Question

Problem 3:(3.7) Using the defining equation for the FT, evaluate the frequency-domain representation (frequency spectrum) for the following signals. Simplify into the form X(ja)-Kga) ejLXja) Sketch the magnitude (|X(juo)) and phase (ZX(j)) spectra for each. -3t (a) x1(t) etut-1) (b) X2(t) e-Itl (c) x3(t) te2u(t)

Answer 1

(a) x_1 (t) = e^-3t u(t - 1) = {e^-3t middot 1, t greaterthanorequalto 1 0, otherwise. The Fourier Transformation F.T of x_1 (t) is X_1 (w) = integral^infinity_t = -infinity x_1 (t) e^-jwt dt = integrla^infinity_t = 1 e^-3t middot e^-jwt dt = integral^infinity_t = 1 e^-(3 + jw)t dt = 1/-(3 + jw) [e^-(3 + jw)t]^infinity_1 = -1/(3 + jw) (0 - e^-(3 + jw)) X_1 (w) = e^-(3 + jw)/(3 + jw) equation(1) \r\n X_1 (w) = e^-3 middot e^-jw/(3 + jw) = Vertical-bar X_1 (w) Vertical-bar middot e^j < X_1 (w) where Vertical-bar X_1 (w) Vertical-bar = Vertical-bar e^-3 e^-jw/(3 + jw) Vertical-bar = e^-3 middot 1/squareroot 9 + w^2 Magnitude spectra and < X_1 (w) = < {e^-3 middot e^-jw/(3 + jw)} = < (e^-3) + < (e^-jw) - < (3 + jw) = 0 degrees + (-w) - tan^-1 (w/3) = -w - tan^-1 (w/3) (b) x_2 (t) = e^- Vertical-bar t Vertical-bar = {e^-t, t greaterthanorequalto 0 e^+ t, t lessthanorequalto 0 Then, F.T of