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I've looked at 10 videos and more webpages concerning the Fourier Transform. I somewhat understand what it does: It finds the frequencies that form a signal. However, it seems that it also changes the domain of a signal from time to frequency. I have some questions here...

- FT maps a function to a function... but only one function? How does the output function represent all those frequencies? Am I missing something?

- Is finding the frequencies that make up a signal the same thing as changing the domain of the signal to frequency?

- What does the imaginary part of an FT mean?

 

As an example of my utter pathetic confusion, I hereby present the sinc function and it's fourier transform: A "rectangle" function... ish. I graphed both of these, but I am completely lost as to how these two functions relate. I have seen the mathematical derivation, but how can I visually justify this?

 

I do somewhat understand how integrating a function multiplied by a backwards moving circle in the complex plane might be really nice... the integral is doing wonderful infinite summations. However, I do 100% understand that any signal can be expressed as a sum of sinusoids... 

 

I'm really lost, so thank you in advance for those who are willing to help me understand FT.

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Guest Dec 31, 2015
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 #1
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It's just as messed up to me as it is to you bro.😪 there are so,e things I don't even pretend to understand XD

Guest Dec 31, 2015
 #2
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Hello there! May I ask if you are a college student, and if so what year? Go through this interactive essay of FT and be patient and read it slowly and leisurely. It just might answer SOME of your questions. The only "Moderator" here that might be able to answer your questions is "Alan" . You will probably not be able to get him until Monday, Jan. 4, 2016. Good luck to you:

http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/

Guest Jan 1, 2016
 #3
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The Wikipedia article on Fourier transforms might help answer some of your questions (https://en.wikipedia.org/wiki/Fourier_transform).  Here is a copy of part of its first paragraph. I've highlighted a couple of parts related to your questions.  

 

The Fourier transform decomposes a function of time (a signal) into the frequencies that make it up, similarly to how a musical chord can be expressed as the amplitude (or loudness) of its constituent notes. The Fourier transform of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is called the frequency domain representation of the original signal.

 

To help understand the relationship between the time domain signal and the corresponding fourier transform domain result, try constructing a wave form yourself by adding together a couple of sine waves of different amplitude and frequencies and then applying a fourier transform to the result.  

 

A signal comprising the sum of pure sinusoids will be zero everywhere except at the frequencies of the sinusoids.  More general signals (e.g. sin(w*t)*e^-t) will have more components in the frequency domain.

Alan  Jan 1, 2016
edited by Alan  Jan 3, 2016
 #4
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[OP]

 

The better explained guide was pretty good to start... but imo the autho tied it up really badly at the end.

 

I found this: http://practicalcryptography.com/miscellaneous/machine-learning/intuitive-guide-discrete-fourier-transform/

 

This guide was actually alot better and understandable for me... It answered virtually all my questions.

Guest Jan 4, 2016

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