Subband- and Wavelet-Based Coding - Digital Signal Processing: Fundamentals and Applications - page 627

wn

0 ()

wn

0 ()

xm

0 ()

yn

0 ()

H

0 ()

↑ G

0 () 2

↓

2

xn ()

xn ()

wn

1 ()

wn

1 ()

xm

1 ()

H

1 ()

↓ G

1 ()

2 ↑

2

yn

1 ()

Analysis stage Synthesis stage

FIGURE 13.8
Two-band filter bank system.

Applying for each band in Figure 13.8 by substituting Equation (13.1) in Equation (13.8) , we have

1
2 G 0 ðzÞðH 0 ðzÞXðzÞþH 0 ðzÞXðzÞÞ

Y 0 ðzÞ¼ (13.9)

1
2 G 1 ðzÞðH 1 ðzÞXðzÞþH 1 ðzÞXðzÞÞ

Y 1 ðzÞ¼ (13.10)

Since the synthesized signal XðzÞ is the sum of Y 0 ðzÞ and Y 1 ðzÞ , it can be expressed as

1
2 ðG 0 ðzÞH 0 ðzÞþG 1 ðzÞH 1 ðzÞÞXðzÞ

XðzÞ¼

1
2 ðG 0 ðzÞH 0 ðzÞþG 1 ðzÞH 1 ðzÞÞXðzÞ

(13.11)

þ

¼ AðzÞXðzÞþSðzÞXðzÞ

For perfect reconstruction, th e recovered signal xðnÞ should be a scaled and delayed version of the
original signal xðnÞ , that is, xðnÞ¼cxðn n 0 Þ . Hence, to achieve a perfect reconstruction, it is

required that

1
2 ðG 0 ðzÞH 0 ðzÞþG 1 ðzÞH 1 ðzÞÞ ¼ 0

SðzÞ¼ (13.12)

1
2 ðG 0 ðzÞH 0 ðzÞþG 1 ðzÞH 1 ðzÞÞ ¼ cz n 0

AðzÞ¼ (13.13)

where c is the constant while n 0 is the delay introduced by the analysis and synthesis filters.
Forcing SðzÞ¼ 0 leads to the following relationship:

G 0 ðzÞ

G 1 ðzÞ ¼ H 1 ðzÞ

(13.14)

H 0 ðzÞ

It follows that

G 0 ðzÞ¼H 1 ðzÞ (13.15)