A unified transform for LTI systems --- presented as a (generalized) frame (Q871464)

Summary: We present a set of functions in \(L^2([0,\infty))\) and show it to be a (tight) generalized frame (as presented by \textit{G. Kaiser} [``A friendly guide to wavelets'', Boston: Birkhäuser (1994; Zbl 0839.42011)]). 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.