Subband- and Wavelet-Based Coding - Digital Signal Processing: Fundamentals and Applications - page 656

By applying and continuing to apply Equation (13.46) , f ðtÞ can be coded at any level we wish.
Furthermore, by recursively applying Equation (13.46) until j ¼ 0, we can obtain signal expansion
using all the mother wavelets plus one scaling function at scale j ¼ 0, that is,

N J 1 N

d j ðkÞ 2 j= 2 jð 2 j t kÞ f ðtÞ z f J ðtÞ¼ c 0 ðkÞfðt kÞþ (13.47)

k ¼ N j ¼ 0 k¼ N

All c j ðkÞ and all d j ðkÞ are called the wavelet coefficients. They are essentially weights for the scaling
function(s) and wavelet functions (mother wavelets). The DWT computes these wavelet coefficients.
On the other hand, given the wavelet coefficients, we are able to reconstruct the original signal by
applying the inverse discrete wavelet transform (IDWT).
Based on the wavelet theory without proof (see Appendix F), we can perform the DWT using the
analysis equations as follows:

N

c j ðkÞ¼ c jþ 1 ðmÞh 0 ðm 2 kÞ (13.48)

m¼ N

N

d j ðkÞ¼ c jþ 1 ðmÞh 1 ðm 2 kÞ (13.49)

m¼ N

where h 0 ðkÞ are the lowpass wavelet filter coefficients listed in Table 13.2 , while h 1 ðkÞ , the highpass
filter coefficients, can be determined by
h 1 ðkÞ¼ð 1 Þ k h 0 ðN 1 kÞ

(13.50)

These lowpass and highpass filters are called the quadrature mirror filters (QMF). As an example, the
frequency responses of the 4-tap Daubechies wavelet filters are plotted in Figure 13.35 .
Next, we need to determine the filter inputs c jþ 1 ðkÞ in Equations (13.48) and (13.49) . In practice,
since j is a large number, the function fð 2 j t kÞ appears to be close to an impulse-like function, that
is, fð 2 j t kÞ z 2 j dðt k 2 j Þ . For example, the Haar scaling function can be expressed as
fðtÞ¼uðtÞuðt 1 Þ , where uðtÞ is the step function. We can easily get fð 2 5

t kÞ¼uð 2 5

t kÞ

uð 2 5 t 1 kÞ¼uðt k 2 5 Þuðt ðk þ 1 Þ 2 5 Þ for j ¼ 5, which is a narrow pulse with a

unit height and a width 2 5 located at t ¼ k 2 5 . The area of the pulse is therefore 2 5 . When
j approaches a larger positive integer, fð 2 j t kÞ z 2 j dðt k 2 j Þ . Therefore, f ðtÞ approximated by the
scaling function at level

j is rewritten as

f ðtÞ z f j ðtÞ¼ N

k ¼ N c j ðkÞ 2 j= 2

fð 2 j t kÞ

(13.51)

¼ / þ c j ð 0 Þ 2 j= 2 fð 2 j tÞþc j ð 1 Þ 2 j= 2 fð 2 j t 1 Þþc j ð 2 Þ 2 j= 2 fð 2 j t 2 Þþ /

z/ þ c j ð 0 Þ 2 j= 2

dðtÞþc j ð 1 Þ 2 j= 2 dðt 1 2 j Þþc j ð 2 Þ 2 j= 2 dðt 2 2 j Þþ /

On the other hand, if we sample f ðtÞ using the same sample interval T s ¼ 2 j (time resolution), the
discrete-time function can be expressed as
f ðnÞ¼f ðnT s Þ¼ / þ f ð 0 T s ÞT s dðt T s Þþf ðT s ÞT s dðt T s Þþf ð 2 T s ÞT s dðn 2 T s Þþ / (13.52)