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A Robbins-Monro  stochastic approximation procedure for identifying a finite memory time-discrete time-stationary linear system from noisy input-output measurements is developed. Two algorithms are presented to sequentially identify the linear system. The first one is derived, based on the minimization of the mean-square error between the unknown system and a model, and is shown to develop a bias which depends only on the variance of the input signal measurement error. Under the assumption that this variance is known a priori, a second algorithm is developed which, in the limit, identifies the unknown system exactly. The case when the covariance matrix of the random input sequence is not positive definite is also considered.