Rakeness in the design of Analog-to-Information Conversion of Sparse and Localized Signals

TL;DR

This paper introduces the concept of rakeness to improve analog-to-information conversion for sparse, localized signals by balancing energy capture and reconstructability, demonstrated on ECG and image data.

Contribution

It proposes a new design criterion called rakeness, optimizing measurement processes for localized signals, and formulates a statistical trade-off with restricted isometry for better reconstruction.

Findings

Rakeness-based design improves measurement quality for localized signals.

The approach enhances SNR in signal reconstruction.

Applications to ECG and image data show significant performance gains.

Abstract

Design of Random Modulation Pre-Integration systems based on the restricted-isometry property may be suboptimal when the energy of the signals to be acquired is not evenly distributed, i.e. when they are both sparse and localized. To counter this, we introduce an additional design criterion, that we call rakeness, accounting for the amount of energy that the measurements capture from the signal to be acquired. Hence, for localized signals a proper system tuning increases the rakeness as well as the average SNR of the samples used in its reconstruction. Yet, maximizing average SNR may go against the need of capturing all the components that are potentially non-zero in a sparse signal, i.e., against the restricted isometry requirement ensuring reconstructability. What we propose is to administer the trade-off between rakeness and restricted isometry in a statistical way by laying down an optimization problem. The solution of such an optimization problem is the statistic of the process generating the random waveforms onto which the signal is projected to obtain the measurements. The formal definition of such a problems is given as well as its solution for signals that are either localized in frequency or in more generic domain. Sample applications, to ECG signals and small images of printed letters and numbers, show that rakeness-based design leads to non-negligible improvements in both cases.