The theme of the previous two chapters will now be extended to the case in which the variables of interest change over time. These variables can be either real-valued vectors (as in Chapter 3), or discrete class variables that only cover a finite number of symbols (as in Chapter 2). In both cases, the variables of interest are called *state variables*.

The design of a state estimator is based on a state *space model* that describes the underlying physical process of the application. For instance, in a tracking application, the variables of interest are the position and velocity of a moving object. The state space model gives the connection between the velocity and the position (which, in this case, is a kinematical relation). Variables, like position and velocity, are real numbers. Such variables are called *continuous states*.

The design of a state estimator is also based on a *measurement model* that describes how the data of a sensory system depend on the state variables. For instance, in a radar tracking system, the measurements are the azimuth and range of the object. Here, the measurements are directly related to the two-dimensional position of the object if represented in polar coordinates.

The estimation of a dynamic class variable, i.e. a *discrete state* variable is sometimes called *mode estimation* or *labelling*. An example is in speech recognition where – for the recognition of a word – a sequence of phonetic classes must be estimated from a sequence of acoustic ...

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