Subband- and Wavelet-Based Coding - Digital Signal Processing: Fundamentals and Applications - page 652

1.5

1

0.5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1.5

1

0.5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1.5

1

0.5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s ec .)

FIGURE 13.31

Haar wavelets in Example 13.5.

EXAMPLE 13.6
Determine h 1 ðkÞ for the Haar mother wavelet.
Solution:
From Equation (13.44) , we can write

p h 1 ð0Þfð2tÞþ p h 1 ð1Þfð2t 1Þ

jðtÞ¼

Hence, we deduce that

p p

h 1 ð0Þ¼1= and h 1 ð1Þ¼1=

Figure 13.32 shows that the Haar mother wavelet is the difference of two scaling functions at scale j ¼ 1.

Notice that the relation between H 0 ðzÞ and H 1 ðzÞ exists and is given by
h 1 ðkÞ¼ð 1 Þ k h 0 ðN 1 kÞ (13.45)

We can verify Equation (13.45) for the Haar wavelet:

h 1 ðkÞ¼ð 1 Þ k h 0 ð 1 kÞ