## 3 Monopulse Ratio Curve Improvement
### Monopulse Ratio Curve Calculation Method  
The monopulse technique forms a sum pattern and a difference pattern based on the beam-forming technique and estimates the location of the communication partner based on the ratio of the signals received by them. Here, the ratio of the sum pattern and the difference pattern is called the monopulse ratio curve, and it is used as an indicator for estimating the communication counterpart's location.  
Typically, the patterns for sum and difference signals are created using a monopulse comparator. However, in the case of an active phased array antenna, a pattern for the sum and difference signals can be formed through a phase-weighted combination of the excited signal of each of the arrayed radiating elements. Equation (2) shows the formula for the sum and difference patterns of the active phased array antenna based on the phase weighting combination, as follows:  
\[P_{\Sigma}=S(\theta,\phi)\sum_{n}\sum_{m}\exp\left(jk\overline{r}\cdot\hat{r} _{(n,m)}\right) \tag{2}\]  
\[P_{\Delta}=S(\theta,\phi)\sum_{n}\sum_{m}\exp\left(jk\overline{r}\cdot\hat{r} _{(n,m)}+jD_{(n,m)}\right) \tag{3}\]  
where \(S(\theta,\phi)\) is the radiation characteristic of a single radiating element (microstrip patch), \(n\) is the radiating element index in elevation reference (four elements), \(m\) is the radiating element index in azimuth reference (eight elements), \(k\) is the propagation constant, \(\overline{r}\) is the unit vector in the \(r\) direction in a spherical coordinate system, \(\hat{r}\) is the position vector of the \((n,m)\)-th radiating element, and \(D\) is the phase weight for the difference pattern synthesis.  
As can be seen in Equation (2), the sum pattern is a pattern without a phase shift weight, it is identical to the radiation characteristics of a single tile without beam steering. The phase weighting \(D\) for forming the difference pattern is set as shown in Equation (4):  
\[D_{(n.m)}=\begin{cases}0,\ \ (1\leq m\leq 4,\ \ \forall n)\\ \pi,\ \ (5\leq m\leq 8,\ \ \forall n)\end{cases} \tag{4}\]  
Figure 4: Radiation characteristics for each tile in (**a**) polar plot and (**b**) rectangular plot.  
Based on the sum and difference patterns, the monopulse ratio curve for estimating the counterpart's location is calculated as shown in Equations (5) and (6). Here, Equation (5) is the formula for the estimated angle, and Equation (6) is the formula for distinguishing the direction (left/right) of the communication counterpart relative to the direction of the broadside for tile, as follows:  
\[an\ =\ \left|\ \frac{\Delta}{\Sigma}\ \right| \tag{5}\]  
\[dir\ =\ \tan^{-1}\biggl{(}-Im\biggl{(}\frac{\Delta}{\Sigma}\biggr{)}\biggr{)} \tag{6}\]  
where \(\sum\) is the sum signal, \(\Delta\) is the difference signal, and \(Im()\) is the imaginary output function.  
The monopulse ratio curve is generally found to have a slope of '1', which provides a 1:1 correspondence between the calculated value of Equations (5) and (6) and the estimated angle. For this purpose, the coefficient (\(c_{r}\)) for monopulse ratio curve control is applied to Equation (5). The optimal monopulse ratio curve control coefficient (\(c_{r}\)) can be calculated using MMSE (Minimum Mean Square Error), as shown in Equation (7):  
\[c_{r}^{opt}\ =\ \underset{c_{r}}{min}\ \mathbb{E}|\phi\ -\ c_{r}\,an|^{2},\ \ (0 \leq\phi\leq\phi_{c}) \tag{7}\]  
where \(\phi_{c}\) is the range required for coefficient optimization.  
The estimated angular accuracy of the monopulse ratio curve with the calculated coefficients is given by the mean absolute error (\(MAE\)), as shown in Equation (8).  
\[MAE\ =\ |\phi-\ c_{r}^{opt}\,an|,\ \ (0\leq\phi\leq\phi_{c}) \tag{8}\]