A systematic approach to the time-domain computation of the impulse response and post-initial conditions of causal LTI systems at the origin

This paper presents a systematic approach to the computation of the impulse response in the time domain at the origin and post-initial conditions of an N ^th-order causal SISO LTI system differential equation. It is shown that the solution and its N-1 derivatives of such a differential equation with a unit impulse input in the time interval 0^- ≤ t ≤ 0⁺ are singularity functions, each containing one stepwise discontinuity term whose magnitude is one of the N post-initial conditions of the differential equation for t ≥ 0⁺. The approach presented is envisaged to provide a simplified tool not only for the computation but also for the teaching of the impulse response.