In this brief paper, a new sharper upper bound for the error ${parallel}x(kT)-x_k{parallel}$ that derives from the procedure of discretization of the solution of a Linear descriptor (regular) differential input system with consistent initial conditions, and Time-Invariant coefficients (LTI) is calculated and fully discussed. Practically speaking, considering numerous applications in engineering (especially in robotics and digital control) and computer science, we are very interested in determining such kind of upper bounds, since they are significant in the design process of the sampling period T.