Introduction to wavelets and wavelet transforms: a primer
Introduction to wavelets and wavelet transforms: a primer
bySidney Burrus, Ramesh A. Gopinath, and Haitao Guo
The goal of most modern wavelet research is to create a set of basis functions (or general expansion functions) and transforms that will give an informative, efficient, and useful description of a function or signal. If the signal is represented as a function of time, wavelets provide efficient localization in both time and frequency or scale. Another central idea is that of multi-resolution where the decomposition of a signal is in terms of the resolution of detail.
1 Introduction to Wavelets
1.1 Wavelets and Wavelet Expansion Systems
A signal or function f(t) can be expressed as a linear decomposition by:
If the expansion is unique, the set is called a basis for the class of functions that can be so expressed.
If the basis is orthogonal, meaning
then the coefficients can be calculated by the inner product
If the basis set is not orthogonal, then a dual basis set exists such that the desired coefficients can be given.
For the wavelet expansion, a two-parameter system is constructed
The set of expansion coefficients are called the discrete wavelet transform (DWT) of f(t).
Where the Fourier series maps a one-dimensional function of a continuous variable into a one-dimensional sequence of coefficients, the wavelet expansion maps it into a two-dimensional array of coefficients.
All so-called first-generation wavelet systems are generated from a single scaling function or wavelet by simple scaling and translation.
Almost all useful wavelet systems also satisfy the multi-resolution conditions. This means that if a set of signals can be represented by a weighted sum of ψ(t - k), then a larger set of signals (including the original) can be represented by a weighted sum of ψ(2t - k).
The lower resolution coefficients can be calculated from the higher resolution coefficients by a tree-structured algorithm called a filter bank.
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