A geometrical approach to sampling signals with finite rate of innovation

Many signals of interest can be characterized by a finite number of parameters per unit of time. Instead of spanning a single linear space, these signals often lie on a union of spaces. Under this setting, traditional sampling schemes are either inapplicable or very inefficient. We present a framework for sampling these signals based on an injective projection operator, which "flattens" the signals down to a common low dimensional representation space while still preserving all the information. Standard sampling procedures can then be applied on that space. We show the necessary and sufficient conditions for such operators to exist and provide the minimum sampling rate for the representation space, which indicates the efficiency of this framework. These results provide a new perspective on the sampling of signals with finite rate of innovation and can serve as a guideline for designing new algorithms for a class of problems in signal processing and communications.