Newland’s Harmonic Wavelets

Abstract

So far, all wavelets have been constructed from dilation equations with real coefficients. However, many wavelets cannot always be expressed in functional form. As the number of coefficients in the dilation equation increases, wavelets get increasingly longer and the Fourier transforms of wavelets become more tightly confined to an octave band of frequencies. It turns out that the spectrum of a wavelet with n coefficients becomes more boxlike as n increases. This fact led Newland (1993a,b) to introduce a new harmonic wavelet ψ(x) whose spectrum is exactly like a box, so that the magnitude of its Fourier transform $\hat{\uppsi }(\upomega )$ is zero except for an octave band of frequencies. Furthermore, he generalized the concept of the harmonic wavelet to describe a family of mixed wavelets with the simple mathematical structure. It is also shown that this family provides a complete set of orthonormal basis functions for signal analysis. This chapter is devoted to Newland’s harmonic wavelets and their basic properties.

Wavelets are without doubt an exciting and intuitive concept. The concept brings with it a new way of thinking, which is absolutely essential and was entirely missing in previously existing algorithms.

Yves Meyer