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Thread: Fourier transform of heaviside functions.

  1. #1
    Member Danshader's Avatar
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    Fourier transform of heaviside functions.

    Well i found this question in my tutorial:
    f(t) = 5[H(t-3) - H(t-11)], find the fourier transform.
    Ans: (-10/w)(sin w)exp(-2iw)

    i am totally blur on how to even start. I tried the general method of fourier transform but got no closer to the answer.

    integrate from 3 to 5 for 5.exp(-iwt)

    Couldn't find any table with Fourier transform for H(t-a).

    All help are appreciated.
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  2. #2
    Senior Member Peritus's Avatar
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    $\displaystyle
    \begin{gathered}
    F\left\{ {5\left[ {u(t - 5) - u(t - 11)} \right]} \right\} = \hfill \\
    \hfill \\
    = 5\int\limits_3^{11} {e^{ - jwt} dt = \frac{5}
    {{jw}}\left( {e^{ - 3jw} - e^{ - 11jw} } \right) = } \hfill \\
    \end{gathered} $


    $\displaystyle \begin{gathered}
    \frac{{10}}
    {w}e^{ - 7jw} \frac{{e^{4jw} - e^{ - 4jw} }}
    {{2j}} = \frac{{10}}
    {w}e^{ - 7jw} \sin (4w) = \hfill \\
    \hfill \\
    = 40e^{ - 7jw} \sin c(4w) \hfill \\
    \end{gathered}
    $

    what we get is a pretty classic result: a square signal in the time domain transforms into sinc in the frequency domain.
    The Fourier transform of a step function includes something called Dirac's delta function you can read more about this in the following article:

    Fourier Transform--Heaviside Step Function -- from Wolfram MathWorld
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  3. #3
    Member Danshader's Avatar
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    ok i understand until line 3 but how did you get from the last term in line 3 to line 4? i wasn't taught sinc functions before.Also can you try getting the answer given for this question. i want to learn how to solve such questions since it has a probability of coming out in exam. thank you
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  4. #4
    Senior Member Peritus's Avatar
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    If you haven't learned about the sinc function then forget about the fourth line.

    Anyway that's the answer I get, maybe it can be transformed into the answer you were given although I'm pretty sure that my answer is correct:

    $\displaystyle
    \frac{{10}}
    {w}e^{ - 7jw} \sin (4w)$

    and if indeed the two answers are equivalent then you won't get points taken off for this answer.
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  5. #5
    Member Danshader's Avatar
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    aight thanks i will try getting the answer required from then answer you give ^-^
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