Results 1 to 3 of 3

Math Help - Fourier Transform help

  1. #1
    Newbie
    Joined
    Feb 2007
    Posts
    5

    Fourier Transform help

    If S(t) is any non-periodic continuous function of t, show that:
    (i) S(t) is real iff F(-omega) = F*(omega)
    (ii) S(t) is pure imaginary iff F(-omega) = -F*(omega)
    (iii) S(t) is even iff F(-omega) = F(omega)
    (iv) S(t) is odd iff F(-omega) = -F(omega)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by superdael View Post
    If S(t) is any non-periodic continuous function of t, show that:
    (i) S(t) is real iff F(-omega) = F*(omega)
    (ii) S(t) is pure imaginary iff F(-omega) = -F*(omega)
    I am note sure what these stars means.
    (iii) S(t) is even iff F(-omega) = F(omega)
    (iv) S(t) is odd iff F(-omega) = -F(omega)
    (iii) and (iv) are similar. I will do (iii) and leave (iv) for you to do. For some reason I got directly the opposite of what you said. Are you sure you are right?

    (Note the converse is basically the same idea).

    EDIT I just realized why I made a mistake. When I made a variable substitution s=-t I forgot to attach the minus sign in front of the integral. I am just lazy to retype the entire thing. So you are right.
    Attached Thumbnails Attached Thumbnails Fourier Transform help-picture6.gif  
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by superdael View Post
    If S(t) is any non-periodic continuous function of t, show that:
    (i) S(t) is real iff F(-omega) = F*(omega)
    (ii) S(t) is pure imaginary iff F(-omega) = -F*(omega)
    (iii) S(t) is even iff F(-omega) = F(omega)
    (iv) S(t) is odd iff F(-omega) = -F(omega)
    The basic method of doing all of these is the same so I will do only one.

    F(omega) = integral_{x=-infty to +infty} f(x)exp(-i omega x) dx

    Now suppose f(x) is real, then:

    F*(omega) = conj[integral_{x=-infty to +infty} f(x)exp(-i omega x) dx]

    ............... = integral_{x=-infty to +infty} conj[f(x)exp(-i omega x)] dx

    ............... = integral_{x=-infty to +infty} f(x)exp(i omega x) dx

    ............... = F(-omega)

    So f real implies F*(omega) = F(-omega).

    Now assume F*(omega) = F(-omega), then:

    integral_{x=-infty to +infty} conj[f(x)exp(-i omega x)] dx

    ............... = integral_{x=-infty to +infty} f(x)exp(i omega x) dx

    or:

    integral_{x=-infty to +infty} f*(x)exp(i omega x)] dx

    ............... = integral_{x=-infty to +infty} f(x)exp(i omega x) dx

    Expand these into a+i b form and equate real and imaginary parts of these
    integrals gives im(f(x) = 0.

    So:

    So F*(omega) = F(-omega) implies f real, which together with the earlier
    paer proves the result.

    RonL
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Fourier Transform
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: October 9th 2011, 01:24 PM
  2. Laplace transform and Fourier transform what is the different?
    Posted in the Advanced Applied Math Forum
    Replies: 8
    Last Post: December 29th 2010, 11:51 PM
  3. Fourier transform
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 26th 2010, 05:08 AM
  4. Replies: 0
    Last Post: April 23rd 2009, 06:44 AM
  5. from fourier transform to fourier series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 1st 2008, 07:35 AM

Search Tags


/mathhelpforum @mathhelpforum