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This article addresses the problem of estimating a multivariate linear system from its output when the input is an unobservable sequence of random vectors with finite-alphabet distribution. By explicitly utilizing the finite-alphabet property, an estimation method is proposed under the traditional inverse filtering paradigm as a generalization of a univariate method that has been studied previously. Identifiability of multivariate systems by the proposed method is proved mathematically under very mild conditions that can be satisfied even if the input is nonstationary and has both cross-channel and serial statistical dependencies. Statistical super-efficiency in estimating both parametric and nonparametric systems is also established for an alphabet-based cost function.