(BQ) Part 2 book "Wavelets and subband coding" has contents: Series expansions using wavelets and modulated bases; continuous wavelet and short time fourier transforms and frames; algorithms and complexity, signal compression and subband coding. | 4 Series Expansions Using Wavelets and Modulated Bases “All this time, the guard was looking at her, first through a telescope, then through a microscope, and then through an opera glass” — Lewis Carroll, Through the Looking Glass Series expansions of continuous-time signals of functions go back at least to Fourier’s original expansion of periodic functions. The idea of representing a signal as a sum of elementary basis functions or equivalently, to find orthonormal bases for certain function spaces, is very powerful. However, classic approaches have limitations, in particular, there are no “good” local Fourier series that have both good time and frequency localization. An alternative is the Haar basis where, in addition to time shifting, one uses scaling instead of modulation in order to obtain an orthonormal basis for L2 (R) [126]. This interesting construction was somewhat of a curiosity (together with a few other special constructions) until wavelet bases were found in the 1980’s [71, 180, 194, 21, 22, 175, 283]. Not only are there “good” orthonormal bases, but there also exist efficient algorithms to compute the wavelet coefficients. This is due to a fundamental relation between the continuous-time wavelet series and a set of (discrete-time) sequences. These correspond to a discrete-time filter bank which can be used, under certain conditions, to compute the wavelet series expansion. These relations follow from multiresolution analysis; a framework for analyzing wavelet bases [180, 194]. The emphasis of this chapter is on the construction of wavelet series. We also discuss local Fourier series and the construction of local cosine bases, which are “good” modulated bases [61]. Note that in this chapter we 209 210 CHAPTER 4 construct bases for L2 (R); however, these bases have much stronger characteristics as they are actually unconditional bases for Lp spaces, 1 < p < ∞ [73]. The development of wavelet orthonormal bases has been quite explosive in the last .
