## 3 Monopulse Ratio Curve Improvement
### Proposed Monopulse Rate Curve Improvement Method Based on the Presented Tile  
An active phased array antenna can shape the beam into a specific shape through the amplitude and phase weighting of the excited signal. In this paper, the beamforming technique was used to improve the monopulse ratio curve. Thus, the sum and difference patterns were synthesized to expand the linear region of the monopulse ratio curve. The array factor of the active phased array antenna with the weighted amplitude and phase of the excited signal is given in Equation (9), as follows:  
\[\mathrm{AF}\,=\,\sum_{n}\sum_{m}W^{a}_{(n,m)}\exp\left(jk\vec{r}\cdot\hat{r}_ {(n,m)}+jW^{p}_{(n,m)}\right) \tag{9}\]  
where \(W^{a}_{(n,m)}\) and \(W^{p}_{(n,m)}\) are the amplitude and phase weighting of the \((n,m)\)-th radiator, respectively. The sum and difference pattern for calculating the monopulse ratio curve based on the AF of a single tile is shown in Equations (10) and (11). \[P_{\Sigma}=S(\theta,\phi)\sum_{n}\sum_{m}W^{a}_{(n,m)}\exp\left(jk\vec{r}\cdot \hat{r}_{(n,m)}+jW^{p}_{(n,m)}\right) \tag{10}\]  
\[P_{\Delta}=S(\theta,\phi)\sum_{n}\sum_{m}W^{a}_{(n,m)}\exp\left(jk\vec{r}\cdot \hat{r}_{(n,m)}+jW^{p}_{(n,m)}+jD_{(n,m)}\right) \tag{11}\]  
Based on Equations (10) and (11), the optimum \(W^{a}\) and \(W^{p}\) should be obtained to stably extend the linear section of the monopulse ratio curve. Thus, a type of heuristic algorithm was used, namely the vegetative propagation by runner (VPR) optimization algorithm [25]. The VPR algorithm is an algorithm that mimics the ecology of plants that reproduce through vegetative organs, and it improves a local optima problem by including the aging phenomena of plants and the continuous surrounding reproduction process. The VPR algorithm identifies the best cost evaluation location by moving and propagating based on the soil quality evaluation value (\(P_{b}\), \(S_{b}\)) obtained through the tap roots and lateral roots formed by each plant, as shown in Equation (12), as follows:  
\[Run^{i}=c_{s}rand()\left(S_{b}\,^{i}-X^{i}\right)\,+\,c_{p}rand()\left(P_{b}-X^ {i}\right) \tag{12}\]  
where \(i\) represents the index of the \(i\)-th plant, \(Run\) represents the position of the vegetative organs, \(c_{p}\) and \(c_{s}\) are the vegetative organ generation coefficient based on the tap root and lateral root, respectively, \(rand()\) represents the uniformly distributed random numbers, and \(X^{i}\) represents the position of the \(i\)-th plant. The position of the next-generation plant is calculated by accumulating the positions of the vegetative organ and the previous generation plant, as shown in Equation (13), as follows:  
\[X^{i}[t+1]\,=\,\,X^{i}[t]+run^{i}[t+1] \tag{13}\]  
where \(t\) is an index representing the generation that corresponds to the number of algorithm iterations. The next-generation plant's tap root is formed at the propagated location on the creeping stem. Therefore, the location of the tap root used for evaluating soil quality is the same as the location of the plant. Lateral roots then form around the tap roots. Therefore, the location of the entourage is expressed in Equation (14), as follows:  
\[lat^{i}\,=\,X^{i}+c_{z}\,randn() \tag{14}\]where \(c_{z}\) is the entourage occurrence coefficient, and \(randn()\) is a normally distributed random number. The VPR algorithm includes plant aging procedures. This procedure simulates the process through which plants absorb fewer nutrients as they age, and it is equivalent to Equation (15). \[P_{b}[t+1]\;=\;P_{b}[t]\exp(c_{t}\,age) \tag{15}\]  
where \(c_{t}\) is the aging coefficient, and \(age\) is the plant aging time, which increases by '1' according to the number of the algorithm iteration. Finally, \(age\) is initialized to '0' when a new, highest-quality location is identified as plants propagate. In this paper, the values corresponding to the plant location in the VPR algorithm are amplitude and phase weighting.