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A fast speech extraction (FSE) method is presented using convex optimization made possible by pause detection of the speech sources. Sparse unmixing filters are sought by L1 regularization and the split Bregman method. A subdivided split Bregman method is developed for efficiently estimating long reverberations in real room recordings. The speech pause detection is based on a binary mask source separation method. The FSE method is evaluated and found to outperform existing blind speech separation approaches on both synthetic and room recorded data in terms of the overall computational speed and separation quality.