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We address the performance analysis of the maximum likelihood (ML) direction-of-arrival (DOA) estimation algorithm in the case of azimuth/elevation estimation of two incident signals using the uniform circular array (UCA). Based on the Taylor series expansion and approximation, we get explicit expressions of the root mean square errors (RMSEs) of the azimuths and elevations. The validity of the derived expressions is shown by comparing the analytic results with the simulation results. The derivation in this paper is further verified by illustrating the consistency of the analytic results with the Cramer-Rao lower bound (CRLB).