# 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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