Question

4) For the periodic signal below, find the compact trigonometric Fourier series and sketch the amplitude and phase spec ra. If either sine or cosine terms are absent in the Fourier series, explain why -T π /4

Please show all steps and show neat work.

thanks

Answer 1

T_0 = pi , w_0 = 2pi/T_0 = 2pi/ pi = 2 and f(t) = 4/pi t By inspection a_0 = 0 because of odd symmetry a_n = 0 (n > 0) hence the fourier series would contain sine terms only b_n = 4/pi int^pi/4_0 4/pi t sin 2nt dt = 2/pi n (2/pi n sin pi n/2 - cos pi n/2) rightarrow f(t) = 4/pi^2 sin 2t + 1/pi sin 4t - 4/9 pi^2 sin6t = 1/2pi sin8t + . . . = 4/pi^2 cos(2t - pi/2) + 1/pi cos(4t - pi/2) + 4/9pi^2 cos (6t + pi/2) + 1/2 pi) + 1/2pi cos (8t + pi/2) + . . . . the plot for c_n and theta_n is show as follows