Question

4. LTI Systems and Erponential Response. (12 pts) (a) (2 pts) Suppose an LTI system has input-output relationship y(t) 2r(t+3). What is the transfer function H(jw) of the given system. Show that H(jw)2. Hint: H(jw(tejdt (b) (5 pts) Suppose an LTI system has input-output relationship y(t)2r(t+3) as Problem 4-(a). Find the output y(t) using the complex exponential response method as discussed in lecture for the input r(t) = ej2t + 2 cos2(t). Hint: cos2(0) 1 (20 cos(26) an d 1-ejot + (c) (5 pts) Suppose an LTI system with the impulse response h(t)3(t5 Find the output y(t) using the complex exponential response method as discussed in lecture for the input x(t)- sin(5t)cos(4t)

Answer 1

y(t) = 2 times (t + 3) to taking fourier transform y(jw) = 2 times (jw) e^jw3 y(w)/x(w) middot 2e^3jw rightdoublearrow H(jw) = 2e^3jw (2) y(t) = 2 times (t + 3) x(t) = e^j2t + 2 cos (t) therefore y(t) = 2e^j2(t + 3) + 2 cos^2 (t + 3) = 2e^j2t middot 2 e^j6 + 4 cos^2 (t + 3) (3) h(t) = 3s (t - 5) x(t) = sin (5t) + cos (4t) y(t) = x' (t) * h(t) = sin (5t) * 3s (t - 5) + cos (4t)*3s(t - 5) = 3 sin 5 (t - 5) + 3 cos [Q t - 5)] = 3 sin (5t - 25) + 3 cos (Q t - 20)