## 2 Derivation of the Azimuth and the Elevation Error under Frequency Mismatch  
Figure 1 illustrates the geometry of an interferometer direction of arrival estimation where \(\mathbf{z}_{m}\) denotes the \(m^{\text{th}}\) sensor location and \(\kappa\) denotes a direction vector for an incident signal from \(\theta=\theta^{(0)}\) and \(\psi=\psi^{(0)}\). A two-dimensional array is adopted for simultaneous estimation of the azimuth and  
elevation. For a uniform circular array (UCA) with \(M\) antennas, the expression in (1) represents the phase difference between the phase at the \(m^{\text{th}}\) sensor location and the phase at the origin for an incident signal from \(\theta=\theta^{(0)}\) and \(\psi=\psi^{(0)}\):  
\[\varphi_{m}=\frac{2\pi r}{\lambda_{RF}}\cos(\theta^{(0)}-\frac{2\pi m}{M})\cos (\psi^{(0)}). \tag{1}\]  
where \(\lambda_{RF}\) denotes the wavelength of an incident signal.  
The derivation of (1) is given in Appendix A. In (1), \(\theta^{(0)}\) and \(\psi^{(0)}\) are the true azimuth and true elevation. \(r\) is the radius of an array antenna system. \(M\) is the number of sensors.  
In the interferometer algorithm using the cosine cost function, the cost function [16] can be written as:  
\[J(\theta,\psi)=\sum_{m=0}^{M-1}\cos\bigg{[}\frac{2\pi r}{\lambda}\cos(\theta- \frac{2\pi m}{M})\cos(\psi)-\frac{2\pi r}{\lambda_{RF}}\cos(\theta^{(0)}-\frac {2\pi m}{M})\cos(\psi^{(0)})\bigg{]}. \tag{2}\]  
The expression in the brackets of Equation (2) is the phase difference between two phase values. The former phase expression in the brackets of Equation (2) is the relative phase at the \(m^{\text{th}}\) sensor location with respect to the phase at the origin for the incident signal from azimuth \(=\theta\) and elevation \(=\psi\) associated with the wavelength of \(\lambda\).  
The latter phase expression in the brackets of Equation (2) is the relative phase at the \(m^{\text{th}}\) sensor location with respect to the phase at the origin for the incident signal from the azimuth \(=\theta^{(0)}\) and elevation \(=\psi^{(0)}\).  
Since \(\cos(x)\) has a local maximum at \(x=0\) for \(-\pi\leq x\leq\pi\), the cost function is maximized when the two phase expressions in the brackets are equal. Therefore, we are concerned with what values of \(\theta\) and \(\psi\) in the former phase expression maximize the expression in (2), which is equivalent to what values of \(\theta\) and \(\psi\) in the former phase expression make the former phase value equal to the latter phase value.  
Figure 1: Sensor locations of the uniform circular array (UCA) and direction vector for an incident signal.  
For the cost function in (2) to be maximized at \(\theta=\theta^{(0)}+\Delta\theta\) and \(\psi=\psi^{(0)}+\Delta\psi\), the partial derivative of the cost function in (2) with respect to the azimuth, \(\theta\), and the partial derivative of the cost function in (2) with respect to the elevation, \(\psi\), should be simultaneously zero at \(\theta=\theta^{(0)}+\Delta\theta\), \(\psi=\psi^{(0)}+\Delta\psi\):  
\[\left.\begin{array}{l}\frac{\partial J(\theta,\psi)}{\partial\theta}\\ \end{array}\right|\begin{array}{l}\theta=\theta^{(0)}+\Delta\theta\\ \psi=\psi^{(0)}+\Delta\psi\end{array}=0. \tag{3}\]  
\[\left.\begin{array}{l}\frac{\partial J(\theta,\psi)}{\partial\psi}\\ \end{array}\right|\begin{array}{l}\theta=\theta^{(0)}+\Delta\theta\\ \psi=\psi^{(0)}+\Delta\psi\end{array}=0 \tag{4}\]  
In this section, it is rigorously derived how \(\Delta\theta\) and \(\Delta\psi\) can be obtained from the constraints of (3) and (4). Note that, \(\Delta\theta\) and \(\Delta\psi\) are the azimuth estimation error and the elevation estimation error, respectively. Therefore, the final objective of this paper is to get expressions of the azimuth error and the elevation error without actually maximizing the cost function in (2) via computationally intensive exhaustive two-dimensional search with respect to \(\theta\) and \(\psi\).