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*[[Signals and systems|Signals and Systems]] |
*[[Signals and systems|Signals and Systems]] |
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==Fourier Transform |
==Fourier Transform Applications== |
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Unfortunately, the Fourier Transform isn't a Transformer. |
Unfortunately, the Fourier Transform isn't a Transformer. |
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[[Image:transformer_roolbar.jpg]] |
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<br>For example, a particular square wave could be represented by the following discrete fourier transform: |
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| ⚫ | <br>Check any of the other pages on this site to find fifty different ways to explain what a Fourier Transform is. If you already know what it is, or you're too lazy to look at the other pages, here's my super trite description: 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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<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 Applications == |
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===The "Fast" Fourier Transform=== |
===The "Fast" Fourier Transform=== |
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<b>What is a Fast Fourier Transform? (FFT)</b><br> |
<b>What is a Fast Fourier Transform? (FFT)</b><br> |
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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. |
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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Revision as of 21:11, 11 October 2007
Fourier Transform Applications
Unfortunately, the Fourier Transform isn't a Transformer.
What is a Fourier Transform?
Check any of the other pages on this site to find fifty different ways to explain what a Fourier Transform is. If you already know what it is, or you're too lazy to look at the other pages, here's my super trite description: 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.
Fourier Transform Applications
The "Fast" Fourier Transform
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.
Cooley-Turkey Algorithm
One of the most popular FFT algorithms is the Cooley-Turkey algorithm. Which I will explain on Friday.