Subband- and Wavelet-Based Coding - Digital Signal Processing: Fundamentals and Applications - page 645

Solution:
Based on Equation (13.37) , we can determine the wavelet at each required scale as follows:
For j ¼ 0, f 0k ðtÞ¼fðt kÞ, only k ¼ 0 is required to cover a 1-second duration.

For j ¼ 1, f 1k ðtÞ¼ p fð2t kÞ, k ¼ 0 and k ¼ 1 are required.

For j ¼ 2, f 2k ðtÞ¼2fð4t kÞ, we need k ¼ 0; 1; 2; 3.
For j ¼ 3, f 3k ðtÞ¼2 p fð8t kÞ, we need k ¼ 0; 1; 2; / ; 7.

Using Figure 13.25 ( a), we obtain the plots shown in Figure 13.26 .

2

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2

1

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2

1

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

0.5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t (sec.)

FIGURE 13.26

Haar father wavelets at different scales and translations.

EXAMPLE 13.3
Sketch the Haar mother wavelet families for four different scales,

j ¼ 0; 1; 2; 3, for a period of 1 second.

Solution:
Based on Equation (13.36) , we have
For j ¼ 0, j 0k ¼ jðt kÞ, k ¼ 0 and k ¼ 1 are required.
For j ¼ 1, j 1k ¼

p jð2t kÞ, k ¼ 0 and k ¼ 1 are required.
For j ¼ 2, j 2k ¼ 2jð4t kÞ, we need k ¼ 0; 1; 2; 3.
For j ¼ 3, j 3k ðtÞ¼2 p jð8t kÞ, we need k ¼ 0; 1; 2; / ; 7.