Question

1. (25 points) A continuous-time periodic signal x (t) is real valued and has a fundamental period T-8. The nonzero Fourier series coefficients for x(t) are Express x(t) in the fornm

Answer 1

The fundamental period of a line period signal x(t) is T = 8 The non - zero touries series coefficient fr x(t) are a_1 = a_- 1 = 1 a_3 = a^x_- 3 = 4 j a_- 3 = (4 j)^/ = -4 j the fundamental frequency of the signal w_0 = pi/T = 2 pi/8 = pi/4 x(t) = Sigma_k = -infinity a_1 e^j k_w j^2 = a_- 3 e^- j 3 (2 pi/T) + a_- 1 e^- j (2 pi/T) + a_- 3 e^j3 (2 pi/T) = -4 j e^j 3 (2 pi/8) + e^- j (2 pi/8) + 4 i e^j 3 (2 pi/8) t of the components which have the same fund metal frequency and the signal x(t) x(t) = [4 e^j 2 pi/8 t + c^- j 2 pi/8 t) + 4 j [e^j 3 (2 pi/8) t x (t) = (4 e^j 2 pi/8 t + e^- j 2 pi/8 t) + 4 j (e^j 6 pi/8 - e^- j 6 pi/8 t)