Necessary and sufficient conditions for LTI systems to preserve signal richness

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.

[1]  P. P. Vaidyanathan,et al.  Staying rich: LTI systems which preserve signal richness , 2005, Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005..

[2]  Anna Scaglione,et al.  Redundant filterbank precoders and equalizers. II. Blind channel estimation, synchronization, and direct equalization , 1999, IEEE Trans. Signal Process..

[3]  P. Vaidyanathan.  Multirate Systems And Filter Banks , 1992 .

[4]  See-May Phoong,et al.  Antipodal paraunitary precoding for OFDM application , 2004, 2004 IEEE International Symposium on Circuits and Systems (IEEE Cat. No.04CH37512).