## 2 Derivation of the Azimuth and the Elevation Error under Frequency Mismatch
### Partial Derivative of the Cost Function with Respect to Elevation
 \tag{30}\]  
If \(\psi^{(0)}\) is equal to zero or \(\frac{\pi}{2}\), the coefficient of the second order term, \(\sin(\psi^{(0)})\cos(\psi^{(0)})\), and the constant term, \(\left(\frac{1}{\lambda}-\frac{1}{\lambda_{RF}}\right)\sin\left(\psi^{(0)} \right)\cos\left(\psi^{(0)}\right)\), are equal to zero. In that case, (30) is no longer a second order equation. For \(\psi^{(0)}=0^{\circ}\), (30) reduces to:  
\[\left(\frac{1}{\lambda}-\frac{1}{\lambda_{RF}}\right)\Delta\psi=0,\]  
\(\Delta\psi\) should be zero unless \(\lambda\) is equal to \(\lambda_{RF}\). For \(\psi^{(0)}=90^{\circ}\), (30) reduces to:  
\[\frac{1}{\lambda}\Delta\psi=0,\]  
from which \(\Delta\psi\) should be zero. If \(\psi^{(0)}\) is not equal to zero or \(\frac{\pi}{2}\), (30) is a second order equation, and the roots of (30) are obtained from the quadratic formula. From (30), the constant term of the quadratic equation is given by:  
\[\left(\frac{1}{\lambda}-\frac{1}{\lambda_{RF}}\right)\sin\left(\psi^{(0)} \right)\cos\left(\psi^{(0)}\right). \tag{31}\]  
If the wavelength associated with the frequency of the incident signal, \(\lambda_{RF}\), is equal to the wavelength associated with the nominal frequency for the calculation of the array manifold, \(\lambda\), (31) reduces to identically zero. In this case, (30) is simplified to:  
\[\left(\Delta\psi\right)^{2}\left(\frac{1}{\lambda}\sin\left(\psi^{(0)}\right) \cos\left(\psi^{(0)}\right)\right)+\Delta\psi\left(\frac{1}{\lambda}\cos^{2} \left(\psi^{(0)}\right)-\frac{1}{\lambda}\sin^{2}\left(\psi^{(0)}\right)- \frac{1}{\lambda_{RF}}\cos^{2}\left(\psi^{(0)}\right)\right)=0 \tag{32}\]  
\[\Delta\psi\left(\Delta\psi\left(\frac{1}{\lambda}\sin\left(\psi^{(0)}\right) \cos\left(\psi^{(0)}\right)\right)+\left(\frac{1}{\lambda}\cos^{2}\left(\psi^{( 0)}\right)-\frac{1}{\lambda}\sin^{2}\left(\psi^{(0)}\right)-\frac{1}{\lambda_{ RF}}\cos^{2}\left(\psi^{(0)}\right)\right)\right)=0. \tag{33}\]  
For \(\lambda_{RF}=\lambda\), from (33), it can be seen that \(\Delta\psi=0\) is always a solution of (30). Since \(\Delta\theta\) is zero from (26), both \(\Delta\theta\) and \(\Delta\psi\) are identically zero for \(\lambda_{RF}=\lambda\), which is consistent with the following  
\[ \begin{array}{l}(\Delta\psi)^{4}\left[\frac{1}{4}\frac{1}{\lambda}\cos(\psi^{(0)}) \sin\left(\psi^{(0)}\right)\right]+\left(\Delta\psi\right)^{3}\left[-\frac{1}{2 \lambda}\cos^{2}(\psi^{(0)})+\frac{1}{2\lambda}\sin^{2}(\psi^{(0)})\right]\\ +\left(\Delta\psi\right)^{2}\left[-\frac{1}{2}\frac{1}{\lambda}\cos(\psi^{(0) })\sin\left(\psi^{(0)}\right)-\frac{1}{\lambda}\sin(\psi^{(0)})\cos(\psi^{(0) })\right]+\Delta\psi\left[-\frac{1}{\lambda}\sin^{2}(\psi^{(0)})\right]=0. \end{array}  \tag{36}\]  
It is clear that \(\Delta\psi=0\) is always one of the solutions of (36) under the approximation of \(\cos(\Delta\psi)\approx 1-\frac{\Delta\psi^{2}}{2}\), \(\sin(\Delta\psi)\approx\Delta\psi\), as well as under the approximation of \(\cos(\Delta\psi)\approx 1\), \(\sin(\Delta\psi)=\Delta\psi\). Consequently, following the same reasoning used to get the elevation estimation error, \(\Delta\psi\), under the approximations of \(\cos(\Delta\psi)\approx 1\), \(\sin(\Delta\psi)=\Delta\psi\), it is true that \(\Delta\theta\) and \(\Delta\psi\) for \(\lambda_{RF}=\lambda\) are given by \(\Delta\theta=0\) and \(\Delta\psi=0\): without frequency offset, the elevation error, \(\Delta\psi\), as well as the azimuth estimation error, \(\Delta\theta\), are identically zero. (35) is a fourth order equation in \(\Delta\psi\). All the real roots of (35) are obtained, and each real root, \(\Delta\psi\), is used for evaluating the cost function in (2) with \(\theta=\theta^{(0)}+\Delta\theta,\psi=\psi^{(0)}+\Delta\psi\). \(\theta=\theta^{(0)}\) should be used since (26) should be true for the partial derivative of the cost function with respect to the azimuth to be zero. Out of the real roots of the fourth order Equation (35), the root maximizing the cost function value is selected as the elevation error, \(\Delta\psi\).