Wavelet Analysis and Synthesis Equations - Digital Signal Processing: Fundamentals and Applications - page 821

Appendix F: Wavelet Analysis and Synthesis
Equations

F. 1 BASIC PROPERTIES
The inner product of two functions is defined as

Z

< x; y >¼ xðtÞyðtÞdt (F.1)

Two functions are orthogonal if

(

A for k ¼ 0

< xðtÞ; xðt kÞ >¼ (F.2)

0 for k s 0

Two functions are orthonormal if

( 1

for k ¼ 0

< xðtÞ; xðt kÞ >¼ (F.3)

0 for k s 0

The signal energy is defined as

Z

2

E ¼ x ðtÞdt (F.4)

Many wavelet families are designed to be orthonormal:

Z

2

E ¼ j ðtÞdt ¼ 1 (F.5)

Z Z Z

2

½ 2 j= 2 jð 2 j t kÞ

2

2 j j

2

ð 2 j t kÞdt E ¼ j

jk ðtÞdt ¼

dt ¼ (F.6)

Let u ¼ 2 j t k .Then du ¼ 2 j dt . Equation (F.6) becomes

Z

2 j j

2

ðuÞ 2 j du ¼ 1

E ¼ (F.7)