Plenty of techniques have been developed to estimate the direction of arrivals (DOAs) of multiple signals. However, regarding coherent signals measured by non-uniform and circular arrays, they still have been remained uncompleted. Therefore, a new approach using an element transposition of the covariance matrix was proposed to estimate DOAs of multiple coherent and non-coherent signals by an arbitrary symmetrical antenna array. The method described mathematically and proven by simulation in the paper was based on a balanced property of phase differences of the antenna array to de-correlate the received signals. The simulations were performed for linear array and circular array. Thus, they indicated that our solution allowed us to estimate and distinguish the DOAs of both non-coherent and coherent signals with higher accuracy and resolution than that of the conventional methods.