Introduction

Finding the best polynomial curve fit in the least squares sense to a set of data was first developed by Gauss during the 18th century when he was only 17. Hundreds of years later Gauss's least squares technique is still used as the most significant criteria for fitting data to curves. In fact the method of least squares is still used as the basis for many curve-fitting programs used on personal computers. All of the filtering techniques that will be discussed in this text are, one way or the other, based on Gauss's original method of least squares. In this chapter we will investigate the least squares technique as applied to signals, which can be described by a polynomial. Then we shall demonstrate via numerous examples the various properties of the method of least squares when applied to noisy measurement data.

It is not always obvious from the measurement data which is the correct-order polynomial to use to best fit the measurement data in the least squares sense. This type of knowledge is often based on understanding the dynamics of the problem or from information derived from mathematical techniques, such as systems identification, that have been previously applied to the problem. If common sense and good engineering judgement are ignored in formulating the least squares problem, disaster can result.

Overview

Before we begin to look at actual examples using the method of least squares, it may be helpful to step back and take a broad view of what we...