We consider the estimation of multiple time-domain sparse filters from echoic mixtures of several unknown sources, when the sources are sparse in the time-frequency domain. We propose a sparse filter estimation framework consisting of two steps: a) a clustering step to group the time-frequency points of mixtures where only one source is active, for each source; b) a convex optimisation step to estimate the filters based on a time-frequency domain cross-relation. We propose a new wideband formulation of a frequency domain cross-relation, besides the one based on classical narrowband approximation. The solutions of the convex optimisation problem, formed using the cross-relation, are characterised. Numerical evaluation shows the benefit of using the wideband cross-relation for sparse echoic filter estimation. Further, the potential of the proposed framework for blind estimation of sparse echoic filters is demonstrated in a controlled experimental setting where in the proposed approach outperforms the state of the art blind filter estimation techniques, when the filters are sufficiently sparse.
