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The class of nonlinear systems treated in this technical note consists of the discrete time nonlinear systems that are formed by the interconnection of linear systems through static nonlinearities with few inputs. This special structure is exploited to reduce the dimension of the integrals involved in the propagation of mean values and covariances, thus permitting accurate calculations. Furthermore, a new quadrature scheme suitable for nonlinear Kalman filtering is introduced. The proposed techniques are applied to a seven-dimensional numerical example. The results show that they can increase the performance significantly.