Scene description
Coverage area metrics are usually considered as a criterion for evaluating the surveillance performance of radar systems.In related works11, 12, 13, 14, 15, 16, the coverage capability of such a system is calculated assuming that the target moves with a single angle of advance. However, in reality, the target heading is unknown and dynamic. Therefore, each possible orientation angle of the target has equal probability within the system. However, the detected target is usually moving towards the radar station within a certain angular range. As a result, all angles within this angular range can be considered to be possible directional angles and have equal probability. Therefore, an angular range is assumed and multiple nose angles within this angular range are considered to calculate the coverage area and used to optimize the performance of the radar system. It is assumed that there are multiple targets within the surveillance range of the radar system. If the system can distinguish between them, the number of targets does not affect the coverage area because the target echoes are treated separately. If they cannot be distinguished from each other, the radar system treats them as one target, and the problem of distinguishing between multiple targets falls outside the scope of antenna placement optimization based on coverage area. Therefore, this paper uses a single target as an example to consider optimizing the coverage area of āāa radar system.
It is assumed that there is. N The radar antennas are distributed over an area Z in a 2D plane. Considering the regional limitations of radar system deployment in practical applications, area Z is set to an irregular polygon. This assumption is similar to the actual radar installation area. Terrain and altitude limitations usually prevent establishing a normal area for antenna installation. However, this design is more difficult to optimize. To the right of Z is the system monitoring area. The surveillance area is divided based on radar range cells. Therefore, the monitoring area is actually a set of grid cells. A schematic diagram of the monitoring and deployment areas is shown in Figure 1.
The deployment area Z shall be divided as follows. U Form grid cells and matrix \(\textbf{Z}_U\). The monitoring area is divided as follows: L Form grid cells and matrix \(\textbf{S}_L\).target can be rotated M The angle and angle matrix are defined as: \(\textbf{A}_M\).of I– The th transmitting antenna is: \({(x_i,y_i)}\) And that j-th receiving antenna is: \({(x_j,y_j)}\).coordinates of I-th grid cell is \({(x_l,y_l)}\). The above antenna and grid cell coordinates satisfy the following conditions:
$$\left\{ {\begin{array}{*{20}l} {\left( {x_{i} ,y_{i} } \right) \in {\mathbf{Z}}_{U} ,1 \le i \le N,} \hfill \\ {\left( {x_{j} ,y_{j} } \right) \in {\mathbf{Z}}_{U} ,1 \le j \le N,} \hfill \\ {\left( {x_{l} ,y_{l} } \right) \in {\mathbf{S}}_{L} ,1 \le l \le L,} \hfill \\ \end{array} } \right.$$
(1)
Each.Then the distance is \({R_{i,l}}\) Between I-th transmitting antenna and I-th grid cell and distance \({R_{j,l}}\) Between j– with the second receiving antenna I-th grid cell can be calculated as follows:
$$\left\{ {\begin{array}{*{20}l} {R_{{i,l}} = \sqrt {(x_{i} – x_{l} )^{2} + (y_ {i} – y_{l} )^{2} } ,} \hfill \\ {R_{{j,l}} = \sqrt {(x_{j} – x_{l} )^{2} + ( y_{j} – y_{l} )^{2} } ,} \hfill \\ \end{array} } \right.$$
(2)
Each.
Target RCS data is given in the form of a two-dimensional matrix \({\delta}\) Which satisfies
$$\Delta = \left\{ {\delta _{{\theta ,\phi }} \in \Delta |0 \le \theta \le 2\pi ,0 \le \phi \le 2\pi } \ That’s right\}.$$
(3)
\(\theta\) means the angle of incidence, \(\phi\) means the angle of reflection. When the target is in it, I-th grid cell and direction meters-th angle \(\alpha_m\)satisfy \(\alpha _m\in \textbf{A}_M, 1\le {m}\le {M}\), all transmit and receive (TR) channels form different angles of incidence and reflection.Angle of incidence \(\Theta _{i,l,m}\) formed by I-th transmitting antenna and reflection angle \(\phi _{j,l,m}\) received by j-th receiving antenna I– Grid cell with rotation angle \(\alpha_m\) can be obtained by
$$\left\{ {\begin{array}{*{20}l} {\theta _{{i,l,m}} = {\text{arctan}}\frac{{y_{i} – y_ {l} }}{{x_{i} – x_{l} }} – \alpha _{m} ,} \hfill \\ {\phi _{{j,l,m}} = {\text{arctan }}\frac{{y_{j} – y_{l} }}{{x_{j} – x_{l} }} – \alpha _{m} ,} \hfill \\ \end{array} } \ I agree. $$
(Four)
Each. Target RCS I-th grid cell and rotation angle \(\alpha_m\) for \(ij\) The TR channel can be obtained as follows: \(\delta _{\theta _{i,l,m},\phi _{j,l,m}}\). As shown in equation (4), different TR channels obtain different RCS values āāof the target.According to the bistatic radar equation, the SNR is \(ij\) TR channel when target is included I– Grid cell with rotation angle \(\alpha_m\) It can be expressed as13
$$\xi _{l}^{{i,j,m}} = \frac{{P_{t} G_{t} G_{r} \lambda ^{2} \delta _{{\theta _{ {i,l,m}} ,\phi _{{j,l,m}} }} }}{{\left( {4\pi } \right)^{3} kT_{o} B_{n} R_{{j,l}}^{2} R_{{i,l}}^{2} }},$$
(Five)
where \({{P}_{t}}\) This is the transmission power of the radar antenna. \({{G}_{t}}\) and \({{G}_{r}}\) denote the gains of the transmitting and receiving antennas, respectively. \(\lambda\) Represents the signal wavelength. k is the Boltzmann constant. \({{T}_{0}}\) Standard room temperature, typically 290K. \({{B}_{n}}\) is the noise bandwidth.
When the target is in it, I-th grid cell, SNR of all grid cells \({{N}^{2}}\) The TR channel can be calculated using the following formula: (Five). The ratio of all echo energy to noise power, \(\xi _l\)calculated as the sum of the echo SNRs. \({{\xi }_{l}^{i,j,m}}\) in all angles on all channels30,]31given by
$$\xi _{l} = \sum\limits_{{i = 1}}^{N} {\sum\limits_{{j = 1}}^{N} {\sum\limits_{{m = 1 }}^{M} {\xi _{l}^{{i,j,m}} } } } .$$
(6)
When the target is in it, IFor the th grid cell, the detection probability of the radar system is expressed as:30,]31
$$\left\{ {\begin{array}{*{20}l} {\overline{{\text{P}}} _{{\text{d}}} = Q_{{J \times J} } \left( {\sqrt {2 \times \xi _{l} } ,\sqrt {2 \times \gamma _{T} } } \right),} \hfill \\ {\overline{{\text{ P}}} _{{{\text{fa}}}} = e^{{ – \gamma _{T} }} \sum\limits_{{n = 0}}^{{N \times N – 1 }} {\frac{{\gamma _{T}^{n} }}{{n!}}} ,} \hfill \\ \end{array} } \right.$$
(7)
where Q means markham Q-function; \(\gamma _T\) Represents the detection threshold. \(\bar{\text {P}}_\text {fa}\) Represents the false alarm probability. \(\xi _l\) can be calculated using the formula. Substitute (6) into the equation. (7) Calculate the detection probability. If the calculated detection probability exceeds the configured detection probability, IThe th grid cell is included in the detection range. When you calculate, \(\xi _l (l=1,2,3,…,L)\) Compute all grid cells and their corresponding detection probabilities and retrieve the grid cells whose calculated detection probabilities exceed a set value. The number of grid cells that meet the requirements is defined as the coverage area of āāthe radar system.
Mathematical model of optimization problem
According to the method of calculating the coverage area, a mathematical model of the optimization problem can be established.Both the antenna installation area Z and the monitoring area are divided into grids, forming a discrete matrix \(\textbf{Z}_U\) and \(\textbf{S}_L\),Each, U Grid cells of antenna placement area Z and L Grid cells within the monitoring area. Note that changes in the placement of the radar antenna have an impact when the target, area Z, and surveillance area are determined. \({R_{i,l}}\), \({R_{j,l}}\) and \({\delta _{\theta _{i,l,m},\phi _{j,l,m}}}\), which further affects the power and coverage area of āāthe echo signal. Therefore, the antenna position should be optimized to maximize the coverage area. The corresponding optimization problem can be expressed as:
$$\mathop {\max }\limits_{{\mathbf{P}}} {\mkern 1mu} {\text{ }}C\left( {\mathbf{P}} \right){\text{ }} s{\text{.}}t{\text{. }}p_{n} \in {\mathbf{Z}}_{U} ,{\text{ }}1 \le n \le N,$$
(8)
where \(\textbf{P} = [{p_1},{p_2},…,{p_N}]\), \({p_n} = ({x_n},{y_n})\) indicates the coordinates of the nth antenna, \(C\left( \textbf{P} \right)\) Represents the coverage area calculated based on antenna placement \(\textbf{P}\).bigger \(C\left( \textbf{P} \right)\) Shows improved monitoring performance under antenna placement \(\textbf{P}\). Therefore, an algorithm to optimize the antenna placement to maximize the coverage area will be considered in the next section.