Fast reconstruction of sparse relative impulse responses via second-order cone programming

The paper addresses the estimation of the relative transfer function (RTF) using incomplete information. For example, an RTF estimate might be recognized as too inaccurate in a number of frequency bins. When these values are dropped, an incomplete RTF is obtained. The goal is then to reconstruct a complete RTF estimate, based on (1) the remaining values, and (2) the sparsity of the relative impulse response, which is the time-domain counterpart of the RTF. We propose two fast algorithms for the RTF reconstruction that solve a second-order cone program (SOCP), and show their advantages over the LASSO formulation previously proposed in the literature. Simulations with speech signals show that in terms of speed and accuracy, the proposed algorithms are comparable with the LASSO solution and considerably faster compared to the generic ECOS solver. The new algorithms are, moreover, easier to control through their parameters, which brings their improved stability when the number of reliable frequency bins is very low (less than 10%).