## 3 Applications
### Simulations
To investigate the change in the estimation error when the distance between the pairs of microphones placed perpendicularly is not equal, three different microphone array geometries were created with \(d_{mx}=0.5-d_{my}=0.5\), \(d_{mx}=0.6-d_{my}=0.374\), and \(d_{mx}=0.374-d_{my}=0.6\), ensuring that the hypotenuses of the triangles formed by the microphones have equal lengths. The simulation results without error are shown in Figure 12a, and with noise are shown in Figure 12b. \begin{table}
\begin{tabular}{c c c c} \hline
**Error Edit** & **Far Field** & **[PERSON] and [PERSON]** & **Proposed Closed Form** \\ \hline
**No** & 0.65 & 0.65 & 0.22 \\
**\(N(0,3)\)** & 8.01 & 7.85 & 4.72 \\ \hline \end{tabular}
\end{table} 
Table 1: The average values of the azimuth angle estimation errors between 0 and 90 degrees in the case of \(d_{m}=0.5\) m, \(d_{k}=100\) m \(Fs=44.1\) kHz. Figure 11: The azimuth angle estimation errors between 0 and 90 degrees in the case of \(d_{m}=0.5\) m, \(d_{k}=100\) m \(Fs=44.1\) kHz—(**a**) without error addition and (**b**) with error addition. Although no clear difference can be seen in Figure 12a, it can be seen in Figure 12b that the three changing conditions are more successful in the 0-30-degree range and that two of them are more successful in the 60-90-degree range. Table 2 shows the total average error values of the three different conditions in the 0-90-degree range. The performance of the elevation angle estimation calculated by Equation (31) was investigated by simulation studies using the first microphone array in Figure 6. The results obtained without adding error, assuming \(d_{m}=0.5\) m, \(d_{k}=100\) m, \(Fs=44.1\) kHz, and azimuth angle \(\theta=45\) are shown in Figure 13a, and obtained by adding noise are shown in Figure 13b. \begin{table}
\begin{tabular}{c c c c} \hline
**Error Edit** & \(d_{x}=0.5\)\(d_{y}=0.5\) & \(d_{x}=0.6\)\(d_{y}=0.374\) & \(d_{x}=0.374\)\(d_{y}=0.6\) \\ \hline
**No** & 0.2197 & 0.2776 & 0.2776 \\ \(N(0,3)\) & 4.7423 & 5.2387 & 5.2351 \\ \hline \end{tabular}
\end{table} 
Table 2: Average values of estimation errors between 0 and 90 degrees for different microphone array geometries. Figure 12: The effect of unequal microphone spacing on the azimuth angle estimation—(**a**) without error addition and (**b**) with error addition. Figure 13: The elevation angle estimation errors between 0 and 30 degrees in the case of \(d_{m}=0.5\) m, \(d_{k}=100\) m, \(Fs=44.1\) kHz, and \(\theta=45\)—(**a**) without error addition and (**b**) with error addition. The average error values of the error graphs shown in Figure 13 are presented in Table 3.  
The far-field approach cannot produce a solution when the expression to be taken the inverse of, especially at right angles, exceeds the range of \(-1\leq\cos^{-1}\left(\frac{v_{s}\times\Upsilon}{d_{m}}\right)\leq 1\) due to measurement errors in the TDoA value. This significantly impairs the measurement sensitivity at right angles. The solution in the far-field approach is also dependent on the speed of sound. The speed of sound varies depending on the weather conditions and affects the measurement results. Additionally, in the far-field approach, the number of samples obtained by the GCC-Phat algorithm is multiplied by \(t_{s}\) time and added to the formula. This increases the number of mathematical and algorithmic operations. Our proposed arrival angle detection method, since it is obtained from the source position equation, the quadrant in which the arrival angle is located, is precisely determined by the _arctanjant2_ function. Studies have been performed to determine the azimuth and elevation angles with the _tanjant_ function, but the quadrant information is still obtained by the algorithm [21; 22; 23]. Our proposed method is obtained as the result of a single function, not as a function of conditions like other methods.