Cooley-Tukey algorithm fast Fourier transform

The discrete FT, eqn [9], can be evaluated in a brute force fashion on a computer using the available sine and cosine functions, eqn [3], but this method is very slow for a large number of points. The FT algorithm of Cooley and Tukey is much faster. The derivation of the Cooley-Tukey algorithm ( fast Fourier transform ) starts by rewriting the exponent in eqn [10] as... 

FCC catalyst, 11 728-729 thermoelectric, 21 555, 556 Cooley-Tukey fast Fourier transform (fft) algorithm, 23 137 Cool flames, 7 442—443 Cooling... 

M67 Fast Fourier transform Radix-2 algorithm of Cooley and Tukey 6700 67B2... 

In the early days, this Fourier transformation was a time-consuming, expensive and difficult task due to limited computer speed and capacity. However, with the advent of the fast Fourier transform algorithm of Cooley and Tukey 6) and the improvement in computers, this problem has been resolved so that real time spectra can be obtained with the transformation time of the order of fractions of seconds. 

This Fourier transform process was well known to Michelson and his peers but the computational difficulty of making the transformation prevented the application of this powerful interferometric technique to spectroscopy. An important advance was made with the discovery of the fast Fourier transform algorithm by Cooley and Tukey 29) which revived the field of spectroscopy using interferometers by allowing the calculation of the Fourier transform to be carried out rapidly. The fast Fourier transform (FFT) has been discussed in several places 30,31). The essence of the technique is the reduction in the number of computer multiplications and additions. The normal computer evaluation requires n(n — 1) additions and multiplications whereas the FFT method only requires (n logj n) additions and multiplications. If we have a 4096-point array to Fourier transform, it would require (4096) (4095) or 16.7 million multiplications. The FFT allows us to reduce this to... 

In practice, one uses a less redundant fast Fourier transform algorithm, e.g.. the Cooley-Tukey algorithm rather than the expression shown above. Possible problems connected with discrete Fourier transfomiation (DFT) include... 

The major advance towards routine use in the mid-infrared region -j j- g. came with a new mathematical method (or algorithm) devised by Cooley and Tukey in 1965 for fast Fourier transformation. This was ... 

Cooley, J.W. and Tukey, J.W. (1955) Fast Fourier transform algorithm. Math. Comput., 19, 297-301. 

The DFT is numerically inefficient and demands many multiplications/divisions. Cooley and Tukey have developed a more efficient algorithm that reduces the number of calculations for N to N loga N. This is the so-called fast Fourier transform (FFT) [75], which is implemented in many programs including Microsoft Excel. However, it requires that the number of data points be a power of 2, that is,... 

Fast Fourier transform (FFT) Title which is now somewhat loosely used to describe any efficient algorithm for the machine computation of the discrete Fourier transform. Perhaps the best known example of these algorithms is the seminal work by Cooley and Tukey in the early 1960s. 

As the number of points to be calculated increases beyond about 10,000, the calculation time for a spectmm can become prohibitive, even for very fast present-day computers, for which that computation can take many hours. Prior to the development of fast, readily available computers, this problem was especially annoying and did not appear to be resolvable until about 1966. At that time, Forman [2] published a paper on the application of the fast Fourier transform technique to Fourier spectrometry. This technique had been described in the literature by Cooley and Tukey [3] one year earlier. This algorithm extended the use of Fourier transform spectrometry to encompass high-resolution data in all regions of the infrared spectmm. It is described in the next section. 

The fast Fourier transform (FFT) is an algorithm that was described by Cooley and Tukey [3] in which the number of necessary computations is drastically reduced when compared to the classical Fourier transform. To understand the algorithm, the interferogram and the spectmm should be regarded as the complex pair [4]... 

A major breakthrough in the area of Fourier transform computation occurred in 1965 with the introduction of the fast Fourier transform algorithm by Cooley and Tukey, This had an immediate impact on Fourier transform spectroscopy allowing for the first time the measurement of extensive spectral intervals at very high resolution. 

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