Model-Based Processing

4Model‐Based Processors

4.1 Introduction

In this chapter, we introduce the fundamental concepts of model‐based signal processing using state‐space models 1,2. We first develop the paradigm using the linear, (time‐varying) Gauss–Markov model of the previous chapter and then investigate the required conditional expectations leading to the well‐known linear Kalman filter (LKF) processor 3. Based on this fundamental theme, we progress to the idea of the linearization of a nonlinear, state‐space model also discussed in the previous chapter, where we investigate a linearized processor – the linearized Kalman filter (LZKF). It is shown that the resulting processor can provide a solution (time‐varying) to the nonlinear state estimation. We then develop the extended Kalman filter (EKF) as a special case of the LZKF, linearizing about the most available state estimate, rather than a reference trajectory. Next we take it one step further to briefly discuss the iterated–extended Kalman filter (IEKF) demonstrating improved performance.

We introduce an entirely different approach to Kalman filtering – the unscented Kalman filter (UKF) that evolves from a statistical linearization, rather than a dynamic model linearization of the previous suite of nonlinear approaches. The theory is based on the concept of “sigma‐point” transformations that enable a much better matching of first‐ and second‐order statistics of the assumed posterior distribution with even higher orders achievable.

Finally, we ...