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Math Help - Fourier Transforms Help

  1. #1
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    Fourier Transforms Help

    I need help with the following problems:



    Question 1:


    a) Find the Fourier Transform of




    b) Determine the limit of this transform as and discuss the result

    Question 2:

    Solve for Y(x) the integral equation





    and verify the solution by direct substitution

    Question 3:

    Use the time-shift property to calculate the Fourier transform of the double pulse defined by








    Question 4:
    Given that




    a) Draw the graph of f(t)
    b) Express f(t) in terms of the Heaviside unit step function
    c) Find the Fourier transform of f(t)

    Thanks in advance for all the valuable help. It is most appreciated.
    Attached Thumbnails Attached Thumbnails Fourier Transforms Help-q1.jpg   Fourier Transforms Help-q2.jpg   Fourier Transforms Help-q3.jpg   Fourier Transforms Help-q4.jpg  
    Last edited by atwinix; September 28th 2007 at 04:14 AM. Reason: Missed detail
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by atwinix View Post
    I need help with the following problems:



    Question 1:


    a) Find the Fourier Transform of

    <br />
\mathcal{F}f(\omega)=\frac{1}{2 \varepsilon}\int_{-\varepsilon}^{\varepsilon}e^{i~\omega x} ~dx = \frac{\sin(\omega \varepsilon)}{\omega \varepsilon}<br />

    (I will leave the last step in evaluating the integral in the above to you,
    it's not difficult).

    b) Determine the limit of this transform as \varepsilon \to 0_{+} and discuss the result
    You are supposed to know that:

    <br />
\lim_{\varepsilon \to 0} \frac{\sin( \varepsilon)}{\varepsilon}=1<br />

    So:

    <br />
\lim_{\varepsilon \to 0}  \frac{\sin(\omega \varepsilon)}{\omega \varepsilon}=1<br />

    RonL
    Last edited by CaptainBlack; September 28th 2007 at 10:02 AM.
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  3. #3
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    More help

    Show that the convolution of a top-hat function with itself is the triangle function. That is

    Hint: Top-Hat function and Triangle function
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by atwinix View Post
    Show that the convolution of a top-hat function with itself is the triangle function. That is

    Hint: Top-Hat function and Triangle function
    Please post new questions in new threads.

    -Dan
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