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 than 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 super-resolution, radar, and ultrasound.

Introduction

We live in an analog world, but we would like our digital computers to interact with it. For example, sound is a continuous-time phenomenon, which can be characterized by the variations in air pressure as a function of time. For digital processing of such real-world signals to be possible, we require a sampling mechanism which converts continuous signals to discrete sequences of numbers, while preserving the information present in those signals.

In classical sampling theory, which dates back to the beginning of the twentieth century [1–3], a bandlimited signal whose maximum frequency is fmax is sampled at or above the Nyquist rate 2fmax.