CHAPTER 4 - Linear Filters—General Properties with Applications to Continuous-Time Processes

Abstract

This chapter discusses the basic properties of linear filters. Linear filters provide an important class of models for physical transformations. For example, to a good degree of approximation, the earth behaves like a linear filter to seismic waves and the ocean to ocean waves. To a more restricted extent, phenomena, such as economic systems and biological systems, behave like linear filters for restricted lengths of time. Linear filters transform time series into new time series where the term time series can be interpreted in the broadest sense as meaning any numerical function of time whether continuous or discrete, random or non-random. Because of this feature, an important topic is the construction of special purpose linear filters to modify data to meet particular objectives or to display specific features of the data. The chapter discusses the methods for combining linear filters, by which a variety of special filters can be obtained. Moreover, it is presented in the chapter how high-pass, band-pass, and notch filters can be constructed from low-pass filters.