There are many ways to define richness of a discrete time signal. In this paper, we consider a particular definition and explore the conditions under which a linear time invariant (LTI) system preserves the richness property. A set of necessary and sufficient conditions has been found. Using this, paraunitary and unimodular matrices can be shown not to preserve richness unless they are constant matrices (or a delayed version in the paraunitary case). A structured proof of the necessary and sufficient conditions is also presented.