## 3 Applications
### Simulations  
The first simulation studies were performed in two-dimensional space using the fifth microphone array shown in Figure 6. In real-time applications, when measuring the TDoA time, hardware time errors of \(\Delta t<t_{s}\), depending on the sampling frequency of the audio recording devices, and \(N_{err}\) measurement sample errors caused by environmental noise as a result of the GCC-Phat algorithm are introduced. These errors are modeled in our simulation study as shown in Figure 4.  
The simulation studies in the XY plane assumed that the distance of the sound source is 100 m. The largest measurement errors in the direction of arrival are at angles below \(10^{\circ}\)[29]. Figure 7 shows the results obtained using different \(d_{m}\) distances in the case of \(d_{k}=100\) m and \(F_{s}=44.1\) kHz. Figure 8 shows the effect of the source position on the direction-of-arrival estimation for near and far distances. The results for the source at 20 m and 500 m distances are shown when the sampling frequency and the microphone spacing are fixed. Figure 6: Orthogonally located sensor pair microphone arrays (**1–4**) for azimuth and elevation angles, (**5**,6**) for only azimuth angle. The error changes of the estimates made with 1 degree increments between 0 and 90 degrees while the \(d_{k}\) distance increases from 5 m to 100 m with 1 m increments, and the  
Figure 8: The effect of the source distance on the direction-of-arrival estimation in the case of \(d_{m}=0.5\) m and \(Fs=44.1\) kHz (**a**) \(d_{k}=20\) m and (**b**) \(d_{k}=500\) m.  
Figure 7: The effect of the distance between microphone pairs on the direction-of-arrival estimation in the case of \(d_{k}=100\) m and \(Fs=44.1\) kHz (**a**) \(d_{m}=0.1\) m, (**b**) \(d_{m}=0.5\) m, (**c**) \(d_{m}=1\) m, and (**d**) \(d_{m}=1.5\) m.  
\(d_{m}\) distance increases from 0.1 m to 1 m with 0.1 m increments are shown in Figure 9. The error values are calculated using the mean absolute error function:  
\[MAE=\frac{1}{N}\frac{N}{0}|\theta-\hat{\theta}| \tag{32}\]  
The expressions \(\theta\) and \(\hat{\theta}\) represent the true and measured angle values, respectively. The mean error decreases below 0.5\({}^{\text{o}}\) after \(d_{m}>0.3\) m and \(d_{k}>35\) m. Additionally, the maximum error values are reached as the \(d_{m}/d_{k}\) ratio approaches one. The results obtained for different sampling frequencies with fixed microphone spacing and source position are shown in Figure 10.  
Figure 10: The effect of sampling frequency on the elevation angle estimation in the case of \(d_{m}=0.5\) m, \(d_{k}=100\) m (**a**) \(F_{8}=22.05\) kHz and (**b**) \(F_{8}=96\) kHz. Figure 9: The change in the absolute average error between 0 and 90 degrees in the case of \(d_{m}\in[0.1,1]\) m, \(d_{k}\in[5,100]\) m \(Fs=44.1\) kHz. The simulation studies that led to the results in Figures 7-10 were investigated under the assumption that the GCC-Phat error in the model shown in Figure 4 is zero. In real-time applications, errors occur in the measurement of TDoA values due to environmental noises, reverberation, and differences in microphone detection patterns. Normally distributed noise with \(N_{err}=N(0,3)\) was added instead of these errors, and the mean absolute error values were obtained by simulation with 1000 repetitions. The simulation results without the added error are shown in Figure 11a, and the simulation results with the added error are shown in Figure 11b. Figure 11a shows that our proposed method produces better results than other methods except for the 55-70-degree range. Figure 11b shows that our proposed method produces stable results than other methods up to 80 degrees. [PERSON]'s method is more successful in the 79-89-degree range. Table 1 shows the total average error values of the methods in the 0-90-degree range.