Fast Fourier Transform analysis

AC Impedance system 368 (EG G PARC, Princeton, NJ) with Fast Fourier Transform analysis was used along with a potentlostat EG G model 273. The potentlostat was coupled to an Apple 11+ computer through an IEEE-488 Interface. A frequency range of 0.01 Hz to 10 Hz was used for many of the experiments. The electrolyte used was a 3 weight percent sodium chloride solution and was prepared using ACS certified chemicals and deionized water. 

Figure 11.11. Determination of exact spaee groups present in a cubosome (diamond-type space group Pn3m) using cryo-TEM tilt experiments and fast Fourier transform analysis, (a) In the untilted particle a hexagonal motif is observed, as evidenced in the intensity representation of the Fourier transform shown in (b). (c) Same particle observed after 20° tilting, showing a different motif in the Fourier transform shown in (d). Adapted from Sagalowicz et al. 2006c.

The pnnaide of the novd ultrasonic spectro%opy utilizing a wide-band polymeric transducer and the FFT (fast Fourier transformation) analysis of a single pulse is introduced and its application to various polymraic matenals is reviewed Ultrasonic analysis of mechanical rdaxation processes and phase transitioiis m solid polymers as wdl as practical non-destructive inspection of defects m composite materials are described... 

Figure 14.10 Fast Fourier transformation analysis on (a) upper sensor signal during cycling loading and (b) lower sensor signal during cycling loading.

Fast Fourier Transformation is widely used in many fields of science, among them chemoractrics. The Fast Fourier Transformation (FFT) algorithm transforms the data from the "wavelength" domain into the "frequency" domain. The method is almost compulsorily used in spectral analysis, e, g., when near-infrared spectroscopy data arc employed as independent variables. Next, the spectral model is built between the responses and the Fourier coefficients of the transformation, which substitute the original Y-matrix. 

If further resolution is necessary one-third octave filters can be used but the number of required measurements is most unwieldy. It may be necessary to record the noise onto tape loops for the repeated re-analysis that is necessary. One-third octave filters are commonly used for building acoustics, and narrow-band real-time analysis can be employed. This is the fastest of the methods and is the most suitable for transient noises. Narrow-band analysis uses a VDU to show the graphical results of the fast Fourier transform and can also display octave or one-third octave bar graphs. 

Most of the early vibration analysis was carried out using analog equipment, which necessitated the use of time-domain data. The reason for this is that it was difficult to convert time-domain data to frequency-domain data. Therefore, frequency-domain capability was not available until microprocessor-based analyzers incorporated a straightforward method (i.e.. Fast Fourier Transform, FFT) of transforming the time-domain spectmm into its frequency components. 

The fast Fourier transform can be carried out by rearranging the various terms in the summations involved in the discrete Fourier transform. It is, in effect, a special book-keeping scheme that results in a very important simplification of the numerical evaluation of a Fburier transform. It was introduced into the scientific community in the mid-sixties and has resulted in what is probably one of the few significant advances in numerical methods of analysis since the invention of the digital computer. 

Fast Fourier Transform Flow Injection Analysis Field Ion Atom Probe Flame-Ionization Detector Field Ion Microscopy... 

Future development of spectroscopic structure-determination methods will depend on the availability of more powerful photon and particle sources as well as advances in photon and particle detectors. Impressive progress has been made in molecular structure determinations based on advances in computation power and in computational algorithms, such as fast Fourier-transform techniques, for nearly every form of spectroscopy and diffraction analysis. Hajdu and co-work-... 

The m/z values of peptide ions are mathematically derived from the sine wave profile by the performance of a fast Fourier transform operation. Thus, the detection of ions by FTICR is distinct from results from other MS approaches because the peptide ions are detected by their oscillation near the detection plate rather than by collision with a detector. Consequently, masses are resolved only by cyclotron frequency and not in space (sector instruments) or time (TOF analyzers). The magnetic field strength measured in Tesla correlates with the performance properties of FTICR. The instruments are very powerful and provide exquisitely high mass accuracy, mass resolution, and sensitivity—desirable properties in the analysis of complex protein mixtures. FTICR instruments are especially compatible with ESI29 but may also be used with MALDI as an ionization source.30 FTICR requires sophisticated expertise. Nevertheless, this technique is increasingly employed successfully in proteomics studies. 

Besides the molecular probe method using gas adsorption,107 162 recently, the TEM image analysis method163"167 has been applied to evaluate the surface fractal dimension of porous materials. The most attractive fact in this method is that the pores in different size ranges can be extracted from the TEM images which include contributions from many different pore sizes by the inverse fast Fourier transform (FFT) operation by selecting the specific frequency range.165 167... 

A uniform approach to trace the analysis and evaluation of electrode kinetics of the Cd(II)/Cd(Hg) system with fast Fourier transform electrochemical instrumentation was presented by Schiewe et al. [34]. 

Only seven published studies exist regarding the effects of a small number of antidepressants on computerized sleep EEG patterns [Jarrett et al. 1988 Kupfer et al. 1991, 1994 Minot et al. 1993). In these studies, only 118 total patients were included, and no confirmatory study was done for any of the drugs. Moreover, the length of the drug administration period varied from 1 day (Jarrett et al. 1988] to 3 years (Kupfer et al. 1994], and the method of computerized analysis was not uniform some studies used period analysis (Jarrett et al. 1988 Kupfer et al. 1991], others used fast Fourier transform (Minot et al. 1993 van Bemmel et al. 1992b, 1993b], and some others used both period analysis and fast Fourier transform (Kupfer et al. 1994). 

Barret, A. M., Normand, M.D., Peleg, M. and Ross, E. (1992). Characterisation of the jagged stress-strain relationships of puffed extrudates using the Fast Fourier Transform and fractal analysis. J. Food Sci. 57, 227-232, 235. 

Orbitrap The newest of the major mass analyzers, the Orbitrap is a hybrid MS consisting of a LIT mass analyzer, or transmission quadmpoles connected to the high-resolution Orbitrap mass analyzer. The Orbitrap utilizes electrical fields between sections of a roughly egg-shaped outer electrode and an inner (spindle) electrode (Chapter 5). Ions orbit between the inner and outer electrodes and their oscillation is recorded on detector plates (Hardman and Makarov, 2003 Hu et al., 2005). As with the FTICR, fast Fourier transform of the raw data is used to convert the data for mass analysis, making the Orbitrap the second major type of FTMS instrument. The resolving power of the Orbitrap is intermediate... 

Frequency Analysis. The Discrete Fourier Transform (and its fast implementation, the Fast Fourier Transform [Brigham, 1974]) (FFT) as well as its cousin, the Discrete Cosine Transform [Rao and Yip, 1990] (DCT) require block operations, as opposed to single sample inputs. The DFT can be described recursively, with the basis being the 2 point DFT calculated as follows ... 

The Phase Vocoder. The Phase Vocoder [Flanagan and Golden, 1966][Gordon and Strawn, 1985] is a common analysis technique because it provides an extremely flexible method of spectral modification. The phase vocoder models the signal as a bank of equally spaced bandpass filters with magnitude and phase outputs from each band. Portnoff s implementation of the Short Time Fourier Transform (STFT) provides a time-efficient implementation of the Phase Vocoder. The STFT requires a fast implementation of the Fast Fourier Transform (FFT), which typically involves bit addressed arithmetic. 

In general, there are two types of compression (1) individual spectra can be compressed and filtered and (2) the entire dataset can be compressed and filtered by representing each of the individual spectra as a linear combination of some smaller set of data, which is referred to as a basis set. In this section, we will address the processing of individual spectra by applying the fast fourier transform (FFT) algorithm and followed this discussion with one on processing sets of spectra with principal component analysis (PCA). 

---