Theoretical issues on LTI systems that preserve signal richness

In this paper, theoretical issues about linear time invariant (LTI) systems that preserve signal richness are explored. This paper considers two particular definitions of signal richness and finds the necessary and sufficient conditions under which an LTI system preserves the richness property. Several examples are presented to clarify the issues involved in the problem. Paraunitary (PU) and unimodular matrices can be shown not to preserve richness unless they are constant matrices (or a delayed version in the PU case). Some richness preserving properties of cascaded systems are also investigated. A structured proof of the necessary and sufficient conditions is presented. The relationship between persistent excitation (PE) and the proposed definitions of richness is also described.