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 degrees. These angle errors, which approach 3 degrees 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.
