MULAN: A Blind and Off-Grid Method for Multichannel Echo Retrieval
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, seismology, 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.
This paper presents a method for estimating continuous-time delays for echos in synthetic multichannel impulse responses from underdetermined observations by doing so in the frequency domain.  The fundamental trick enabling this technique to work is the "annihilating filter technique", by which a convolution in the frequency domain with a second-order polynomial is able to cancel out signals that are exponential in frequency, such as delays.  This technique also applies for weighted sums of such signals, such as impulse responses made up of echoes.  It is able to work on irregularly sampled frequencies (e.g., a subset of the frequencies computed by a discrete Fourier transform).  Experiments are performed on synthetic impulse responses made up of pure delays with no additional noise and show that the proposed approach is significantly more accurate in identifying the locations and amplitudes of a known number of echoes than a LASSO-based approach and a standard cross-relation method.  Other experiments show that the method is still able to identify the time of 46% of 7 impulses within 1us from 2-microphone recordings and that this ability improves with the number of microphones.  Quality: The paper is technically sound.  Its logic is relatively easy to follow, well presented, and seems to be correct.  It discloses (but perhaps doesn't highlight) the assumptions that it makes in terms of the compositions of the impulse responses being analyzed (sums of pure impulses) and the necessity of providing the correct number of impulses in the impulse response. Experimental evaluation shows that it works very well within these constraints.  Under-determined impulse response estimation is a hard problem and this approach is able to perform quite well for a constrained, but meaningful, subset of conditions.  Clarity: The paper is very clearly written. It provides a thorough background to impulse responses and their estimation using clear notation and diagrams.  The descriptions of "four important bottlenecks" of existing methods seems a little over-emphasized, in that the first two (non-sparse and on-grid) seem to be quite similar, and the third (strongly rely on knowledge of filter length) is similar to the proposed method's strong reliance on knowledge of the number of echoes.  I believe that I would be able to reproduce the proposed approach from its description in this paper.  Originality: The paper does a very good job of summarizing existing approaches to the problem.  This approach of finite rate of innovation solutions to impulse estimation has recently been applied to other problems, but as far as I can tell has not been applied to impulse response estimation before.  Significance: While the subset of noise-free recordings of a known number of perfectly specular and frequency-independent echoes is somewhat limited, it also provides a solid model of more realistic signals, thus paving the way for further development.  It seems like this model should be robust to some amount of added noise, although it is not clear how robust it is to model mismatch or imperfect reflections.  The proposed approach addresses a difficult task in a better way than previous work, thus advancing the state-of-the-art in a demonstrable way.  The derived algorithm is a simple alternating minimization, where each step is a closed-form eigenvalue decomposition.  The fact that it can be used on an arbitrary subset of frequencies from a DFT is also interesting and facilitates follow-up work.   Minor comments: * Line 112: "the later" should be "the latter" * Equation (22): should Q be equal to the sum of the Toep_0(a_m)Diag(x_m) terms instead of their concatenation?   I have read the authors' response.