EURASIP Journal on Applied Signal Processing 
Volume 2006 (2006), Article ID 91604, 9 pages
doi:10.1155/ASP/2006/91604

A Unified Transform for LTI Systems—Presented as a (Generalized) Frame

Arie Feuer,1 Paul M. J. Van den Hof,2 and Peter S. C. Heuberger2

 1Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
2Delft Center for Systems and Control, Delft University of Technology, Delft 2628 , The Netherlands
Received 19 August 2004; Revised 30 May 2005; Accepted 31 May 2005

Abstract

We present a set of functions in L2([0,∞)) and show it to be a (tight) generalized frame (as presented by G. Kaiser (1994)). The analysis side of the frame operation is called the continuous unified transform. We show that some of the well-known transforms (such as Laplace, Laguerre, Kautz, and Hambo) result by creating different sampling patterns in the transform domain (or, equivalently, choosing a number of subsets of the original frame). Some of these resulting sets turn out to be generalized (tight) frames as well. The work reported here enhances the understanding of the interrelationships between the above-mentioned transforms. Furthermore, the impulse response of every stable finite-dimensional LTI system has a finite representation using the frame we introduce here, with obvious benefits in identification problems.