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The diffusion behavior driven by bounded noise under the influence of a coupled harmonic potential is investigated in a two-dimensional coupled-damped model. With the help of the Laplace analysis we obtain exact descriptions for a particle's two-time dynamics which is subjected to a coupled harmonic potential and a coupled damping. The time lag is used to describe the velocity autocorrelation function and mean square displacement of the diffusing particle. The diffusion behavior for the time lag is also discussed with respect to the coupled items and the amplitude of bounded noise.