Fast Fourier Transform - definition. What is Fast Fourier Transform
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%ما هو (من)٪ 1 - تعريف

O(N LOGN) DIVIDE AND CONQUER ALGORITHM TO CALCULATE THE DISCRETE FOURIER TRANSFORMS
Fast Fourier Transform; Fast fourier transform; Fast Fourier Transforms; IFFT; FFT; Arithmetic complexity of the discrete Fourier transform; FFT complexity; FFT algorithm; Arithmetic complexity of the discrete fourier transform; Fast fourier; Fast Fourier; Approximations of the fast Fourier transform; Applications of the fast Fourier transform; Interaction algorithm; Inverse fast fourier transform; Multidimensional fast Fourier transform
  • An example FFT algorithm structure, using a decomposition into half-size FFTs
  • A discrete Fourier analysis of a sum of cosine waves at 10, 20, 30, 40, and 50 Hz
  • Time-based representation (above) and frequency-based representation (below) of the same signal, where the lower representation can be obtained from the upper one by Fourier transformation

Fast Fourier Transform         
<algorithm> (FFT) An algorithm for computing the {Fourier transform} of a set of discrete data values. Given a finite set of data points, for example a periodic sampling taken from a real-world signal, the FFT expresses the data in terms of its component frequencies. It also solves the essentially identical inverse problem of reconstructing a signal from the frequency data. The FFT is a mainstay of numerical analysis. Gilbert Strang described it as "the most important algorithm of our generation". The FFT also provides the asymptotically fastest known algorithm for multiplying two polynomials. Versions of the algorithm (in C and Fortran) can be found on-line from the GAMS server {here (http://gams.nist.gov/cgi-bin/gams-serve/class/J1.html)}. ["Numerical Methods and Analysis", Buchanan and Turner]. (1994-11-09)
Fast Fourier transform         
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa.
Fourier-transform ion cyclotron resonance         
INSTRUMENT IN MASS SPECTROMETRY
Fourier Transform Mass Spectrometry; FTICR; FTMS; Fourier Transform Ion Cyclotron Resonance; Fourier transform mass spectrometry; Stored waveform inverse Fourier transform; Fourier transform ion cyclotron resonance mass spectrometer; Fourier transform ion cyclotron resonance mass spectrometry; Fourier transform ion cyclotron resonance mass spectrometers; FT-ICR; Fourier transform ion cyclotron resonance; Fourier-transform mass spectrometry
Fourier-transform ion cyclotron resonance mass spectrometry is a type of mass analyzer (or mass spectrometer) for determining the mass-to-charge ratio (m/z) of ions based on the cyclotron frequency of the ions in a fixed magnetic field. The ions are trapped in a Penning trap (a magnetic field with electric trapping plates), where they are excited (at their resonant cyclotron frequencies) to a larger cyclotron radius by an oscillating electric field orthogonal to the magnetic field.

ويكيبيديا

Fast Fourier transform

A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT from O ( N 2 ) {\textstyle O\left(N^{2}\right)} , which arises if one simply applies the definition of DFT, to O ( N log N ) {\textstyle O(N\log N)} , where N {\displaystyle N} is the data size. The difference in speed can be enormous, especially for long data sets where N may be in the thousands or millions. In the presence of round-off error, many FFT algorithms are much more accurate than evaluating the DFT definition directly or indirectly. There are many different FFT algorithms based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory.

Fast Fourier transforms are widely used for applications in engineering, music, science, and mathematics. The basic ideas were popularized in 1965, but some algorithms had been derived as early as 1805. In 1994, Gilbert Strang described the FFT as "the most important numerical algorithm of our lifetime", and it was included in Top 10 Algorithms of 20th Century by the IEEE magazine Computing in Science & Engineering.

The best-known FFT algorithms depend upon the factorization of N, but there are FFTs with O(N log N) complexity for all N, even for prime N. Many FFT algorithms depend only on the fact that e 2 π i / N {\textstyle e^{-2\pi i/N}} is an N-th 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.