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An efficient time domain shooting and bouncing ray (TDSBR) method is proposed to analyze the transient near-field scattering from an electrically large complex object illuminated by a far-field antenna source. The time-domain far-field incident sources can be derived by a convolution of pulsed excitation with the inverse Fourier transform of frequency domain (FD) far-fields radiated from an antenna. In order to obtain accurate near-field scattering results, time domain physical optics (TDPO) near-field integral representations are proposed and reduced to closed-form expressions to improve computing efficiency by applying locally expanded Green function approximation. Different from the plane wave incident situation, in the case of antenna radiation, the incident fields of each individual facet in the ray path are represented by the radiation fields from the equivalent mirror antenna derived in this paper. Since the radiation fields naturally contains the time delay term, the complicated processing of the time delay problem can be avoided by solving the equivalent mirror antenna radiation fields. It is simpler and more accurate than the traditional time domain geometrical optics (TDGO) method. Numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.