A fully digital model for Kalman filters

The Kalman filter is a mathematical method, whose purpose is to process noisy measurements in order to obtain an estimate of some relevant parameters of a system. It represents a valuable tool in the GNSS area, with some of its main applications related to the computation of the user PVT solution and to the integration of GNSS receivers with INS or other sensors. The Kalman filter is based on a state space representation, that describes the analyzed system as a set of differential equations that establishes the connections between the inputs, the outputs and the state variables of the analyzed system. In the continuous time domain there exists a large class of physical processes with a time evolution well described by means of stochastic differential equations. A typical problem is the need for an equivalent system in the discrete time, due to the discrete nature of the data to be processed. In the literature, it is quite common to solve this problem in the continuous time domain and to approximate the solution using a Taylor series approximation, to obtain an approximate discrete time version of the continuous time problem. By the way, other methods exist, based on the possibility to transform a continuous-time system to a discrete-time system by means of transformations from the Laplace complex plane to the z plane. These methods are widely used in the digital signal processing community, for example, to design digital filters from their analog counterparts. The main advantage of this approach is that it is very easily implemented by applying some mechanical rules. Moreover the nature of the approximation introduced by the Laplace-z transformation is a-priori known and clearly readable in the frequency domain. In the following the classical methods based on the Taylor approximation and on the Laplace-z transformations will be analyzed and compared.