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In recent years, many sparse estimation methods, also known as compressed sensing, have been developed. However, most of these methods presume that the measurement matrix is completely known. We develop a new blind maximum likelihood method — the expectation-sparse-maximization (ESpaM) algorithm — for models where the measurement matrix is the product of one unknown and one known matrix. This method is a variant of the expectation-maximization algorithm to deal with the resulting problem that the maximization step is no longer unique. The ESpaM algorithm is justified theoretically. We present as well numerical results for two concrete examples of blind channel identification in digital communications, a doubly-selective channel model and linear time invariant sparse channel model.