Distributional System Representations on Bandlimited Signals

In this paper we analyze the distributional convergence behavior of time-domain convolution type system representations on the Paley-Wiener space
PW
_π
¹
. Two convolution integrals as well as the discrete counterpart, the convolution sum, are treated. It is shown that there exist stable linear time-invariant (LTI) systems for which the convolution integral representation does not exist because the integral is divergent, even if the convergence is interpreted in a distributional sense. Furthermore, we completely characterize all stable LTI systems for which a convolution representation is possible by giving a necessary and sufficient condition for convergence. The classical and the distributional convergence behavior are compared, and differences between the convergence of the convolution integral and the convolution sum are discussed. Finally, the results are illustrated by numerical examples.