Random Processes: Filtering, Estimation, and Detection

7 Optimum Linear Filters: The Wiener Approach

7.1 OPTIMUM FILTER FORMULATION

In a general sense we think of a filter as a black box that takes information in via a continuous signal x(t) or discrete time signal x[n], processes it and produces a continuous output signal y(t) or discretete time output y[n]. Conceptually this is illustrated in Figure 7.1. For example, the input signal might be the sum of a desired signal plus noise, and we would like to obtain as an output a cleaner version of the desired signal. In discussing optimum filters to perform a particular processing, we must be concerned with the following questions: What do we wish to have for the output? How are we going to process the input to obtain the desired output? What are we going to use as a performance measure for determining optimality? And what are the known statistical properties of the input and output processes? Each of these questions are discussed in turn.

7.1.1 What Is to Be Estimated?

Although the desired output can be almost anything this chapter will concentrate primarily on the following main problems: prediction, filtering, smoothing, and estimation of properties.

The process of prediction determines an estimate of a process X(t) at time λ units in the future by processing X(t) at only the current time and times in the past.

The process of filtering gives as an output an estimate of a given signal process from the input of another process that is somehow related to the signal process.

By smoothing ...