Detalhes do Documento

This paper develops a rigorous theory of convolution for anti-causal linear time-invariant (LTI) systems on isolated nonuniform time scales. We resolve a fundamental inconsistency in the existing Bohner–Guseinov shift operator by introducing a modified boundary-value problem, and show that the shift defined via bilateral delta Laplace inversion uniquely satisfies this formulation. Explicit shift formulas for the impulse, unit step, ramp, and exponential functions are derived. Convolution is then defined for anti-causal systems, and the main result establishes that the bilateral delta Laplace transform of the convolution of two functions factors as the product of their individual transforms, under explicit uniform-convergence conditions. As an application, we characterise all LTI systems on isolated time scales via their impulse response and derive the transfer function relation. First- and second-order anti-causal systems are analysed in detail, and the effect of non-uniform graininess on the impulse response is quantified numerically for a jittered grid.