A convolution formula for time-invariant linear communication systems

Abstract

For every linear and time-invariant time-discrete (communication) system T : l∞ → l∞, formally, the following convolution formula can be derived: \input amssym.def $$(Tf)(n)=\sum_{k \in {\Bbb Z}} h(n-k) f(k),\quad n \in {\Bbb Z}, f \in l^{\infty},$$ where h = Tδ is the delta impulse response. This paper is concerned with the question under which assumptions linear and time-invariant time-discrete systems T : l∞ → l∞ can be characterized by this formula. For this purpose we derive a convolution formula in a more general situation which also leads to a well-known convolution formula in the time-continuous case. Copyright © 1999 John Wiley & Sons, Ltd.