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A new formulation of the problem of identification of the impulse response of a discrete time linear time invariant system from a finite set of finite length noisy I-O measurements is given. This transforms the problem to an input estimation problem of a new type. We present a solution to this input estimation problem in the framework of maximum a posteriori (MAP) estimation. This involves an optimization problem of a special type which we solve largely based on some recent results in linear system theory related to system structure and the geometric and polynomial theories of linear systems which are algebraic in nature. Furthermore, the solution to the identification problem we consider in this paper also provides a solution to the problems of 1) Deadbeat output control with internal stability by minimum norm input, 2) Disturbance decoupling with internal stability by minimum norm input, and 3) Exact model matching of linear systems by minimum norm compensators.