Quantitative analysis of EEG signals: Time-frequency methods and ...

4.2 Theoretical Background
4.2.1 Continuous Wavelet Transform
A wavelet family ab is a set of elemental functions generated by dilations and translations
of a unique admissible mother wavelet (t):
ab(t) = jaj ;1=2
t ; b
where a b 2 R, a 6= 0 are the scale and translation parameters respectively.
a
(27)
As a
increases the wavelet becomes more narrow and by varying b, the mother wavelet is
displaced in time. Thus, the wavelet family gives a unique pattern and its replicas at
dierent scales and with variable localization in time.
The continuous wavelet transform of a signal X(t) 2 L 2 (R) (nite energy signals) is
dened as the correlation between the signal and the wavelet functions ab , i.e.
(W X)(a b) = jaj ;1=2 Z 1
;1
X(t) ( t ; b
a ) dt = < X(t) ab > (28)
where denotes complex conjugation. Then, the dierent correlations
< X(t) ab > indicates how precisely the wavelet function locally ts the signal at
every scale a. Since the correlation is made with dierent scales of a single function,
instead of with complex sinusoids characterized by their frequencies, wavelets give a
time-scale representation.
4.2.2 Dyadic Wavelet Transform
The continuous Wavelet Transform maps a signal of one independent variable t onto
a function of two independent variables a b. This procedure is redundant and not
ecient for algorithm implementations. In consequence, it is more practical to dene
the Wavelet Transform only at discrete scales a and discrete times b. One way toachieve
this, is by choosing the discrete set of parameters fa j = 2 j b jk = 2 j kg, with j k 2 Z.
Replacing in eq. 27 we obtain the discrete wavelet family
jk(t) = 2 ;j=2 ( 2 ;j t ; k ) j k 2 Z (29)
that forms a basis of the Hilbert space L 2 (R), and whose correlation with the signal is
called Dyadic Wavelet Transform.
38