###### Abstract  
This study proposes a closed-form direction-of-arrival (DoA) solution derived from multilateration equations for microphone arrays of co-centered and orthogonal pairs. The generalized cross-correlation phase transform (GCC-Phat) algorithm is used to obtain the time-difference-of-arrival (TDoA) values. Simulation studies have shown the success of our proposed method compared to existing DoA methods in the literature by varying the sampling frequency of the sound signal, inter-microphone distances, and the source distance. The results from the simulation are validated by the measurements from our experiments. Our proposed solution gives better results than the far-field solution against the angle error, which is more pronounced at incidence angles smaller than \(15^{\circ}\). These angle errors, which approach \(3^{\circ}\) using the far-field method, are reduced to less than 0.5 degrees using our proposed solution. Our solution also gives more stable results against TDoA measurement errors. Our proposed solution achieves a 66% improvement for azimuth angle and 5.88% improvement for elevation angle compared to the simulation results in the absence of TDoA measurement error, outperforming the far-field approach. When normally distributed sampling error is added to TDoA measurements, with a standard deviation of three samples, our proposed solution achieves a 41% improvement for azimuth angle and a 5.44% improvement for elevation angle. In our field measurements, an absolute mean error of 0.94 degrees was observed with our proposed method for azimuth angle. It is shown to be a more stable and faster solution method for real-time applications. estimation based on the signal correlation matrix. The most widely used methods in this class are MUSIC [5] and ESPRIT [6]. Indirect methods include approaches that perform location estimation based on the TDoA information between microphones [7; 8; 9]. In recent years, there has been a development of deep learning-based methods as well [10; 11; 12]. Direct localization methods are frequently used in applications where the location of the sound is known and the sound source is one of the multiple candidate sound sources. These methods are computationally demanding. The direction of arrival (DoA) of a sound source with unknown location and whose distance to the reference point of the microphone array is much greater than the inter-microphone distance of the microphone array is calculated using the far-field approach. In the far-field approach, it is assumed that the sound waves propagate to the microphones in parallel lines [1; 13]. It is frequently utilized to detect the DoA of sound sources with unknown locations. The far-field approach method uses TDoA times between microphones. The most frequently used method in the literature for obtaining TDoA values is the GCC method, which uses the PHAT weighting function [14]. The details of the method will not be explained in this paper. If the distance value between the sound source and the reference point of the microphone array is close to the distance value between the microphones, the sound source can be localized using the triangulation method or the multilateration method using the TDoA values [15; 16; 17; 18]. In the model used to detect the location of weapons that can fire ammunition with a Mach number greater than one, the distance can be estimated using the angle of arrival and the time difference between the shock wave and the muzzle burst, and the location can thus be determined [19]. Although different microphone geometries have been used in studies, uniform linear (ULA) [13] and uniform circular (UCA) [20] distributed, L-shaped, and spherical microphone arrays are more common. Among these microphone arrays, the L-shaped [21; 22] microphone arrays and the circular [3; 23], spherical [24; 25; 26; 27], and spatial [23] microphone arrays with a multiple of four co-dispersed microphones in the same plane have co-centered and orthogonal pairs of microphones.