Question

IF Let x(t) Show that e 20" σ>0, and let (o) be the Fourier transform of x(t) .

Answer 1

Given x(t) = e^-t^2/2 sigma^2, sigma > 0 It is a gaussian pulse and we know that the fourier transform of a gaussian pulse is gaussian pulse let a = 1/2 sigma^2 then x(t) = e^-at^2 differentiate both side with respect to t d/dt x(t) = d/dt (e^-at^2) therefore d/dt x(t) = -2 at e^-at^2 d/dt x(t) = -2 at x(t) here we are using fourier transform to transform equation (1), for that we will have to we some fourier transform property for x(t) x(omega) d/dt x(t) j omega times (omega) t x(t) j d/d omega times (omega) By using the above properties convert equation (1) using transform \r\n therefore d/dt x(t) = -2 at x(t) j omega times (omega) = (-2) [j d/d omega times (omega)] omega times (omega) = -2a d/d omega times (omega) d/d omega times (omega) + omega/2a times (omega) = 0 comparing it with differential equation {dy/dx + p(x) middot y = Q9x)}. Therefore here