  In this paper, we study the problem of estimating the state of a dynamic
state-space system where the output is subject to quantization. We compare some
classical approaches and a new development in the literature to obtain the
filtering and smoothing distributions of the state conditioned to quantized
data. The classical approaches include the Extended Kalman filter/smoother in
which we consider an approximation of the quantizer non-linearity based on the
arctan function, the quantized Kalman filter/smoother, the Unscented Kalman
filter/smoother, and the Sequential Monte Carlo sampling method also called
particle filter/smoother. We consider a new approach based on the Gaussian sum
filter/smoother where the probability mass function of the quantized data given
the state is modeled as an integral equation and approximated using
Gauss-Legendre quadrature. The Particle filter is addressed considering some
resampling methods used to deal with the degeneracy problem. Also, the sample
impoverishment caused by the resampling method is addressed by introducing
diversity in the samples set using the Markov Chain Monte Carlo method. In this
paper, we discuss the implementation of the aforementioned algorithms and the
Particle filter/smoother implementation is studied by using different
resampling methods combined with two Markov Chain algorithms. A numerical
simulation is presented to analyze the accuracy of the estimation and the
computational cost.
