Distributional time-domain system representations

In this paper we analyze the convergence behavior of convolution-type system representations for the Paley-Wiener space PWπ¹. We completely characterize all stable linear time-invariant (LTI) systems for which we have convergence in the distributional sense by giving a necessary and sufficient condition for convergence. Furthermore, we prove that there are stable LTI systems and signals in PWπ¹ for which the convolution integral and the convolution sum diverge even in a distributional sense. In signal processing, distributions are often used to show convergence. Surprisingly, here we are in a situation where distributions cannot be used to justify convergence.