Subband- and Wavelet-Based Coding - Digital Signal Processing: Fundamentals and Applications - page 659

We will verify this expression later. Applying the wavelet analysis equations, we have
c 1 ðkÞ¼ N
m¼ N

c 2 ðmÞh 0 ðm 2kÞ

d 1 ðkÞ¼ N
m¼ N

c 2 ðmÞh 1 ðm 2kÞ

Specifically,

p 2

c 1 ð0Þ¼ N
m¼ N

c 2 ðmÞh 0 ðmÞ¼c 2 ð0Þh 0 ð0Þþc 2 ð1Þh 0 ð1Þ¼2 1

p þ 1 1 p ¼ 3

2

c 1 ð1Þ¼ N
m¼ N

1
2

1

p þ 0 1 p ¼ 1

c 2 ðmÞh 0 ðm 2Þ¼c 2 ð2Þh 0 ð0Þþc 2 ð3Þh 0 ð1Þ¼

p

2

d 1 ð0Þ¼ N
m¼ N

c 2 ðmÞh 1 ðmÞ¼c 2 ð0Þh 1 ð0Þþc 2 ð1Þh 1 ð1Þ¼2 1 1

1 2

p þ 1 p

¼ p

1
2

1

N

1 ¼ 1

2 p þ 0 p p

d 1 ð1Þ¼ c 2 ðmÞh 1 ðm 2Þ¼c 2 ð2Þh 1 ð0Þþc 2 ð3Þh 1 ð1Þ¼

m¼ N

Using the subband coding method in Figure 13.36 yields
>> x0¼rconv([1 1]/sqrt(2),[2 1 0.5 0])
x0 ¼ 2.1213 0.3536 0.3536 1.4142
>> c1¼x0(1:2:4)
c1 ¼ 2.1213 0.3536
>> x1¼rconv([1 1]/sqrt(2),[2 1 0.5 0])
x1 ¼ 0.7071 1.0607 0.3536 1.4142
>> d1¼x1(1:2:4)
d1 ¼ 0.7071 0.3536

where the MATLAB function rconv() for filter operations with the reversed filter coefficients is listed in Section
13.8 . Repeating for the next level, we have
c 0 ðkÞ¼ N
m¼ N

c 1 ðmÞh 0 ðm 2kÞ

d 0 ðkÞ¼ N
m¼ N

c 1 ðmÞh 1 ðm 2kÞ

Thus

p
2

c 0 ð0Þ¼ N
m¼ N

c 1 ðmÞh 0 ðmÞ¼c 1 ð0Þh 0 ð0Þþc 1 ð1Þh 0 ð1Þ¼ 3 1

p þð 1 p Þ 1 p ¼ 5

4

2

p
2

1 d 0 ð0Þ¼ N

m¼ N

c 1 ðmÞh 1 ðmÞ¼c 1 ð0Þh 1 ð0Þþc 1 ð1Þh 1 ð1Þ¼ 3 1 1

2

¼ 7
4

p þ

p 2

p