## 3 Antenna Analysis and DOA Estimation  
For impedance characteristic observation, we simulated input impedances for TE10 and TE20 modes, as shown in Figure 10. The resistance and reactance of the TE10 mode are 67.7 \(\Omega\) and 7.1 \(\Omega\), and for TE20 mode, the resistance and reactance of 51.3 \(\Omega\) and 15.0 \(\Omega\) are obtained, respectively. Figure 11a,b show the E-field distributions of the proposed antenna excited by SIWs with different modes, which are observed at \(81\times 51\) points in the y = 0 mm plane. Inside the SIWs, the electric-field distributions of the TE\({}_{10}\) and TE\({}_{20}\) modes are clearly observed, which is achieved by the structural differences of the SIWs. Each E-field with different modes in the SIWs radiates the sum and difference patterns through the single patch antenna. The multi-mode SIW feeder allows the sum and difference patterns to be radiated directly without the pattern comparator of conventional monopulse systems. As a result, the complexity of the system can be significantly decreased by reducing the number of radiators and by simplifying the waveguide pattern comparator. To examine the DOA estimation of the proposed antenna, we simulate and measure \(\Delta\)/\(\Sigma\) in terms of the incident angle \(\theta\), as represented in Figure 12. Solid, dashed, and dotted lines indicate the measured, simulated, and ideal results of the \(\Delta\)/\(\Sigma\) values for the DOA estimation. The simulated and measured results significantly deviate from the ideal results due to the asymmetric microstrip feeding line structure in TE\({}_{20}\) SIW, which causes distortions on the \(\Delta\) radiation pattern. Thus, to compensate for the deviations of the DOA estimations, we derive the fitting curve \(F(\theta)\) using a quadratic equation with the coefficients of \(k_{1}\) and \(k_{2}\), as written in Equation (1). The coefficients of \(k_{1}\) and \(k_{2}\) correspond to the left (\(\theta<0\)) and right (\(\theta\geq 0\)  
Figure 8: Simulated and measured maximum gains in the upper hemisphere of the proposed antenna: (a) TE\({}_{10}\) mode; (b) TE\({}_{20}\) mode.  
Figure 9: Two-dimensional radiation patterns of the proposed antenna: (a) TE\({}_{10}\) mode; (b) TE\({}_{20}\) mode.  
sides of the fitting curve, and the determined values of \(k_{1}\) and \(k_{2}\) are 0.019 and 0.022, respectively. A calibration function \(C(\theta)\) is then defined by calculating the difference between the \(\Delta/\Sigma(\theta)_{\text{ideal}}\) and \(F(\theta)\) in Equation (2). Finally, the \(\Delta/\Sigma(\theta)_{\text{w/cal}}\) of the proposed antenna with calibration can be achieved by compensating the deviation with \(C(\theta)\) in order to minimize the root mean square error (RMSE) of the DOA estimations as suggested in Equation (3).  
\[F(\theta)=\left\{\begin{array}{l}F_{1}(\theta)=k_{1}\theta^{2}\ (\theta<0)\\ F_{2}(\theta)=k_{2}\theta^{2}\ (\theta\geq 0)\end{array}\right., \tag{1}\]  
\[C(\theta)=\Delta/\Sigma(\theta)_{\text{ideal}}-F(\theta), \tag{2}\]  
\[\Delta/\Sigma(\theta)_{\text{w/ cal.}}=\Delta/\Sigma(\theta)_{\text{w/o cal.}}+C(\theta). \tag{3}\]  
Figure 11: Near-field distributions: (**a**) TE\({}_{10}\) mode; (**b**) TE\({}_{20}\) mode.  
Figure 10: Input impedance of the proposed antenna: (**a**) TE\({}_{10}\) mode; (**b**) TE\({}_{20}\) mode.  
Figure 13 shows the root mean square error (RMSE) with and without the calibration for the DOA estimation according to the incident angle \(\theta\). The RMSE values are averaged at incident angles between \(-0.5\) and \(0.5\). Through this calibration, the average RMSE of \(0.07\) is obtained, which is \(0.1\) lower than that of the uncalibrated result.