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The state estimation for linear discrete-time systems with non-Gaussian state and output noise is a challenging problem. In this paper, we derive the suboptimal quadratic estimate of the state by means of a recursive algorithm. The solution is obtained by applying the Kalman filter to a suitably augmented system, which is fully observable. The augmented system is constructed as the aggregate of the actual system, and the observable part of a system having as state the second Kronecker power of the original state, namely the quadratic system. To extract the observable part of the quadratic system, the rank of the corresponding observability matrix is needed, which is a difficult task. We provide a closed form expression for such a rank, as a function of the spectrum of the dynamical matrix of the original system. This approach guarantees the internal stability of the estimation filter, and moreover, permits a reduction in the computational burden.