Performance Bounds and Design Criteria for Estimating Finite Rate of Innovation Signals

TL;DR

This paper establishes fundamental limits and design criteria for estimating finite rate of innovation signals from noisy data, analyzing the impact of sampling methods and signal structure on estimation accuracy and stability.

Contribution

It provides a fundamental estimation accuracy bound, analyzes noise effects on FRI techniques, and proposes a method for optimal sampling kernel design based on the Karhunen--Loève transform.

Findings

Fundamental limit on estimation accuracy regardless of sampling method

Noise significantly affects certain FRI recovery techniques

Optimal sampling kernels can be designed using a generalized Karhunen--Loève transform

Abstract

In this paper, we consider the problem of estimating finite rate of innovation (FRI) signals from noisy measurements, and specifically analyze the interaction between FRI techniques and the underlying sampling methods. We first obtain a fundamental limit on the estimation accuracy attainable regardless of the sampling method. Next, we provide a bound on the performance achievable using any specific sampling approach. Essential differences between the noisy and noise-free cases arise from this analysis. In particular, we identify settings in which noise-free recovery techniques deteriorate substantially under slight noise levels, thus quantifying the numerical instability inherent in such methods. This instability, which is only present in some families of FRI signals, is shown to be related to a specific type of structure, which can be characterized by viewing the signal model as a union of subspaces. Finally, we develop a methodology for choosing the optimal sampling kernels based on a generalization of the Karhunen--Lo\`eve transform. The results are illustrated for several types of time-delay estimation problems.