Haar Wavelets

Abstract

A Haar wavelet is the simplest type of wavelet. In discrete form, Haar wavelets are related to a mathematical operation called the Haar transform. The Haar transform serves as a prototype for all other wavelet transforms. Studying the Haar transform in detail will provide a good foundation for understanding the more sophisticated wavelet transforms which we shall describe in the next chapter. In this chapter we shall describe how the Haar transform can be used for compressing audio signals and for removing noise. Our discussion of these applications will set the stage for the more powerful wavelet transforms to come and their applications to these same problems. One distinctive feature that the Haar transform enjoys is that it lends itself easily to simple hand calculations. We shall illustrate many concepts by both simple hand calculations and more involved computer computations. 1.1 The Haar transform In this section we shall introduce the basic notions connected with the Haar transform, which we shall examine in more detail in later sections.