Subband- and Wavelet-Based Coding - Digital Signal Processing: Fundamentals and Applications - page 650

5 5

4

0

3

2 -5

0 0.5 1 0 0.5 1

10 50

0 0

-10 -50

0 0.5 1 0 0.5 1

50 50

0 0

-50 -50

0 0.5 1 0 0.5 1

Time (sec.) Time (sec.)

FIGURE 13.29

Approximations using Haar scaling functions and mother wavelets.

where the mother wavelet coefficients d j ðkÞ can also be determined by the inner product:

Z

d j ðkÞ¼ f ðtÞj jk ðtÞ ¼ f ðtÞ 2 j= 2 jð 2 j t kÞdt (13.42)

Figure 13.29 demonstrates the function approximation ( Figure 13.28 ) with the base scaling function at
resolution j ¼ 0, and mother wavelets at scales j ¼ 0 ; 1 ; 2 ; 3 ; 4. The combined approximation
( J ¼ 5) using Equation (13.41) is shown in Figure 13.30 .

13.5 MULTIRESOLUTION EQUATIONS
There are two very important equations for multiresolution analysis. Each scaling function can be
constructed by a linear combination of translations with the doubled frequency of a base scaling
function fð 2 tÞ , that is,

N

p 2

fðtÞ¼ h 0 ðkÞfð 2 t kÞ (13.43)

k ¼ N