Thesis - Instituto de Telecomunicações

4.2. ELECTRODERMAL ACTIVITY 75the process based on linear systems theory [97]. Taking the SCR signal as the impulsiveresponse of a system with transfer function H(s), with the impulsive input representing thetriggering stimulus of the SCR (see figure 4.5), we tried to identify the transfer functionthat would provide the desired observed SCR behavior. Given that SC ′ = 0 at t o , andtaking, for simplicity, t o = 0, a single pole model, with multiplicity n higher than one,seemed plausible. We therefore looked for transfer functions of the formH(s) =α(s + b) n . (4.27)Figure 4.5: Proposed SCR model.With this model we assume that the overlapping of EDA events is a linear operation,and that SCL is a constant added to the signal. In the next sections we derive the completemodel, and also provide the mechanism to use this isolated model to find overlapping eventsand compute the tonic level (SCL) of the entire signal with multiple SCR events.EDA ModelAccording to our proposal, an isolated SCR is modeled as the output of a linear systemwith transfer function H(s) given by equation 4.27, to an impulsive-type input, representingthe triggering stimulus of the SCR. The identification of the model order, n, is based onmatching the corresponding impulsive response h(t), given by the inverse Laplace transform( )h(t) =L −1 α(s + b) n= αtn−1 e −bt(n − 1)! u(t) =atk e −bt u(t) =f(t)u(t), (4.28)with the observed morphological signal features, detailed in the previous section, inparticular the evenly spaced temporal marks t o to t 4 . In the previous equation, u(t) refersto the unitary step function (presented in equation 4.22), k = n − 1 and a = α k! .