Kalman Filter: continuous state space, discrete observations

This is just an idea. How can we model the kalman filter to get the state representation in continuous space when the observations to the system are actually from the discrete space. The discrete observation, for example, can be sequence of digits (0-9) where the digits themselves do not have numerical meaning but the sequence of observations are correlated. I know we can try Hidden Markov Model (HMM) for this, but I would like to get the continuous (real-valued vector) space state representation rather than the discrete space as in HMM.