Subband- and Wavelet-Based Coding - Digital Signal Processing: Fundamentals and Applications - page 626

in ()

1.0

n

FIGURE 13.7

Impulse train with a period 4 samples.

Hence, using the inverse of discrete Fourier transform, iðnÞ can be expressed as

M 1
k ¼ 0 IðkÞe j

M 1
k ¼ 0 e j

1
M

1
M

2 pkn
M

2 pkn
M

iðnÞ¼ ¼ (13.5)

Substituting Equation (13.5) into Equation (13.3) leads to

M 1
k ¼ 0 wðnÞe j

1
M

2 pkn
M

wðnÞ¼ (13.6)

App lyi ng the z-transform in Equation (13.6) , we achieve the fundamental relationship between WðzÞ
and WðzÞ :

n

M 1 N

n¼ 0 wðnÞe j

M 1 N

n¼ 0 wðnÞ

1
M

1
M

2 pkn
M

z n ¼ e j

2 pk
M

WðzÞ¼ z

k ¼ 0 k ¼ 0

M 1
k ¼ 0 W

1
M

(13.7)

e j

2 pk
M

¼ z

h i

1
M

2 pðM 1 Þ
M

2 p 0
M

2 p 1
M

e j

e j e j ¼ z þ W z þ / þ W z

W

Equation (13.7) indicates that the signal spectrum WðzÞ before the synthesis filter is an average of the
various modulated spectrum WðzÞ . Notice that both WðzÞ and WðzÞ are at the original sampling rate f s .
We will use this result for further development in the next section.

13.2 SUBBAND DECOMPOSITION AND TWO-CHANNEL PERFECT
RECONSTRUCTION QUADRATURE MIRROR FILTER BANK
To explore Equation (13.7) , let us begin with a two-band case as illustrated in Figure 13.8 .
Substituting M ¼ 2 in Equation (13.7) , it follows that

1
k ¼ 0 Wðe j

X

1
2

1
2 ½WðzÞþWðzÞ

2 pk
2

WðzÞ¼ zÞ¼ (13.8)