What is the purpose of the exp[-(t^2)/2] term in Fourier transforms?

In summary, the conversation discusses the confusion surrounding the application of Fourier transforms to functions, specifically the role of the exp[-(t^2)/2] term in the wave equation. The speaker also mentions the importance of understanding Fourier series in order to fully understand Fourier transforms.
  • #1
cytochrome
166
3
I need more help understanding Fourier Transforms. I know that they transform a function from the time domain to the frequency domain and vice versa, but the short cuts to solve them just straight up confuse me.

http://www.cse.unr.edu/~bebis/CS474/Handouts/FT_Pairs1.pdf

This list of relations makes sense, but it's so hard for me to apply this to actual functions for some reason... I'd almost rather just do the integrals.

For example, every wave is of the form (except for the first exponential, which can always be different)

f(t) = exp[-(t^2)/2]exp(i*w*t)

The exp(i*w*t) part is the plane wave, correct? What is exp[-(t^2)/2] called? This term localizes the wave to keep it from being a plane wave, but I don't know what to call it.

Do you take the Fourier transform of the whole f(t), or just the localizing term?

Can someone please help me clear this up?
 
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  • #2
cytochrome said:
For example, every wave is of the form (except for the first exponential, which can always be different)

f(t) = exp[-(t^2)/2]exp(i*w*t)


The exp(i*w*t) part is the plane wave, correct? What is exp[-(t^2)/2] called? This term localizes the wave to keep it from being a plane wave, but I don't know what to call it.

Where are you getting this part from? I suspect you are misunderstanding something with this. Fourier transforms allow us to decompose a function into contributions of individual waves of the form

[tex]F(\omega)e^{i\omega t}[/tex]

[itex]F(\omega)[/itex] can be any function (well, practically any). A given function [itex]f(t)[/itex] determines [itex]F(\omega)[/itex] and vice versa.

If you want to truly understand Fourier transforms, I recommend getting a good grasp of Fourier series first.
 

1. What is the Fourier Transform Thread?

The Fourier Transform Thread is a mathematical tool used to analyze the frequency components of a time-domain signal. It decomposes a signal into its individual frequency components, allowing for a better understanding of its underlying patterns and characteristics.

2. How does the Fourier Transform work?

The Fourier Transform takes a time-domain signal and converts it into its respective frequency-domain representation. This is done by decomposing the signal into a sum of sinusoidal functions with different frequencies, amplitudes, and phases.

3. What is the importance of the Fourier Transform in science?

The Fourier Transform is used in a wide range of scientific fields, including physics, engineering, and signal processing. It allows scientists to analyze and understand complex signals, such as sound waves and electromagnetic waves, and make predictions about their behavior.

4. Are there different types of Fourier Transforms?

Yes, there are several types of Fourier Transforms, including the Discrete Fourier Transform, Fast Fourier Transform, and Inverse Fourier Transform. Each type has its own specific use and application.

5. What are some real-world applications of the Fourier Transform?

The Fourier Transform has many practical applications, such as in image processing, data compression, and digital signal processing. It is also used in fields like astronomy, chemistry, and biology to analyze and interpret various types of data.

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