Fourier Transform of a Triangular Voltage Pulse

In summary, the problem given involves calculating the Fourier transform of a function in a second year physics course. The solution involves factoring (1 - cos(ωτ)) in terms of sin(ωτ/2) and identifying the celebrated 'sinc' function. The amplitude, width, and center of the transform can then be determined by understanding the behavior of the sinc function. However, if the sinc function has not been taught, it is recommended to learn it as it is a simple concept.
  • #1
SeannyBoi71
84
0

Homework Statement


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So this is a physics problem, but this question doesn't really have to do with the "physics" part of it as much as simply calculating the Fourier transform. (This is a second year physics course and our prof is trying to briefly teach us math tools like this in learning quantum mechanics).

Homework Equations



[tex] \tilde{g}(\omega) = \frac{1}{\sqrt2\pi} \int g(t) e^{-i \omega t} dt [/tex]

The Attempt at a Solution


I have done the calculation of g(ω) several times and got an answer

[tex] \frac{2}{(\tau \omega ^2 \sqrt2 \pi)} (1 - cos(\omega \tau)) [/tex]

I believe it is right, but since the work to get it is extensive I don't want to type it up unless someone thinks I made an error. My actual concern is that I have a problem sketching the transform. I graphed it on Wolfram so I have a general idea, but I really have no idea how to find the amplitude, width, and whether it should be centred at ω=0 or at a k0 value. Any insight would be greatly appreciated.
 
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  • #2
SeannyBoi71 said:
[tex] \frac{2}{(\tau \omega ^2 \sqrt2 \pi)} (1 - cos(\omega \tau)) [/tex]
Try factoring [itex](1 - cos(\omega \tau))[/itex] in terms of sin(ωτ/2) and identify the celebrated 'sinc' function
 
  • #3
I had to google 'sinc function' to find out what it was... We have not been taught this so surely there is a way to find the attributes of the transform by simply looking at it. Our prof mentioned something about finding the general root of the transform (here it would be whenever the cos term is 1), but I don't know how to relate that to the centre or anything
 
  • #4
SeannyBoi71 said:
I had to google 'sinc function' to find out what it was... We have not been taught this so surely there is a way to find the attributes of the transform by simply looking at it. Our prof mentioned something about finding the general root of the transform (here it would be whenever the cos term is 1), but I don't know how to relate that to the centre or anything

If you haven't been taught sinc function, then learn it, it's simple enough. Your expression can be written as the square of the sinc function, where behavior of sinc is well understood. I certainly don't know what your prof had in mind, but the sinc function is centered at zero.
 

Related to Fourier Transform of a Triangular Voltage Pulse

1. What is a Fourier Transform?

A Fourier Transform is a mathematical tool used to decompose a complex signal into its individual frequency components. It allows us to analyze and understand the frequency content of a signal.

2. How is a Fourier Transform applied to a triangular voltage pulse?

A Fourier Transform can be applied to a triangular voltage pulse by taking the integral of the pulse over an infinite time period and then dividing by the period of the pulse. This will result in a frequency domain representation of the pulse, showing the amplitude and phase of each frequency component.

3. What information can be obtained from a Fourier Transform of a triangular voltage pulse?

A Fourier Transform of a triangular voltage pulse can provide information about the frequency content of the pulse, including the dominant frequencies and their relative amplitudes. It can also reveal any distortions or noise present in the signal.

4. What is the relationship between the time domain and frequency domain representations of a triangular voltage pulse?

The time domain representation of a triangular voltage pulse shows the amplitude of the signal over time, while the frequency domain representation shows the amplitude of each frequency component present in the signal. The two representations are related by the Fourier Transform.

5. How is the inverse Fourier Transform used in relation to a triangular voltage pulse?

The inverse Fourier Transform can be used to reconstruct the original signal from its frequency domain representation. In the case of a triangular voltage pulse, the inverse Fourier Transform would convert the frequency domain representation back into the time domain representation of the pulse.

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