Spectral Transformations - Biomedical Signal Analysis: Contemporary Methods and Applications - page 39

Ψ ( ω )

ψ

(t/a)

0 < a < 1

0 < a < 1

t

ω

Ψ ( ) ω

ψ

(t/a)

a > 1

a > 1

t

ω

(b)

(a)

Figure 2.2
Wavelet in time and frequency domains: (a) scale parameter 0 <a< 1, (b) scale
parameter a> 1.

frequency resolution of the WT can easily be detected.
The foundation of the WT is based on the scaling property of the
Fourier transform. If

ψ ( t ) ←→ Ψ( ω )

represents a Fourier transform pair, then we have

√ a Ψ t

←→ √ a Ψ( aω )

1

(2.22)

a

with a> 0 being a continuous variable. A contraction in the time domain
produces an expansion in the frequency domain, and vice versa. Figure
2.3 illustrates the corresponding resolution cells in the time-frequency
domain. The figure makes visual the underlying property of wavelets:
they are localized in both time and frequency. The functions e jωt are