Description

The Wavelet system expansion is a tool used to decompose a function into a set of weighted basis functions that are localized in time and frequency. It is analogous to the Fourier series expansion, which represents a signal ft f t by a summation of complex sinusoids weighted by a set of coefficients. The complex sinusoids form a basis for the function space that ft f t is in. However, sinusoids have infinite support and infinite energy, but many signals of interest have finite energy and support. It does not seem practical to decompose a finite energy signal into a set of infinite energy basis functions.

A Wavelet system expansion can be written as the following

where φt φ t is the mother scaling function, ψt ψ t is the mother wavelet function, c j , k c j , k are the scaling or coarse resolution coefficients, and d j , k d j , k are the wavelet or high resolution coefficients. For an orthogonal wavelet system, the coefficients can be calculated by their inner products and where φ j , k t=2j2φ2jt-k φ j , k t 2 j 2 φ 2 j t k and ψ j , k t=2j2ψ2jt-k ψ j , k t 2 j 2 ψ 2 j t k . The variable j∈ℤ j denotes the scale of the scaling and wavelet functions and the variable k∈ℤ k denotes the shift. The expansion can be thought of as a coarse approximation of the signal ft f t , which contains the low-frequency energy, plus the finer details of the signal, which contains the high-frequency energy.

Wavelet and Scaling Functions

The scaling coefficients in Equation 1 represent the function ft f t projected on the function space  j 0  j 0 spanned by the scaling functions φ j = j 0 , k φ j = j 0 , k for all k∈ℤ k . As jj increases, the scaling space increases as well. The wavelet functions ψ j = j 0 , k ψ j = j 0 , k span the difference between  j 0  j 0 and  j 0 + 1  j 0 + 1 which is called  j 0  j 0 . The space spanned by  j 0  j 0 contain functions that are a coarse approximation of ft f t . By adding the wavelet functions in  j 0  j 0 , finer approximations of ft f t are added to the space. As higher scale wavelet functions are added to the space, the span of the space increases as shown in Figure 1.

Figure 2 shows the haar wavelet and scaling functions for different scales and shifts. The Haar mother scaling function and mother wavelet function can be defined as

As the scale increases, the scaling and wavelet functions have shorter support and capture finer details of the signal. The scaling functions at a lower scale can be expressed as a weighted sum of scaling functions at the next scale. For the case of the Haar system, the following relationship holds:

By requiring for any wavelet system that φt∈  0 φ t  0 , the space spanned by integer shifts of φt φ t , and φt∈  1 φ t  1 , the space spanned by integer shifts of φ2t φ 2 t , φt φ t can be expressed in terms of weighted sum of integer shifts of φ2t φ 2 t , or more formally as

This is also known as the recursion equation and is fundamental to the theory of the scaling functions. hn h n is known as the scaling filter and is a set of coefficients used to weight the scaling functions at the higher scale. The scaling filter completely determines the scaling function. hn h n must satisfy a set of necessary conditions in order for a given φt φ t to be a solution to Equation 7. There are also a set of separate set of sufficient conditions on hn h n to guarantee a solution to the basic recursion equation. The recursion equation can be written in the frequency domain as where Hω H ω is the Fourier transform of hn h n . The frequency domain equivalent of the recursion equation makes the significance of the scaling filter more apparent. Hω H ω completely describes the scaling function Φω Φ ω , where Φ0 Φ 0 is the eigenfunction of Hω H ω .