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An intuitive brute force way of computing a Fourier Transform is just adding the function up one at a time starting at n=0 or n=1 like this: |
An intuitive brute force way of computing a Fourier Transform is just adding the function up one at a time starting at n=0 or n=1 like this: |
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:<math>\sum_{i=m}^n x_i = x_m + x_{m+1} + x_{m+2} +\cdots+ x_{n-1} + x_n. </math> |
:<math>\sum_{i=m}^n x_i = x_m + x_{m+1} + x_{m+2} +\cdots+ x_{n-1} + x_n. </math> |
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What an FFT does is rearrange 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. |
What an FFT does is rearrange 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. The faster computation time makes it a great way to perform analysis for DSP applications like quantum mechanics, linear network analysis, probability, and antenna analysis. |
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<br><b> Cooley-Tukey Algorithm </b> |
<br><b> Cooley-Tukey Algorithm </b> |
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Revision as of 09:47, 19 October 2007
What is a Fourier Transform?
Unfortunately, the Fourier Transform isn't a Transform-er.
(This handsome fellow is named Roolbar.)
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: A Fourier Transform is a bunch of sinusoids of different frequencies and time offsets added together create a just about any function you want. Also, you can say that a Fourier Transform is the way of representing a function in the frequency domain instead of the time domain.
Instead of describing the Fourier Transform itself once again, the following pages describe how Fourier Transforms are analyzed and utilized in a computer.
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 is just adding the function up one at a time starting at n=0 or n=1 like this:
What an FFT does is rearrange 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. The faster computation time makes it a great way to perform analysis for DSP applications like quantum mechanics, linear network analysis, probability, and antenna analysis.
Cooley-Tukey Algorithm
Three popular algorithms are the Cooley Tukey algorithm (radix-r algorithms), the Good-Thomas Algorithm, and the Winograd FFT algorithm. The Cooley Tukey Algorithm Radix-2 algorithm is the simplest and most popular of all. The algorithm separates a given discrete Fourier transform of size into two equal transform of size recursively. Here is the definition of the algorithm given on wikipedia's page.
where is an integer ranging from to .
