MULAN: A Blind and Off-Grid Method for Multichannel Echo Retrieval
This addresses the basis-mismatch problem in fields like sonar and acoustics, offering a significant improvement over existing two-step methods that are limited by sampling rates.
The paper tackles the blind echo retrieval problem, where unknown source signals with delays and attenuations are measured by multiple sensors, by proposing a method that operates directly in the parameter space of echo locations and weights, achieving near-exact retrieval and outperforming conventional methods by several orders of magnitude in precision.
This paper addresses the general problem of blind echo retrieval, i.e., given M sensors measuring in the discrete-time domain M mixtures of K delayed and attenuated copies of an unknown source signal, can the echo locations and weights be recovered? This problem has broad applications in fields such as sonars, seismol-ogy, ultrasounds or room acoustics. It belongs to the broader class of blind channel identification problems, which have been intensively studied in signal processing. Existing methods in the literature proceed in two steps: (i) blind estimation of sparse discrete-time filters and (ii) echo information retrieval by peak-picking on filters. The precision of these methods is fundamentally limited by the rate at which the signals are sampled: estimated echo locations are necessary on-grid, and since true locations never match the sampling grid, the weight estimation precision is impacted. This is the so-called basis-mismatch problem in compressed sensing. We propose a radically different approach to the problem, building on the framework of finite-rate-of-innovation sampling. The approach operates directly in the parameter-space of echo locations and weights, and enables near-exact blind and off-grid echo retrieval from discrete-time measurements. It is shown to outperform conventional methods by several orders of magnitude in precision.