Question

The Laplace transform of a causal periodic signal can be found from the knowledge of the Laplace transform of its first cycle alone. 4. a) If the Laplace transform of f(t) shown in Fig. 4 a) is F(s), shown that G(s), the Laplace transform of g(t) shown in Fig. 4 b) is given by: F(s) G(s) = Re(s) 〉 0
Please complete parts a and b of question 4 with explanation. all figures have been pasted, thank you.

Answer 1

Given Data: The time shift theorem is useful in determining the transform of periodic functions let the function x (t) be a casual periodic wave function of wave form which satisfies the condition g (t) = f (t + nT) for all t greaterthanorequalto 0, where T is period of the function and n = 0, 1, 2, Now G (s) = integral^infinity_0 f (t) e^-st dt This can be written as G (s) = integral^T_0 f (t) e^-s dt + integral^2T_t f (t) e^-st + + integral^(n + 1) T_nT f (t) e^-st dt + = integral^T_0 f (t) e^-st dt + integral^2T_t f (t + T) e^-st + + integral^(n + 1) T_nT f (t + nT) e^-st dt + \r\n since g (t) is periodic we have so G (s) = integral^T_0 f (t) e^-st + e^-st integral^T_0 f (t) e^-st dt + + e^st integral^T_0