Subband- and Wavelet-Based Coding - Digital Signal Processing: Fundamentals and Applications - page 660

The MATLAB verifications follow:
>> xx0¼rconv([1 1]/sqrt(2),c1)
xx0 ¼ 1.2500 1.2500
>> c0¼xx0(1:2:2)
c0 ¼ 1.2500
>> xx1¼rconv([1 1]/sqrt(2),c1)
xx1 ¼ 1.7500 1.7500
>> d0¼xx1(1:2:2)
d0 ¼ 1.7500
Finally, we pack the wavelet coefficients w 2 ðkÞ at j ¼ 2 together as

5
4

7
4

1 1

w 2 ðkÞ¼½c 0 ð0Þd 0 ð0Þd 1 ð0Þd 1 ð1Þ ¼

p p

2

Then the function can be expanded using one scaling function and three mother wavelet functions:
f ðtÞ z f 2 ðtÞ¼ N
k ¼ N

c 0 ðkÞfðt kÞþ 1
j ¼0

N
k ¼ N

j=2 j t kÞ

d j ðkÞ2 jð2

¼ 5

4 fðtÞþ 7 4 jðtÞþjð2tÞ 1 2 jð2t 1Þ

Figure 13.38 shows the plots for each function and the combined function to verify that f ðtÞ does have amplitudes
of 4, 2, 1, and 0.

2

0

-2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2

0

-2

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

6

- -2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (sec.)

FIGURE 13.38

The signal reconstructed using the Haar wavelets in Example 13.7.