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A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse.
FFTs are of great importance to a wide variety of applications, from digital signal processing and solving partial differential equations to algorithms for quick multiplication of large integers.
This article describes the algorithms, of which there are many; see discrete Fourier transform for properties and applications of the transform.
Let x0, ...., xN-1 be complex numbers.
The DFT is defined by the formula Evaluating these sums directly would take O(N 2) arithmetical operations.
An FFT is an algorithm to compute the same result in only O(N log N) operations.
In general, such algorithms depend upon the factorization of N, but (contrary to popular misconception) there are FFTs with O(N log N) complexity for all N, even for prime N. Many FFT algorithms only depend on the fact that is an Nth primitive root of unity, and thus can be applied to analogous transforms over any finite field, such as number-theoretic transforms.
Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a 1/N factor, any FFT algorithm can easily be adapted for it as well.
Source: Wikipedia (http://en.wikipedia.org/wiki/Fast_fourier_transform)
Here are the best overviews and tutorials for fft:
- tutorial, 5 stars
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FFT Online Tutorial
The FFT Interactive Tutorial is a self-paced, multimedia-based technical tutorial designed to help you understand the fundamentals of frequency-domain measurements.
Online tutorials contain audio, video, quizzes, and links for a full interactive experience.
This tutorial teaches you about the fast Fourier transform (FFT) in addition to some derived measurements, such as the power spectrum and power spectral density.
- overview, 4 stars
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Fourier Transform @ Wikipedia
In mathematics, the Fourier transform, named in honor of French mathematician Joseph Fourier, is a certain linear operator that maps functions to other functions.
Loosely speaking, the Fourier transform decomposes a function into a continuous spectrum of its frequency components, and the inverse transform synthesizes a function from its spectrum of frequency components.
A useful analogy is the relationship between a series of pure notes (the...
- overview, 4 stars
-
Fast Fourier transform @ Wikipedia
A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse.
There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory; this article gives an overview of the available techniques and some of their general properties, while the specific algorithms are described in subsidiary articles linked...
To browse all sites for 'FFT,' click
here
.
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