Subband- and Wavelet-Based Coding - Digital Signal Processing: Fundamentals and Applications - page 642

A translated version of the function jðtÞ with a shifted time constant b is defined as jðt bÞ .
Figure 13.23 shows several translated versions of the wavelet. A scaled and translated function jðtÞ is
given by jððt bÞ=aÞ . This means that jððt bÞ=aÞ changes frequency and time shift. Several
combined scaling and translated wavelets are displayed in Figure 13.24 .
Besides these two properties, a wavelet function must satisfy admissibility and regularity
conditions (vanishing moment up to a certain order). Admissibility requires that the wavelet
(mother wavelet) have a bandpass-limited spectrum and a zero average in the time domain,
which means that wavelets must be oscillatory. Regularity requires that wavelets have some
smoothness and concentration in both time and frequency domains. This topic is beyond the
scope of this topic and the details can be found in Akansu and Haddad (1992). There exists
a pair of wavelet functions: the father wavelet (also called the scaling function) and mother
wavelet. Figure 13.25 shows a simplest pair of wavelets: the Haar father wavelet and mother
wavelet.
To devise an efficient wavelet transform algorithm, we let the scale factor be a power of two,
that is,

a ¼ 2 j (13.33)

Note that the larger the index j , the smaller the scale factor a ¼ 2 j . The time shift becomes

b ¼ k 2 j ¼ ka

(13.34)

1

0

-1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

1

0

-1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

1

0

-1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (sec.)

FIGURE 13.22

Scaled wavelet functions.