Question

2.7.5 The impulse response of a continuous-time LTI system is given by (a) What is the frequency response H (w) of this system? (b) Find and sketch |H(w) (c) Is this a lowpass, bandpass, or highpass filter, or none of those? 2.7.6 The impulse response of a continuous-time LTI system is given by h(t) = δ(t-2) (This is a delay of 2.) (a) What is the frequency response H (w) of this system? (b) Find and sketch the frequency response magnitude, H () (c) Find and sketch the frequency response phase, ZH' (w) (d) Is this a lowpass, bandpass, or highpass filter, or none of those?

Answer 1

Given h(t) = delta(t) - e^-t u(t) let h(t) = h_1 (t) + h_2 (t) where h_1 (t) = delta(t) and h_2 (t) = -e^-t u(t) let fourier transform of h(t) rightleftarrow H(w) Therefore h(t) = h_1 (t) + h_2 (t) H(w) = H_1 (w) + H_2 (w) (i) h_1 (t) = delta(t) we know X(w) = integral^infinity_-infinity x(t) e^-jwt dt H_1 (w) = integral^infinity_-infinity h_1 (t) e^-jwt dt {we know integral^infinity_-infinity x(t) delta(t) = x(0) H_1 (w) = integral^infinity_-infinity delta(t) e^-jwt dt H_1 (w) = e^-jw(0) H_1 (w) = 1 (ii) h_2 (t) = e^-t u(t) = {-e^-t, t > 0 0, t < 0 H_2 (w) = integral^infinity_-infinity h_2 (t) e^-jwt dt = -integral^infinity_-infinity e^-t u(t) e^-jwt dt = -integral^infinity_-infinity e^-t e^-jwt dt \r\n H_2 (w) = -integral^infinity_0 e^-(1 + jw)t dt H_2 (w) = -1/1 + jw Therefore H(w) = H_1 (w) + H_2 (w) = 1 - 1/1 + jw H(w)