Question

7. (1 mark) Suppose that we have the systems K, H2, H3 and H4, whose responses to a complex exponential input e2t to are given by j2t 11 H2 ej2t te 124. Indicate which of these systems cannot be LTI

Answer 1

Given that e^j 2t rightarrow H_1 2 e^j 2t H_1 (e^j2t) = 2e^j 2t clearly the null vector 0 is not lie in h_1 because H 2 e^k 2t = 0 = > t rightarrow 0 Here null vector is not lie in H_1 therefore H_1 is not < I_1 (ii) H_2 (e^j 2t) = te^j 2t Let e^j 2t, e^j 2t_2 EH_2 therefore H_2 (e^j 2t_1) = t_1 e^j 2t_1 H_2 (e^j_2 t_2) = t_2 e^j_2 t_2 Now H_2 (C_1 e^j_2 t_1 + C_2 e^j_2 t_2) = e_1 t_1 e^j_2T_1 + c_2t_2 e^j_2 t_2 = C_1 H_2 (e^J_2 t_1) + c_2 H_2 (e^j_2 t_2) Here it is L T_1 therefore H_i is not L T_1