Distributional system representations on bandlimited signals

Ullrich J. Monich, Holger Boche

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


In this paper we analyze the distributional convergence behavior of time-domain convolution type system representations on the Paley-Wiener space PWπ1. 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.

Original languageEnglish
Article number5482100
Pages (from-to)4557-4571
Number of pages15
JournalIEEE Transactions on Signal Processing
Issue number9
StatePublished - Sep 2010
Externally publishedYes


  • Bandlimited signal
  • convergence
  • convolution
  • distribution
  • system representation


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