Subband- and Wavelet-Based Coding - Digital Signal Processing: Fundamentals and Applications - page 641

FIGURE 13.21

Wavelet transform amplitudes.

while the vertical axis is index j , which is inversely proportional to the scale factor ( a ¼ 2 j ). As will
be discussed, the larger the scale factor (the smaller the index j ), the smaller the frequency value. The
amplitudes of the wavelet transform are displayed according to the intensity. The brighter the intensity,
the larger the amplitude. The areas with brighter intensities indicate the strongest resonances between
the signal and the wavelets of various frequency scales and time shifts. In Figure 13.21 , the four
different frequency components and the discontinuities of the sinusoids are displayed as well. We can
further observe the fact that the finer the frequency resolution, the coarser the time resolution. For
example, we can clearly identify the start and stop times for 80-, 180-, and 350-Hz frequency
components, but frequency resolution is coarse, since index j has larger frequency spacing. However,
for the 0.8-Hz sinusoid, we have fine frequency resolution (small frequency spacing so we can see the
0.8-Hz sinusoid) and coarse time resolution as evidenced by the way in which the start and stop times
are blurred.
The CWT is defined as

Z N

Wða; bÞ¼ f ðtÞj ab ðtÞdt (13.31)

N

where Wða; bÞ is the wavelet transform and j ab ðtÞ is called the mother wavelet, which is defined as

t b
a

1

j ab ðtÞ¼

p j

(13.32)

The wavelet function consists of two important parameters: scaling a and translation b . A scaled
version of the function jðtÞ with a scale factor of a is defined as jðt=aÞ . Consider a base function
jðtÞ¼ cos ðutÞ when a ¼ 1. When a > 1, jðtÞ¼ cos ðut=aÞ is a scaled function with a frequency
less than u rad/s. When a < 1, jðtÞ¼ cos ðut=aÞ has a frequency larger than u . Figure 13.22 shows
the scaled wavelet functions.