Fourier transform of exponential function

In summary, the problem statement is giving a function and a corresponding Fourier transform, the given data is the function and the known data is the inverse Fourier transform. The problem is trying to find the Fourier transform of a function given only its inverse Fourier transform. The attempt at a solution uses the scaling property of Fourier transforms to rescale time and frequency to get to the Fourier integral. The solution is to use the FT of the sample function to find the Fourier transform.
  • #1
Peter Alexander
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1. The problem statement, all variables, and given/known data
Task begins by giving sample function and a corresponding Fourier transform $$f(t) = e^{-t^2 / 2} \quad \Longleftrightarrow \quad F(\omega) = \sqrt{2 \pi} e^{-\omega^2 / 2}$$ and then asks to find the Fourier transform of $$f_a(t) = e^{-a t^2}$$

Homework Equations


The equation for Fourier transform, according to our textbook, is $$F(\omega) = \int_{-\infty}^{\infty} e^{i \omega t} f(t) d\omega$$ If I understand correctly, multiple notations exist, but this one was given to us.

The Attempt at a Solution


Normally the computation is pretty straightforward, simply plug ##f(t)## into the equation, perform some integration and done. However, this one gave me loads of troubles! A solution, according to output from Mathematica and our textbook solutions is $$F(\omega) = \sqrt{\frac{\pi}{a}} e^{-\frac{\omega ^2}{4 a}}$$ Some research later, I've found out that Fourier transform is linked to Dirac delta function, but this concept hasn't been discussed on lectures nor mentioned in the textbook. My suspicion is that link between exponential function and trigonometric functions (Euler's formula) can lead me a step closer to a solution.

Using Mathematica, I've seen that the integrand ##e^{i \omega t} f_a(t)## can be expressed as $$e^{t (-a t+i \omega )} = \cosh \left(a t^2-i t \omega \right)-\sinh \left(a t^2-i t \omega \right)$$ but since both integrals do not converge on ##(-\infty, \infty)## they cannot be computed. This claim was reafirmed when I tried integrading both functions in Mathematica.

At that point, I'm completely stuck. I thought about checking the given examples and trying to guess the value of Fourier transform this way, but even then I couldn't produce a solution. Would somebody be kind enough to help me?
 
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  • #2
You need to rescale time such that ##f(t')## gives you ##f_a(t)##. Then rescale the frequency to ##\omega'## to recover the Fourier integral and then you can find ##F(\omega')## using the FT of the sample function.

Edit: To clarify what I meant, you want to find
$$
\int_{-\infty}^{\infty} e^{i \omega t} e^{-at^2} d\omega
$$
You know that
$$
\int_{-\infty}^{\infty} e^{i \omega t} e^{-t^2/2} d\omega = \sqrt{2\pi} e^{-\omega^2/2}
$$
so modify this equation by changing ##t## and ##\omega## until you get to the integral above.
 
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  • #3
Are you familiar with the scaling property of Fourier transformations? If so, you should use it (and maybe review the proof). You should become familiar with the use of any property that has been covered to this point in your class.
 
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  • #4
I actually solved it, can't believe it was that easy!
I would like to thank both of you for your time and effort helping me understand this problem.
 
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1. What is the Fourier transform of an exponential function?

The Fourier transform of an exponential function is a complex function that represents the frequency components of the original exponential function. It is defined as the integral of the original function multiplied by a complex exponential function.

2. How is the Fourier transform of an exponential function calculated?

The Fourier transform of an exponential function can be calculated using the Fourier transform formula, which involves an integral. Alternatively, it can also be calculated using the properties of the Fourier transform, such as the time shifting and frequency shifting properties.

3. What is the relationship between the Fourier transform of an exponential function and the Fourier transform of a sinusoidal function?

The Fourier transform of an exponential function can be seen as a special case of the Fourier transform of a sinusoidal function. This is because an exponential function can be expressed as a combination of sinusoidal functions with different frequencies.

4. Can the Fourier transform of an exponential function be used in signal processing?

Yes, the Fourier transform of an exponential function is commonly used in signal processing to analyze signals in the frequency domain. It is particularly useful for analyzing signals with exponential decay or growth.

5. Are there any real-world applications of the Fourier transform of an exponential function?

Yes, the Fourier transform of an exponential function has many real-world applications, such as in image processing, audio processing, and data compression. It is also used in fields such as physics, engineering, and economics to analyze and model various phenomena.

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