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*[[Signals and systems|Signals and Systems]] |
*[[Signals and systems|Signals and Systems]] |
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==Fourier Transform Applications== |
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===The "Fast" Fourier Transform=== |
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====Cooley-Turkey Algorithm ==== |
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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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<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> |
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What is a Fast Fourier Transform? (FFT)<br> |
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That's alot of numbers seemingly out of the blue, at first observance. I bet your questions are: |
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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. |
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<br> |
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=== 1. Telling me the difference between a transformer and a Fourier Transform hasn't helped me finish my assignment. What else can you tell me? === |
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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. |
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Wait till Friday. That's when the deadline for this assignment is, so hopefully I'll panic before then and get something done on this page in addition to the spectacular transformers picture. |
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<br> |
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One of the most popular FFT algorithms is called the Cooley-Turkey algorithm. Which I will explain on Friday |
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=== 2.Can you show me some really easy plug-in formulas, so I can get my homework done faster? === |
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Yes. <br> |
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=== 3.Why would I bother getting one? I don't know what to do with it. === |
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<br> Fourier Transforms are awesome! They allow continuous functions to be represented by ones and zeros, which means we can |
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implement functions on the computer. Fourier Transforms form the basis of signal processing. They allow us another way to transform |
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a math problem so it's alot easier to solve. |
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Revision as of 22:04, 11 October 2007
Fourier Transform Applications
The "Fast" Fourier Transform
Cooley-Turkey Algorithm
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 square wave could be represented by:
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