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Fourier transform of rectangular pulse (Waves)

  1. May 9, 2009 #1
    1. The problem statement, all variables and given/known data

    F(w) is the Fourier transform of f(t). Write down the equation for F(w) in terms of f(t).
    A rectangular pulse has height H and total length t0 in time. Show that as a function of w, the amplitude density is propertional to sinc(wt0/2).

    2. Relevant equations

    F(w) = integral from -infinity to +infinity of: f(t)exp(-iwt)dw

    3. The attempt at a solution

    integral from -t0/2 to +t0/2 of: h*exp(-iwt)dw

    I have access to the solution to this problem, which says that it should be:
    integral from -t0/2 to +t0/2 of: h*exp(-iwt)dt,
    but I don't understand why I'm integrating wrt t now, when the definition says w.

    Could somebody please explain this?

    Thanks in advance.
     
  2. jcsd
  3. May 9, 2009 #2

    Cyosis

    User Avatar
    Homework Helper

    Where do you get that definition from? Think about it, you want to find a function [itex]F(\omega)[/itex], but if you calculate the integral you've written down as the "definition" then the integration boundaries will be inserted into [itex]\omega[/itex] after the integration. As as a result you won't have a function with variable [itex]\omega[/itex].

    The correct definition is (normalization conventions can be different):
    [tex]
    F(\omega)}=\int_{-\infty}^\infty e^{-i \omega t} dt
    [/tex]
     
  4. May 9, 2009 #3
    Actually I got that definition from the solution to the question. It makes a whole lot more sense now, thanks for your reply!
     
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