Fourier transform algorithms

When performing optical simulations of laser beam propagation, using either the modal representation presented before, or fast Fourier transform algorithms, the available number of modes, or complex exponentials, is not inhnite, and this imposes a frequency cutoff in the simulations. All defects with frequencies larger than this cutoff frequency are not represented in the simulations, and their effects must be represented by scalar parameters. 

Fast Fourier transform algorithms are used to isolate the harmonics sampled. The individual phase delays are obtained from ... 

FFT Fast Fourier Transform algorithms 33 are used to obtain the frequency domain representation of the sample and reference waveforms sampled. 

There are two ways to collect FLIM data freqnency-domain or time-domain data acqnisition (Alcala et al. 1985 Jameson et al. 1984). Briefly, in freqnency domain FLIM, the fluorescence lifetime is determined by its different phase relative to a freqnency modulated excitation signal nsing a fast Fourier transform algorithm. This method requires a frequency synthesizer phase-locked to the repetition freqnency of the laser to drive an RF power amplifier that modulates the amplification of the detector photomultiplier at the master frequency plus an additional cross-correlation freqnency. In contrast, time-domain FLIM directly measures t using a photon connting PMT and card. 

The discrete Fourier transform (DFT) of the data is evaluated to take advantage of the considerable speed and accuracy of the fast-Fourier-transform algorithm as calculated by modern digital computers. For most... 

When using the fast-Fourier-transform algorithm to calculate the DFT, inverse filtering can be very fast indeed. By keeping the most noise-free inverse-filtered spectral components, and adding to these an additional band of restored spectral components, it is usually found that only a small number of components are needed to produce a result that closely approximates the original function. This is an additional reason for the efficiency of the method developed in this research. 

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... 

Figure 1. Schematic representation of a cubic trapped ion cell commonly used in FTMS. Coherent motion of ions in the cell induces an image current in the receiver plates. The time domain signal is subjected to a Fourier transform algorithm to yield a mass spectrum.

Several major points should be mentioned. In FT-IR interferograms are recorded, and the infrared spectra computed from the interferograms, via a fast Fourier transform algorithm introduced relatively recently (4). It is tire replacement of the monochromator of earlier spectrometers by an interferometer which is primarily responsible for the improved performance of FT instruments. 

Another important feature to note is that the Fourier Transform algorithm consists of an integral, going from -oo to + oo. This would imply that the mirror moves over an infinite distance. In practice, however, the mirror movement is restricted in the order of centimetres. 

The actual Fourier transform is a digital calculation, so not all frequencies are tested. In fact, the number of frequencies tested is exactly equal to the number of time values sampled in the FID. If we start with 16,384 complex data points in our FID (16,384 real data points and 16,384 imaginary data points), we will end up with 16,384 data points in the real spectrum (the imaginary spectrum is discarded). Another difference from the above description is that the actual Fourier transform algorithm used by computers is much more efficient than the tedious process of multiplying test functions, one by one, and calculating the area under the curve of the product function. This fast Fourier transform (FFT) algorithm makes the whole process vastly more efficient and in fact makes Fourier transform NMR possible. 

Whatever the excitation, the transformation of the response from the frequency to the time domain (Fig. 11.21) is done with the inverse Fourier transform, normally as the FFT (fast Fourier transform) algorithm, just as for spectra of electromagnetic radiation. Remembering that the Fourier transform is a special case of the Laplace transform with... 

EEG is quantitatively analyzed by spectral analysis using a fast Fourier transform algorithm. Spectra are calculated from epochs of 4 sec duration. The mean total spectral power between 1.5 and 64 Hz and the power in 5 subfrequency bands (1.5 4, 5-8, 8.5-12.5, 13-35 and 36-64 Hz) are calculated for the periods during which the treadmill is on. 

Sleep-wake cycle analysis is based on power spectral analysis. A fast Fourier transform algorithm calculates from the raw EEG signal total power and the power of subfrequency bands for each recorded channel from 512 data points. Single power spectra are then averaged to give a spectral point covering 20 sec of real time EEG. Under visual inspection of the spectral point-time curve, thresholds are placed... 

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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