Subband- and Wavelet-Based Coding - Digital Signal Processing: Fundamentals and Applications - page 657

1.5

H1(z)

H0(z)

1

0.5

0

0 0.5 1 1.5 2 2.5 3

Frequency in radians

FIGURE 13.35
Frequency responses for 4-tap Daubechies filters.

Hence, comparing Equation (13.51) with the discrete-time version in Equation (13.52) , it follows that

c j ðkÞ 2 j= 2

¼ f ðkÞT s (13.53)

Substituting T s ¼ 2 j in Equation (13.53) leads to
c j ðkÞ¼ 2 j= 2

f ðkÞ (13.54)

With the obtained sequence c j ðkÞ using sample values f ðkÞ , we can perform the DWT using Equations
(13.48) and (13.49) . Furthermore, Equations (13.48) and (13.49) can be implemented using a dyadic
tree structure similar to the subband coding case. Figure 13.36 depicts the case for j ¼ 2.
Note that the reversed sequences h 0 ðkÞ and h 1 ðkÞ are used in the analysis stage. Similarly, the
IDWT (synthesis equation) can be developed (see Appendix F) and expressed as

N N

c jþ 1 ðkÞ¼ c j ðmÞh 0 ðk 2 mÞþ d j ðmÞh 1 ðk 2 mÞ (13.55)

m¼ N m¼ N

Finally, the signal amplitude can be rescaled by

f ðkÞ¼ 2 j= 2 c j ðkÞ

(13.56)

An implementation for j ¼ 2 using the dyadic subband coding structure is illustrated in Figure 13.37 .