## 5 Conclusions  
In this paper, for the interferometer DOA estimation algorithm, explicit expressions of the azimuth estimation error and elevation estimation error due to the frequency mismatch were derived and validated by numerical results. The main contribution of this paper was to derive the azimuth and elevation error analytically when the frequency of the array manifold was different from the carrier frequency of the incident signal source. The derivation in this paper was verified by comparing the simulation-based estimation error with the analytically-derived estimation error. It was derived that the analytic azimuth estimation error was identically zero even when the frequency of the incident signal was different from the nominal frequency used in the calculation of the array manifold, regardless of the true azimuth, \(\theta^{(0)}\), and the true elevation, \(\psi^{(0)}\). Since no approximation was used in deriving (26), the azimuth estimation error was exactly, not approximately, equal to zero even when the actual frequency of the incident signal was not equal to the nominal frequency employed in the calculation of the array manifold, which was confirmed in the upper plots of Figures 2 and 3 in the numerical results. For the elevation estimation error, to improve the agreement between the analytically-derived results and the simulation-based results, \(\cos(\Delta\psi)\approx 1-\frac{\Delta\psi^{2}}{2},\sin(\Delta\psi)\approx \Delta\psi\), rather than \(\cos(\Delta\psi)\approx 1\), \(\sin(\Delta\psi)\approx\Delta\psi\), could be adopted. Figure 10: Analytically-derived elevation error and simulation-based error with respect to the true elevation of the incident signal (\(f_{RF}=\frac{c}{\lambda_{RF}}\) = 1040 MHz and 1050 MHz). Figure 9: Analytically-derived elevation error and simulation-based error with respect to the true elevation of the incident signal (\(f_{RF}=\frac{c}{\lambda_{RF}}\) = 1020 MHz and 1030 MHz). It was illustrated in the lower plots of Figures 2b and 3a,b that the analytically-derived elevation estimation errors based on the approximations of \(\cos(\Delta\psi)\approx 1-\frac{\Delta\psi^{2}}{2},\sin(\Delta\psi)\approx\Delta\psi\) showed better agreement with the simulation-based elevation estimation error than the analytically-derived elevation estimation errors based on the approximations of \(\cos(\Delta\psi)\approx 1\), \(\sin(\Delta\psi)\approx\Delta\psi\). For a given approximation, the agreement of the simulation-based results and the analytically-derived results improved with the decrease of \(|\Delta\psi|\), which was confirmed by comparing the lower plots of Figures 2b and 3a,b. The performance due to the frequency mismatch reinforced once again the necessity of the wide-band DOA estimation algorithm. However, an interferometer using the narrow-band method may still be effective in the DOA of a signal source with a narrow bandwidth or a high elevation angle. [PERSON] performed the MATLAB simulation and wrote the initial draft. [PERSON] and [PERSON] derived the mathematical formulation of the proposed scheme. In addition, [PERSON] checked the numerical results and corrected the manuscript. All authors read and agreed to the published version of the manuscript. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07048294). The authors gratefully acknowledge the support from the Electronic Warfare Research Center at Gwangiu Institute of Science and Technology (GIST), originally funded by the Defense Acquisition Program Administration (DAPA) and the Agency for Defense Development (ADD). The authors declare no conflict of interest.