**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 ...**

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