Multi-channel blind deconvolution with short filters

This work considers the problem of recovering of an input signal and N filters from the circular convolutions of the signal with each filters. This problem is referred to as multi-channel blind deconvolution. The input signal is assumed to be random while the filters are assumed to be short. This problem can be reformulated as a low-rank matrix recovery problem and solved by nuclear norm minimization. Numerical experiments show the effectiveness of this algorithm under a certain generative model of the channels. Under this generative model and in the noiseless case, the proposed algorithm achieves successful recovery as soon as the oversampling factor is greater than or equal 1. It is also robust in the presence of noise. The hard constraint on the filters length imposed by our model is however a source of ill-posedness. In this work, sufficient and necessary conditions on the filters that guarantee well-posedness are given and proved. The derived identifiability conditions were already known. To our knowledge, this is however the first time that such conditions are linked to the null space of the Hessian of the cost function around the ground truth. Two applications of multi-channel blind deconvolution are then discussed. In the context of communications systems, multi-channel blind deconvolution can be used to perform multi-channel blind equalization. A complete blind equalization system is described and its performance is assessed in numerical simulations. In the context of geophysical imaging, multi-channel blind deconvolution can be used to perform Green’s functions retrieval. Unfortunately, with the proposed model and algorithm, this last application suffers from ill-posedness.