Thesis - Instituto de Telecomunicações

4.2. ELECTRODERMAL ACTIVITY 71(eq. 4.18); t β and is found as the point at which the first order derivative is zero and thesecond order derivative is negative (eq. 4.19). Finally, the decay zon! e corresponds to thesucceeding part of the signal, where the first derivative gets negative again (eq. 4.20).f ′ (t) < 0 ⇐ t < t α (4.16)f ′ (t) = 0 ∧ f ′′ (t) > 0 ⇐ t = t α (4.17)f ′ (t) > 0 ⇐ t α > t > t β (4.18)f ′ (t) = 0 ∧ f ′′ (t) < 0 ⇐ t = t β (4.19)f ′ (t) < 0 ⇐ t > t β . (4.20)This model is associated with a single isolated event, in which case t α corresponds tot 0 indicated in figure 3.6, and t β corresponds to t max . An implementation of this model isfound in a study by Storm [226]. The proximal zone of the derivative zeros is fitted witha quadratic polynomial to obtain a better approximation to the location of the zeros. Theparameters expressed in table 3.4 can easily be extracted by knowing the two instants t αand t β .The major difficulties with this view of the EDA signal is the inability of retrievingcorrect parameters when the signals overlap, or when an event is so smooth that does nothave a peak or valley.Sigmoid-exponential ModelLim et al. [146] proposed a set of models of increasingcomplexity, based on a sigmoid-exponential function, f s , defined ase −(t−to)/t df s (t) =g(1 + ((t − t o )/t r ) −2 ) 2 u(t − t 0). (4.21)In the previous equation, u(t) refers to the unitary step function⎧⎨ 0 (t ≤ 0)u(t) =⎩ 1 (t>0), (4.22)