Question

(b) (2 pts) (t) is given as r(t) e sin(t) Find X(jw). Show that X(jw) = 25 + (w- 1)225(w+1)2 (c) (4 pts) x(t) is given as x(t)-π inc(t) cos(nt). Find X(jw) (d) (4 pts) 2(t) is given as 2(t) e Áil+ 3) + e' ỗ(t-3). Find X (jw). Simplify the answer as (e) (4 pts) 2(t) is given as r(t) = rect(2(t )) reetgehj)). Hint: use Fourier Transform pair: sine(t)艹rect( ) much as possible Find X(jw). Simplify the answer as much as possible. Hint: use Fourier Transform pair rect(t) sinc (w/2n)

Answer 1

X(t) = e^-5 Vertical-bar t Vertical-bar sin (t) e^-5 Vertical-bar t Vertical-bar rightarrow 2 times 5/(5)^2 + w^2 { e^-a Vertical-bar t Vertical-bar rightarrow 2a/a^2 + w^2 e^-5 Vertical-bar t Vertical-bar rightarrow 10/25 + w^2 e^5 Vertical-bar t Vertical-bar sin (t) rightarrow 1/2pi 10/25 + w^2 times [pi/j (delta (w - 1) + delta (w + 1)] modulation property X (jw) rightarrow 1/2pi (pi/j {10/25 + (w - 1)^2 - 10/25 (w + 1)^2} {shifting property s(t) rightarrow X (jw) x(t plusminus t_0) rightarrow e^jw t_0 times (jw) x(t) rightarrow x(w) x(w) times delta (w - w_0) rightarrow x (w - w_0) X(jw) = -5j/25 + (w - 1)^2 + 5j/25 + (w + 1)^2 {Modulation property x(t) rightarrow X (jw) x(t) sin w_0 t rightarrow x (j w) * pi/j {delta (w - w_0) - delta (w + w_0)} FT {x(t) m(t) } = 1/2pi times (jw) * M (jw) \r\n x(t) = pi sin c(t) cos pi t sin (t) rightarrow rect (w/2pi) cos pi t rightarrow pi [delta (w - pi) + delta (w + pi)] FT [x(t) middot m(t)] = 1/2pi (x (w) * M(w)) x (jw) = 1/2pi rect (w/2pi) * [pi delta (w - pi) + delta (w +