1 Sampling at the Rate of Innovation: Theory and Applications

Parametric signals, such as streams of short pulses, appear in many applications including bio-imaging, radar, and spread-spectrum communication. The recently developed finite rate of innovation (FRI) framework, has paved the way to low rate sampling of such signals, by exploiting the fact that only a small number of parameters per unit of time are needed to fully describe them. For example, a stream of pulses can be uniquely defined by the time-delays of the pulses and their amplitudes, which leads to far fewer degrees of freedom then the signal’s Nyquist rate samples. This chapter provides an overview of FRI theory, algorithms and applications. We begin by discussing theoretical results and practical algorithms allowing perfect reconstruction of FRI signals from a minimal number of samples. We then turn to treat recovery from noisy measurements. Finally, we overview a diverse set of applications of FRI theory, in areas such as superresolution, radar and ultrasound.