Cooley algorithm

Cooley J W and ] W Tukey 1965. An Algorithm for the Machine Calculation of Complex Fourier Series Aiathemalics of Computation 19 297-301. 

J. W. Cooley and J.W. Tukey, An algorithm for the machine calculation of complex Fourier series. Math. Comput., 19 (1965) 297-301. 

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

Previously, we referred to the FFT. Its development was a giant step that enabled the efficient computation of discrete Fourier transforms. The FFT algorithm and its variations have revolutionized signal analysis and made interferometric infrared spectroscopy practical. Both NMR spectra and mass spectra are also now computed from data that are acquired in their Fourier transform domain. The rediscovery of this algorithm by Cooley and Tukey (1965) is responsible for its current widespread use. Summaries of its properties and pitfalls are provided by Bergland (1969), Brigham (1974), and Bracewell (1978). 

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

I RADIX-2 ALGORITHM OF COOLEY AND TUKEY I tttttt ttltt t ttt ttttttt ttilt ttttttttttttt... 

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... 

Fourier-transform n.m.r. spectroscopy did not become a practical tool until the development of efficient algorithms for computerized, fast, Fourier transformation (f.f.t.) The first such algorithm to be widely appreciated was described by Cooley and Tukey156 in 1965, although Cooley and coworkers157 have pointed out that a somewhat... 

Chum HL, Ratcliff, Schroeder HA, Sopher DW (1984) Electrochemistry of biomass-derived materials Characterization, fractionation, and reductive electrolysis of ethanol extracted explosively depressurized aspen lignin J Wood Chem Technol 4 505-532 Compton DAC, Young JR, Kollar RG, Mooney JR, Grasselh JG (1987) In McClure GL (ed) Computerized quantitative infrared analysis ASTM, Philadelphia, 36-57 Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex Fourier series Math Comput 19 297-301... 

An algorithm developed by Cooley and Tukey simplified this extremely time-consuming calculation, bringing it within the capability of modern microcomputers. Today, the transformation takes only a few seconds, after which the frequency-domain spectrum Fj can be plotted. (The frequency-domain spectrum corresponding to Figure 3.16 will be discussed in Chapter 5.)... 

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... 

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