Abstract: For an unbounded domain, a local high-order transmitting boundary is implemented based on the scaled boundary finite element method and the continued-fraction expansion. An improved continued-fraction expansion is employed to solve the dynamic stiffness matrix equation of an unbounded domain. Compared to the previous approach, it is numerically more robust for large-scale systems and arbitrarily high orders of expansion. The equation of this boundary is a system of first-order ordinary differential equations in the time domain. The stability of the high-order boundary depends on the general eigenproblem of the coefficient matrices. Possible spurious modes can be eliminated using the spectral shifting technique. The accuracy and the robustness of the high-order transmitting boundary are verified by two numerical examples.