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In this paper we describe an algorithm for estimating the parameters of a linear, discrete-time system, in state-space form, having quantized measurements. The estimation is carried out using the maximum likelihood criterion. The solution is found using the expectation maximization (EM) algorithm. A technical difficulty in applying this algorithm for this problem is that the a posteriori probability density function, found in the EM algorithm, is non-Gaussian. To deal with this issue, we sequentially approximate it using scenarios, i.e., a weighted sum of impulses which are deterministically computed. Numerical experiments show that the proposed approach leads to a significantly more accurate estimation than the one obtained by ignoring the presence of the quantizer and applying standard estimation methods.