Question

Let x(t) a periodic signal with period To such that x(t)-sin(coot) for。st for To/2 s t s To. To2 and x(t)-0 a) Plot x(t) b) Expand x(t) in trigonometric Fourier series (sine/cosine). c) Calculate the average power of x().

Answer 1

Given periodic signal x(t) = {sin w_0 t 0 lessthanorequalto l lessthanorequalto T/2 0 T/2 lessthanorequalto t lessthanorequalto T_0 w_0 T = 2 pi (a) plot x(t) (b) Expand x(t) in trigonometric Fourier series: - a_0 = 1/T integral_0^T x(t) dt = 1/T integral_0^T/2 sin w_0 t = 1/T [-cos w_0 t]_0^T/2 a_0 = 1/T [-cos w_0 T/2 + cos 0] = 1/T [-cos pi + cos 0] = 2/T a_0 = 2/T a_n = 2/T integral_0^T/2 2/2 sin w_0 t cos n w_0 t dt 2 sin A cos B = sin(A + B) + sin(A - B) a_n = 2/2T integral_0^T/2 [sin w_0 (n + 1)t + sin w_0(1 - n)t]dt a_n = 2/2T integral_0^T/2 sin w_0 (1 + n)t dt + integral_0^T/2 sin w_0 (1 - n)t dt a_n = 2/2T {[-cos w_0(1 + m)t/w_0(1 + n)]_0^T/2 + [-cos w_0 (1 - n)t/w_0 (1 - n)]_0^T/2} a_n = 2/2T {[-cos w_0 (1 + n)t/w_0 (1 + n)