Читать книгу EEG Signal Processing and Machine Learning - Saeid Sanei - Страница 68

The Morlet–Grossmann definition of the continuous WT for a 1D signal f(t) is:

(4.20)

where (.)^* denotes the complex conjugate, is the analyzing wavelet, a (>0) is the scale parameter (inversely proportional to frequency) and b is the position parameter. The transform is linear and is invariant under translations and dilations, i.e.:

(4.21)

and

(4.22)

The last property makes the WT very suitable for analyzing hierarchical structures. It is similar to a mathematical microscope with properties that do not depend on the magnification. Consider a function W(a,b) which is the WT of a given function f(t). It has been shown [18, 19] that f(t) can be recovered according to:

Figure 4.4 TF representation of an epileptic waveform in (a) for different time resolutions using the Hanning window of (b) 1 ms, and (c) 2 ms duration.

(4.23)

where

(4.24)

Although often it is considered that ψ(t) = ϕ(t), other alternatives for ϕ(t) may enhance certain features for some specific applications [20]. The reconstruction of f(t) is subject to having C_ϕ defined (admissibility condition). The case ψ(t) = ϕ(t) implies , i.e. the mean of the wavelet function is zero.