We first study the a posteriori probability density function of the state of a discrete-time system given the measurement data. By applying the Bayesian law to the state and measurement equations of the stochastic system, the a posteriori density is obtained in closed-form and computed recursively for arbitrary i.i.d. state noise and any discrete-type measurement noise (or multi-level digital signal). Then, our effort concentrates on the estimation of impulsive noise which interferes with the multi-level signal of interest. By considering the L
P-norm performance criterion, where 0<p⩽2, the corresponding estimator is obtained. Using a new damping function scheme, the performance of the estimator is improved even further. As an example, a highly impulsive state process driven by noise with symmetric α-stable distribution is estimated and then removed from the measurement data
