Thesis - Instituto de Telecomunicações

76 CHAPTER 4. SIGNAL PROCESSING AND FEATURE EXTRACTIONGiven the function f(t) =at k e −bt u(t) above, its first and second order derivatives aregiven byf ′ (t) =( kt − b )at k e −bt u(t), (4.29)andf ′′ (t) = ( b 2 − 2bk/t + k(k − 1)/t 2) at k e −bt u(t) (4.30)respectively. According to the constraint of zero first-order derivative at t o , taken for simplicityas t o = 0, we conclude that k ≥ 1 (and thus, n>1). A particular value for k isobtained using the discovered relationship in the time intervals in equation 4.26. In fact,from equation 4.26, and considering t o = 0, we havet 2 =2t 1 . (4.31)Computing the zeros of f ′ (t)f ′ (t) = 0 ⇒ t = k b∨ t =0, (4.32)and of f ′′ (t)f ′′ (t) = 0 ⇒ t = k ± √ kband according to the definitions of t 1 and t 2 , we have∨ t =0, (4.33)t 2 = k b , (4.34)t 1 = k − √ k. (4.35)bBy simple manipulation of equations 4.31, 4.34 and 4.35, we get k = 4. The function f(t)and its first and second order derivatives are then expressed by equations 4.36 to 4.38,where the evenly spaced intervals, defined in equation 4.26, are evident (t 1 = 2 b , t 2 = 4 b , and