6 - The Discrete Wavelet Packet Transform

Summary

Introduction

In Chapter 4 we discussed the discrete wavelet transform (DWT), which essentially decomposes a time series X into coefficients that can be associated with different scales and times. We can thus regard the DWT of X as a ‘time/scale’ decomposition. The wavelet coefficients for a given scale T_j ≡ 2^J−1 tell us how localized weighted averages of X vary from one averaging period to the next. The scale T_j gives us the effective width in time (i.e., degree of localization) of the weighted averages. Because the DWT can be formulated in terms of filters, we can relate the notion of scale to certain bands of frequencies. The equivalent filter that yields the wavelet coefficients for scale T_j is approximately a band-pass filter with a pass-band given by [l/2^j+1, 1/2^j]. For a sample size N = 2^J, the N - 1 wavelet coefficients constitute - when taken together - an octave band decomposition of the frequency interval [1/2^J+1, 1/2], while the single scaling coefficient is associated with the interval [0, 1/2^J+1]. Taken as a whole, the DWT coefficients thus decompose the frequency interval [0, 1/2] into adjacent individual intervals.

In this chapter we consider the discrete wavelet packet transform (DWPT), which can be regarded as any one of a collection of orthonormal transforms, each of which can be readily computed using a very simple modification of the pyramid algorithm for the DWT.