Convex Optimization Approach for Stable Decomposition of Stream of Pulses

TL;DR

This paper presents a convex optimization method for accurately estimating delays and amplitudes in a stream of pulses, with error bounds proportional to noise and conditions for clustering around true delays, applicable in ultrasound and radar.

Contribution

It introduces a convex optimization framework for stable pulse stream decomposition that accounts for noise and separation conditions, improving estimation accuracy.

Findings

Recovery error is proportional to noise level.

Estimated delays cluster around true delays under mild conditions.

Positivity of amplitudes removes the need for separation conditions.

Abstract

This paper deals with the problem of estimating the delays and amplitudes of a weighted superposition of pulses, called stream of pulses. This problem is motivated by a variety of applications, such as ultrasound and radar. This paper shows that the recovery error of a tractable convex optimization problem is proportional to the noise level. Additionally, the estimated delays are clustered around the true delays. This holds provided that the pulse meets a few mild localization properties and that a separation condition holds. If the amplitudes are known to be positive, the separation is unnecessary. In this case, the recovery error is proportional to the noise level and depends on the maximal number of delays within a resolution cell.