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This paper studies a novel decomposition technique, suitable for blind separation of linear mixtures of signals comprising finite-length symbols. The observed symbols are first modeled as channel responses in a multiple-input-multiple-output (MIMO) model, while the channel inputs are conceptually considered sparse positive pulse trains carrying the information about the symbol arising times. Our decomposition approach compensates channel responses and aims at reconstructing the input pulse trains directly. The algorithm is derived first for the overdetermined noiseless MIMO case. A generalized scheme is then provided for the underdetermined mixtures in noisy environments. Although blind, the proposed technique approaches Bayesian optimal linear minimum mean square error estimator and is, hence, significantly noise resistant. The results of simulation tests prove it can be applied to considerably underdetermined convolutive mixtures and even to the mixtures of moderately correlated input pulse trains, with their cross-correlation up to 10% of its maximum possible value.