Reflections on sampling-filters for compressive sensing and finite-innovations-rate models

This paper revisits sampling-filters for signals having a finite rate of innovations. Such filters arise in many applications including digital communications and compressive sensing, and mulitchannel versions of these systems have been considered in the past. The main focus of this paper is on sampling-filters that result in perfect reconstruction (PR), or zero-forcing (ZF). Conditions for existence of these filters are expressed both in terms of bandwidth requirement and in the framework of Riesz basis. Many practical advantages induced by the Riesz basis property are also pointed out. When the sampling filters for PR exist, they are in general not unique. Optimum filters that minimize the effect of noise are discussed and compared with energy compaction filters, which are suboptimal.