Question

Q4. (8) For the LTI system given in the following figure, find: (a) Y(o) if X(o) is given by: x(a)) = 1+ω (b) Find y(t) X(o) Y(c) |H(ω) = cos40| Hint: Find x(o) and H(ω). Then find Y(o) and then y(t)

Answer 1

DATA: X(w) = 2/1 + w^2, H(w) = cos 4w a) CONVOLUTION IN TIME DOMAIN RESULTS IN MULTIPLICATION IN FREQUENCY DOMAIN therefore Y(w) = H(w) X(w) Y(w) = cos 4w middot 2/1 + w^2 b) e^-at v(t) rightarrow^F T 1/a + jw e^at v(t) rightarrow^F T 1/a - jw USING PROPERTY OF LINEARITY e^-a(t) rightarrow^F T 2a/a^2 + w^2 HERE x(w) = 2/1 + w^2 iscongruentto 2 middot 1/1 + w^2 therefore a = 1 therefore x(t) = e^-(t) - (i) Now given H(w) = cos 4w x(t) rightarrow F T x(w) x(t - t_0) rightarrow^F T e^-j wt_0 x(w) x(t + t_0) rightarrow e^jw t_0 x(w) x(t - t_0) + x(t + t_0)/2 rightarrow^F T e^jw t_0 + e^-jw t_0/2 = cos wt_0 Here given cos 4w therefore t_0 = 4 y(t) = e^- Vertical-bar t - 4 Vertical-bar + e^-t - 4 Vertical-bar /2