On Sampled Sequence Representations at Discontinuities and Their Impact on Discrete Convolution

If one compares the continuous-time convolution outputs with their sampled discrete counterparts, one may observe slight differences even when the sampling process itself is otherwise straightforward. This issue becomes noticeable when one or both continuous-time signals have a discontinuity at the sampling instant, such as t = 0. In this paper, we revisit this issue and explain its root causes: the treatment of midpoint values at discontinuities and the sampling-period scaling that appears when a continuous-time convolution is approximated in discrete time. Although the midpoint rule is not new, we show how this classical result can be used systematically to construct sampled sequence representations that are consistent with inverse-transform reconstruction at discontinuities. Based on this viewpoint, we derive midpoint-consistent sampled sequence representations for quite a few representative functions, and we show the corresponding implications for sampled convolution formulae. Several examples are used to compare conventional discrete formulae with midpoint-consistent sampled formulae and with samples of the continuous-time results. The proposed formulation is intended for sampled continuous-time signals at discontinuities; it is not meant to replace standard native discrete-time conventions used in digital signal processing.