Wavelet Analysis and Synthesis Equations - Digital Signal Processing: Fundamentals and Applications - page 822

Both father and mother wavelets are orthonormal at scale j :
Z

( 1

k ¼ n

f jk ðtÞf jn ðtÞdt ¼ (F.8)

0 otherwise

( 1

Z

k ¼ n

j jk ðtÞj jn ðtÞdt ¼ (F.9)

0 otherwise

F. 2 ANALYSIS EQUATIONS
When a function f ðtÞ

j þ 1, it can be

is approximated using the scaling functions only at scale

expressed as

N

c j ðkÞ 2 j= 2 fð 2 j t kÞ f ðtÞ¼

k ¼ N

Using the inner product,

Z

f ðtÞ 2 j= 2 fð 2 j t kÞdt c j ðkÞ¼< f ðtÞ; f jk ðtÞ >¼ (F.10)

Note that

N

p 2

fðtÞ¼ h 0 ðnÞfð 2 t nÞ (F.11)

n¼ N

Substituting Equation (F.11) into Equation (F.10) leads to

Z

f ðtÞ 2 j= 2 N
n¼ N

p h 0 ðnÞf½ 2 ð 2 j t kÞndt

c j ðkÞ¼< f ðtÞ; f jk ðtÞ >¼

Z

N

f ðtÞ 2 ðjþ 1 Þ= 2 h 0 ðnÞfð 2 ðjþ 1 Þ t 2 k nÞdt c j ðkÞ¼

n¼ N

Let m ¼ n þ 2 k . Interchange of the summation and integral leads to

Z

N

f ðtÞ 2 ðjþ 1 Þ= 2 h 0 ðm 2 kÞfð 2 ðjþ 1 Þ t mÞdt c j ðkÞ¼

m¼ N

Z

N

c j ðkÞ¼ f ðtÞf ðjþ 1 Þm ðtÞdt h 0 ðm 2 kÞ (F.12)

m¼ N