Question

2. (12 points) Apply the result from part 1 to determine the response of a lowpass filter. a) (4 points) Determine the fundamental frequency and non-zero complex exponential Fourier series coeffi- cients of the periodic signal 2π f(t) =-2-5 sin(2nt) + 10 cos(Grt + "") and sketch the Fourier magnitude spectum D versus w and the Fourier phase spectrum LD versus w (b) (2 points) Use Parseval's theorem for the exponential Fourier series to find the power of the signal f(t) c) (4 points) The signal f (t) from part (a) is passed through an ideal low-pass filter whose frequency response function is to produce an output signal y(t). Find an expression for the resulting sinusoidal steady-state response y(t) of the system and express your result as the sum of rcal-valued sinusoidal term(s) and if necessary, a DC offset (d) (2 points) Use Parseval's theorem for the exponential Fourier series to find the power of the signal y(t).

Answer 1

(a) f (t) = -2 - 5 sin (2 pi t) + 10 cos (6 pi t + 2 pi/3) w_0_1 = 2 pi, w_0_2 = 6 pi: w_0 = GCD (w_0_1, w_0_2) = GCD (2 pi, 6 pi) Fundamental frequency = 1/T_0 = f_0 = w_0/2 pi = 2 pi/2 pi = 1 Hz f_0 = 1 HZ: T_0 = fundamental period = 1 sec complex - Exponential F . s coeff = D_n f (t) = sigma^infinity_n = -infinity D_jn e^jnw_0 t = sigma^infinity_n = -infinity