Helena Peic Tukuljac, Antoine Deleforge, Remi Gribonval
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 location and weights be recovered? This problem has broad applications in fields such as sonars, seismology, ultrasounds or room acoustics. It belongs to the broader class of blind channel identification problems, which have been intensively studied in signal processing. All existing methods proceed in two steps: (i) blind estimation of sparse discrete-time filters and (ii) echo information retrieval by peak picking. 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 also strongly limited. This is the so-called basis-mismatch problem in compressed sensing. We propose a radically different approach to the problem, building on top of 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 magnitudes in precision.