Signal model and observation model are commonly used to describe a dynamic system model in system identification or estimation such as Kalman filtering. The signal model is usually described by a linear dynamical equation driven by generating noise. The observation model is composed of a linear transformed signal and an additive white Gaussian noise. In this thesis, we set the generating noise to be a white binary sequence. This discrete generating noise makes the generating signal to be discrete. In contrast, the conventional generating signal is continuous. Discrete signal is simpler than the continuous signal. However, there still are too many states for this discrete signal. Therefore, defining the states and reducing the number of states are important in our work. In this thesis, we apply the tree structure to define the states. The number of states is reduced by focusing on the most probable working states. Afterwards, we apply two methods to recover the white sequence using the observation data. One is the Viterbi method; the other is Extended Kalman filter. Both methods are based upon the concept of signal states. Finally, we compare the error rates with the signal generated by continues phase modulation method.

In this thesis, we compute the distance spectrum for non-causal generating signals and compare the different spectrum effects for different non-causal systems. The non-causal system is the system which the present output is determined by the future and the past. The distance spectrum is the list of the difference measures of any two signals and search through all the possible event paths by trellis as much as possible. We use the method of the line difference to compute the distance spectrum of non-causal generating signal systems by defining the line and the line difference to find the distance for every pair of signals. Using this method, we have computed the distance spectrum for non-causal generating signals. Finally, we compare the different spectrums for different non-causal systems of different coefficients.