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Thread: Fourier transform of a signal model

  1. #1
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    Fourier transform of a signal model

    I am having a discussion with another person about some data that I am looking at. The data can be roughly modeled as a ramp with an additive sinusoid. My friend is arguing that we can model this as:
    f(t) = a t+sin(2 \pi \omega t)

    where a,t,\omega \in \Re^+

    He then argues to use the Fourier transform to find the spectrum.

    However, I am arguing the following points:
    1) The function  f is not periodic, as required by the Fourier transform
    2) For the Fourier transform to exist, a sufficient condition is that the function is absolutely integrable. The function, as defined above, including the integration limits as defined by the support of  t , is not absolutely integrable.
    3) When my friend is computing the above Fourier transform above, the result is actually the form for a 'sawtooth' plus sinusoid. He is choosing (limiting) the limits of integration, in this case, to  (0, T) , so by computing the Fourier transform, he is implicitly making the signal periodic with period  T .

    Are my arguments correct?

    Assuming what I have said is correct, I expect the discussion will then go as follows: real world signals (such as the output of an amplifier or A/D converter) saturate at some value. So the signal doesn't increase indefinitely. So is it reasonable to say that the "form" of the Fourier transform is still correct? That is, can anything be said about the form of the Fourier transform whose signal is 1) not periodic and 2) violates the existence condition? That is, could limiting arguments be made?

    Thank you
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  2. #2
    MHF Contributor

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    Re: Fourier transform of a signal model

    The Fourier transform (as opposed to Fourier series) does not require that the function be periodic.
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  3. #3
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    Re: Fourier transform of a signal model

    I suppose I should also limit the support as follows:  0 < a < 1 and  \omega is in the audio range.
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  4. #4
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    Re: Fourier transform of a signal model

    the fourier sum produces the periodic extension of the signal being modeled.

    take a look at the following sheet

    Fourier transform of a signal model-clipboard01.jpg

    the first graph is of $f(t)$ and then 1 period of $f(t)$ with the rest being zero.

    the second graph shows $f(t)$ with the (even) periodic extension of $f(t)$ using 1 period.

    the periodic extension of non periodic signals can be created with whatever period you like.

    You should note that in this case the periodic extension has discontinuities at the period boundaries.

    If you did deep into Fourier theory you'll find that the reconstructed signal converges to the average of the left and right limits at discontinuities.
    Last edited by romsek; Feb 2nd 2017 at 11:10 AM.
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