Question

Question 3 Determine the Exponential Fourier Series of the periodic signal shown in Figure 1 F(t) 1 -2 1 2 Figure 1 (10 marks)

Answer 1

F(t) = sigma_n = -infinity^infinity F_n e^jn w_o t F_n = 1/T_o integral_-T_0/2^T_0/2 f(t) e^-jn w_o t dt From figure T_0 = 2, w_0 = 2 pi/T = pi F_n = 1/2 [integral_1^1 f(t) e^-jn w_0t dt] F_n = 1/2 [integral_-1^0(0) e^-jn w_0 t dt + integral_0^1 1 . e^- jn w_o t dt] F_n = 1/2 integral_0^1 e^-j n w_ot dt F_n = 1/2 [e^-j n w_o t/- jn w_0]_0^1 F_n = 1/2 [e^-jn w_0/-jn w_o + 1/jn w_o] F_n = 1/2 [e^-j n pi/-j n pi = 1/j n pi] F_n = 1/2 n pi j [-e^- jn pi + 1] \r\n F_n = 1/2 n pi j [-e^- j n pi + 1] cos (n pi) - j sin (n pi) F_n = 1/2 n pi j[-(-1) + 1] F_n = 1/2 n pi j(2) = 1/n pi j f(t) = sigma_n = -infinity^infinity F_n e^jn w_0 t f(t) = sigma_n = -infinity^infinity 1/ jn pi e^j n pi_0 t