The definition of discretization proposed previously by the authors for continuous-time signals is extended to include an impulse and its derivatives, and then applied to descriptor systems that produce such signals. This definition provides bases for the continuous-time impulses to have their discrete-time counterparts, by taking values at multiple sampling points into account, rather than a single point that has traditionally been used for ordinary functions. Theorems are developed based on the definition, one of which provides a useful formula for choosing a sufficient initial-condition for a discrete-time descriptor system such that its initial response corresponds, in the sense of the proposed discretization definition, to the continuous-time original. This solves the long-standing issue of ambiguity surrounding discontinuous initial responses and that of inconsistent initial conditions that have obscured the validity of discrete-time descriptor models and of their numerical computations. (C) 2017 Elsevier Ltd. All rights reserved.