Subband- and Wavelet-Based Coding - Digital Signal Processing: Fundamentals and Applications - page 651

80

60

40

20

0

-20

-40

-60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s ec .)

FIGURE 13.30

Signal coded using the wavelets at resolution J ¼ 5.

where h 0 ðkÞ is a set of scaling function coefficients (wavelet filter coefficients). The mother wavelet
function can also be built by a sum of translations with the double frequency of the base scaling
function fð 2 tÞ , that is,

N

p 2

jðtÞ¼ h 1 ðkÞfð 2 t kÞ (13.44)

k¼ N

where h 1 ðkÞ is another set of wavelet filter coefficients. Let us verify these two relationships via
Example 13.5 below.

EXAMPLE 13.5
Determine h 0 ðkÞ for the Haar father wavelet.
Solution:
From Equation (13.43) , we can express
fðtÞ¼ p h 0 ð0Þfð2tÞþ p h 0 ð1Þfð2t 1Þ

Then we deduce that

h 0 ð0Þ¼h 0 ð1Þ¼1= p

Figure 13.31 shows that the Haar father wavelet is the sum of two scaling functions at scale j ¼ 1.