## 2 Derivation of the Azimuth and the Elevation Error under Frequency Mismatch
### Partial Derivative of the Cost Function with Respect to Elevation
In summary, the expressions for the estimation errors of \(\Delta\theta\) and \(\Delta\psi\), under the approximations of \(\sin(\Delta\psi)\approx\Delta\psi\) and \(\cos(\Delta\psi)\approx 1-\frac{\Delta\psi^{2}}{2}\), are given by (26) and a root out of the four roots of (35)  
From (35), the coefficients of the fourth order equation in \(\Delta\psi\) are dependent on \(\psi^{(0)},\lambda\), and \(\lambda_{RF}\), not on \(\theta^{(0)}\), which is also true under the approximations of \(\sin(\Delta\psi)\approx\Delta\psi\) and \(\cos(\Delta\psi)\approx 1\). Under the approximations of \(\sin(\Delta\psi)\approx\Delta\psi\) and \(\cos(\Delta\psi)\approx 1-\frac{\Delta\psi^{2}}{2}\), the elevation estimation error, \(\Delta\psi\), is dependent on how different the wavelength of the incident signal, \(\lambda_{RF}\), is from the nominal wavelength used for the calculation of the array manifold, \(\lambda\). The elevation error, \(\Delta\psi\), is also dependent on the true elevation of the incident signal, \(\psi^{(0)}\), not the true azimuth of the incident signal, \(\theta^{(0)}\). Consequently, from (26) and (34), the dependence of the elevation estimation error on the true elevation, the wavelength of the incident signal, and the nominal wavelength for the array manifold is the same both for the approximations of \(\sin(\Delta\psi)\approx\Delta\psi,\cos(\Delta\psi)\approx 1\) and for the approximations of \(\sin(\Delta\psi)\approx\Delta\psi,\cos(\Delta\psi)\approx 1-\frac{\Delta\psi^{2}}{2}\). How accurate the first order [PERSON] approximation and the second order approximation are highly depends on how large the elevation estimation error, \(\Delta\psi\), is: the accuracy of the first order approximation, \(\cos(\Delta\psi)\approx 1\), degrades as \(\cos(\Delta\psi)\approx 1\) increases. For improvement of the accuracy of the [PERSON] approximation, the second order approximation, \(\cos(\Delta\psi)\approx 1-\frac{\Delta\psi^{2}}{2}\), can be applied. Note that the difference between the first order [PERSON] approximation and the second order [PERSON] approximation is \(\frac{\Delta\psi^{2}}{2}\), implying that the superiority of the second order approximation over the first order approximation is dominant for large \(\Delta\psi\):  
For small elevation estimation error, adopting only the first order [PERSON] approximation results in the agreement between the simulation-based results and the analytically-derived results. Inconsistency between the simulation-based results and the analytically-derived results increases with the increase of the absolute value of \(\Delta\psi\), implying that the second order [PERSON] approximation should be adopted for the agreement of the simulation-based results and the analytically-derived results when the absolute value of elevation estimation error, \(\Delta\psi\), is large. For small \(\Delta\psi\), since \(\frac{\Delta\psi^{2}}{2}\) is small, the superiority of the second order approximation over the first order approximation is very small, implying that the second order approximation need not be adopted for small elevation estimation error.