population \(\mathbf{Fp}=[\mathbf{fp}_{1},\cdots,\mathbf{fp}_{j},\cdots,\mathbf{fp}_{j}]\) with the population size is \(J\), the stop condition \(\xi\). In the \(k\)-th iteration:
1: Use initial population \(\mathbf{Fp}\) and Equation (30) to calculate \(w^{(1)}\).
2:while\(w^{(k)}>\xi\)
3: select the dominant individual \(\mathbf{Fp}_{1}\) to crossover and mutation for generation new population \(\mathbf{F\ddot{p}_{1}}\)
4: introduction random new individuals \(\mathbf{F\ddot{p}_{1}}\) and fuse with \(\mathbf{F\ddot{p}_{1}}\) to form a new population \(\mathbf{F_{p}}=[\mathbf{F\ddot{p_{1}}},\mathbf{F\ddot{p_{1}}}]\)
5: Use the formed population \(\mathbf{Fp}\) and Equation (30) to calculate the minimum fitness value \(w^{(k)}\) of the \(k\)-th iteration.
6:endwhile output: the fitness value \(w^{(k)}\) and the optimized PRF group \(\mathbf{fp}\).
```  
**Algorithm 1**Genetic algorithm optimization