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*[[Signals and systems|Signals and Systems]] |
*[[Signals and systems|Signals and Systems]] |
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==Fourier Transform Applications== |
==Fourier Transform Introduction and focus on Applications== |
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Unfortunately, the Fourier Transform isn't a Transformer. If it was, you would have seen it in the movie that came out lately. [[Image:transformer_roolbar.jpg]] |
Unfortunately, the Fourier Transform isn't a Transformer. If it was, you would have seen it in the movie that came out lately. [[Image:transformer_roolbar.jpg]] |
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<br>One way to explain a Fourier Transform is to say it's a bunch of sinusoids added to create a just about any function you want. Another way to describe it is to say it's a way of representing a function in the frequency domain instead of the time domain. |
<br>One way to explain a Fourier Transform is to say it's a bunch of sinusoids added to create a just about any function you want. Another way to describe it is to say it's a way of representing a function in the frequency domain instead of the time domain. |
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<br>For example, a square wave could be represented by: |
<br>For example, a particular square wave could be represented by the following discrete fourier transform: |
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<math>x_{\mathrm{square}}(t) = \frac{4}{\pi} \sum_{k=1}^\infty {\sin{\left ((2k-1)2\pi ft \right )}\over(2k-1)} </math> |
<math>x_{\mathrm{square}}(t) = \frac{4}{\pi} \sum_{k=1}^\infty {\sin{\left ((2k-1)2\pi ft \right )}\over(2k-1)} </math><br> |
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== Fourier Transform Applicaitons == |
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What is a Fast Fourier Transform? (FFT)<br> |
<b>What is a Fast Fourier Transform? (FFT)</b><br> |
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It's an algorithm that can compute the discrete Fourier transform faster than other algorithms. In digital systems, continuous Fourier Transforms are sampled, turning them into discrete Fourier Transforms which then can be computed and manipulated using Digital Signal Processing. |
It's an algorithm that can compute the discrete Fourier transform faster than other algorithms. In digital systems, continuous Fourier Transforms are sampled, turning them into discrete Fourier Transforms which then can be computed and manipulated using Digital Signal Processing. |
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Revision as of 22:08, 11 October 2007
Fourier Transform Introduction and focus on Applications
Unfortunately, the Fourier Transform isn't a Transformer. If it was, you would have seen it in the movie that came out lately. 
One way to explain a Fourier Transform is to say it's a bunch of sinusoids added to create a just about any function you want. Another way to describe it is to say it's a way of representing a function in the frequency domain instead of the time domain.
For example, a particular square wave could be represented by the following discrete fourier transform:
Fourier Transform Applicaitons
The "Fast" Fourier Transform
Cooley-Turkey Algorithm
What is a Fast Fourier Transform? (FFT)
It's an algorithm that can compute the discrete Fourier transform faster than other algorithms. In digital systems, continuous Fourier Transforms are sampled, turning them into discrete Fourier Transforms which then can be computed and manipulated using Digital Signal Processing.
An intuitive brute force way of computing a Fourier Transform means rearranging the the summation so that you don't compute the transform in sequential order - you group similar elements together and simplify before combining them. This cuts down the adding and multiplying, thus cutting computation time down by about 100 times.
One of the most popular FFT algorithms is called the Cooley-Turkey algorithm. Which I will explain on Friday