Help finding ths Fourier transform

In summary, the conversation discusses finding the Fourier transform of a function using direct computation and the convolution theorem. The speaker first attempts to solve it directly using Euler's identity but realizes it is a dead end. They then consider using the Lorentzian transform but are not familiar with it. They then try the convolution approach, but are unable to integrate one of the terms. A possible solution is suggested using partial fractions and the exponential integral function.
  • #1
gony rosenman
11
4

Homework Statement


find the Fourier transform of the following function in two ways , once using direct computation , and second using the convolution therom .

Homework Equations


JTbfgV4bPmqUm6G66
JTbfgV4bPmqUm6G66
Acos(w0t)/(d2+t2)

The Attempt at a Solution


I tried first to solve directly . used Euler's identity and got
∫e-it(w0+w)/(d2+t2) + ∫eit(w0-w)/(d2+t2)

but I think it was a deadend , unsolvable integral .

I got a hint from an internet source saying to use the Lorentzian transform but I am not familiar with it

then I tried going first from the convolution approach , which means I find the separate F transform of cos(w0t) and of 1/(d2+t2) and then calculate the convolution of them in order to get the FT of the multiplication oh them .

I couldn't do that as well because I didn't manage to integrate e-iwt/(d2+t2) and so can't find that Fourier transform as well

any direction ,or explained hint would be greatly appreciated!
 
Last edited:
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  • #2
This integral can be done using partial fractions, expanding the denominator as (d+t) * (d-t). It then gives a sum of terms of the exponential integral (Ei) function. Or you could just look it up.
 

1. What is the Fourier transform?

The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies. It converts a signal from its original domain (usually time or space) to a representation in the frequency domain.

2. Why is the Fourier transform important?

The Fourier transform is an essential tool in many fields of science and engineering. It allows us to analyze signals and systems in terms of their frequency components, which can provide valuable insights into their behavior and characteristics.

3. How do I calculate the Fourier transform?

The Fourier transform can be calculated using various mathematical methods, such as integration or convolution. In practice, there are also many software tools and libraries available that can perform the calculation for you.

4. What are some applications of the Fourier transform?

The Fourier transform has many applications in fields such as signal processing, image processing, data compression, and quantum mechanics. It is also used in various real-world applications, such as audio and video encoding, medical imaging, and wireless communication.

5. Are there any limitations to the Fourier transform?

The Fourier transform is a powerful tool, but it does have some limitations. It assumes that the signal is periodic and infinite, and it can struggle with signals that have discontinuities or sudden changes. Additionally, it is not always suitable for analyzing non-stationary signals, which have varying frequency components over time.

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