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Math Help - Fourier Transform of Pulse with Gaussian Envelope

  1. #1
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    Fourier Transform of Pulse with Gaussian Envelope

    I am trying to analytically calculate the fourier transform of the following signal:

    x(t) = exp-(t/t0)^2 * cos(w0(1-qt)t)

    if anyone has any ideas on how to simplify this in the transform to something 'intergratable' that would be brilliant. I just get a load of exponential terms with imaginary componants that wont simplify!
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    Grand Panjandrum
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    Quote Originally Posted by jamieross1987 View Post
    I am trying to analytically calculate the fourier transform of the following signal:

    x(t) = exp-(t/t0)^2 * cos(w0(1-qt)t)

    if anyone has any ideas on how to simplify this in the transform to something 'intergratable' that would be brilliant. I just get a load of exponential terms with imaginary componants that wont simplify!
    What is qt, is this a constant or q times t?

    If it is a constant, then write the cos in exponential form and the FT is then the sum of two shifted scaled copies of the FT of exp(-(t/t_0)^2)

    CB
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    It is q*t I'm afraid. Any ideas for an approach if this is the case?
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    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by jamieross1987 View Post
    I am trying to analytically calculate the fourier transform of the following signal:

    x(t) = exp-(t/t0)^2 * cos(w0(1-qt)t)

    if anyone has any ideas on how to simplify this in the transform to something 'intergratable' that would be brilliant. I just get a load of exponential terms with imaginary componants that wont simplify!
    A time function of the type...

    \displaystyle x(t)= \varphi(t)\ \cos \omega_{0}\ (1-q t)\ t (1)

    ... is known as chirp pulse and is widely used, for example, in radar signal processing. That means that the FT of the signal (1) is well explained in specialized licterature...

    Kind regards

    \chi \sigma
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    Grand Panjandrum
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    Quote Originally Posted by chisigma View Post
    A time function of the type...

    \displaystyle x(t)= \varphi(t)\ \cos \omega_{0}\ (1-q t)\ t (1)

    ... is known as chirp pulse and is widely used, for example, in radar signal processing. That means that the FT of the signal (1) is well explained in specialized licterature...

    Kind regards

    \chi \sigma
    In fact this is a Linear Frequency Modulated (LFM) pulse, but a Gaussian envelope is never used (having all of time as its support), the envelope always has a finite union of finite interval for its' support.

    CB
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