Subband- and Wavelet-Based Coding - Digital Signal Processing: Fundamentals and Applications - page 653

1

0

-1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

0

-1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

0

-1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (sec.)

FIGURE 13.32
Haar wavelets in Example 13.6.

p

0

h 1 ð 0 Þ¼ð 1 Þ h 0 ð 1 0 Þ¼h 0 ð 1 Þ¼ 1 =

p 2

1

h 1 ð 1 Þ¼ð 1 Þ h 0 ð 1 1 Þ¼h 0 ð 0 Þ¼ 1 =

This means that once we obtain the coefficients of h 0 ðkÞ , the coefficients h 1 ðkÞ can be determined via
Equation (13.45) . We do not aim to obtain wavelet filter coefficients here. The topic is beyond the
scope of this topic and the details are given in Akansu and Haddad (1992). Instead, some typical filter
coefficients for Haar and Daubechies are given in Table 13.2 .
We can apply the Daubechies-4 filter coefficients to examine multiresolution Equations (13.43) and
(13.44) . From Table 13.2 , we have

h 0 ð 0 Þ¼ 0 : 4830 ; h 0 ð 1 Þ¼ 0 : 8365 ; h 0 ð 2 Þ¼ 0 : 2241 ; h 0 ð 3 Þ¼ 0 : 1294

We then expand Equation (13.43) as

p 2 p 2 p 2 p 2

fðtÞ¼ h 0 ð 0 Þfð 2 tÞþ h 0 ð 1 Þfð 2 t 1 Þþ h 0 ð 2 Þfð 2 t 2 Þþ h 0 ð 3 Þfð 2 t 3 Þ