Question

P4- Fourier Transform (20 points) For a stable LTIC system with transfer function, h(t), Find the zero-state response if the input x(t)=δ70(t)(The problem shuld be solved using Fourier Transform) h(t) 上一 -27- TO TOE

Answer 1

Y(t) = x(t) times h(t) y(w) = X(w) H(w) X(w): - x(t) is periodic signal so you need to find Fourier transform of a periodic signal. Let x(t) = sigma_n = -infinity^infinity C_n e^jn w_0 t Applying Fourier transform x(w) = sigma_n = -infinity^infinity C_n 2pi delta(w - nw_0) property F.T of e^jw_0 t e^jw_0 t Doubleheadarrow 2pi delta (w - w_0) where C_n rightarrow Fourier Series coefficients. C_n: - C_n = 1/T integral_-T/2^T/2 x(t) e^-jn w_0 t dt = 1 /T_0 integral_-T/2^T/2 delta (t) e^-jn w_0 t dt \r\n C_n = 1/T_0 integral_-T/2^T/2 delta (t) e^-jn w_0 (0) dt. = 1/T_0 (1) [because integral_-infinity^infinity delta(t) dt = 1 C_n = 1/T_0 X(w) = sigma_n = -infinity^infinity C_n 2pi delta (w - nw_0) X(w) = sigma_n = -infinity^infinity 1/T_0 2pi delta (w - nw_0) H(w): - A rect (t/T) Doubleheadarrow AT S_a (WT/2) H(w) = (1) T_0 Sa (WT_0/2) Y(w) = X(w) H(w) = sigma_n = -infinity^infinity 1/T_0 2pi delta(w - n w_0) times T_0 Sa (wT_0/2)