Question

4. The Fourier transform of a rectangular pulse 1 비 r/2 0 otherwise is given by (a) Use pr(t) and properties of the Fourier transform to find the Fourier transform, D(w), of d(t) shown below, in terms of P(. First state the approach that you are using to find D(), then show all of the details. d(t)

Answer 1

Given P_T (t) = {1, vert t vert lessthanorequalto T/2 0, otherwise Given P_T (w) = T sin c (Tw/2 pi) Given d(t) as To find D(w): First calculate derivative of d(t) and then use time integration property to find D(w) Time integration property x(t) rightarrow^F.T x(w) integral x(t) dt rightarrow^F.T x(w)/Jw + pi Delta (w) times (0) [x(0) = integral x(t) dt] \r\n (Derivative of ramp is step: with amplitude as slope of ramp) Z(f) = (1) sin c(w(1)/2 pi) [rectangular function & sin c functions are fourier transform pairs] z(t) = sin c (w/2 pi) y(f) = c^jw/2 sin c (w/2 pi), Time shifting property fourier transformer (AT) sin c (F) Area of rectangular function width of t rectangular function rightwardsdoublearrow Now d(t) is integration of y(t) Apply time integration property on y(t). y(t) rightarrow^F.T. y(w) integral y(t) dt rightarrow^F.T y(w)/jw + pi Delta (w) y(0) Now F[integral y (t) dt] = D(w) = e^jw/2 sin (w/2 pi)/jw + pi Delta (w)(w) Because y(0) = integral y(t) dt = 1 downwardsdoublearrow Area D(w) = e^jw/2 sin c (w/2 pi)/jw + pi Delta (w)