FFT analysis

Mathematical techniques allow us to quantify total displacement caused by all vibrations, to convert the displacement measurements to velocity or acceleration, to separate this data into its components using FFT analysis, and to determine the amplitudes and phases of these functions. Such quantification is necessary if we are to isolate and correct abnormal vibrations in machinery. 

Hanning The Hanning correction provides the best capture of the individual frequency components of a signature. However, this weighting factor may distort the actual amplitude of the frequency components. Nevertheless, it is used for routine monitoring using FFT analysis. 

Flat-top weighting is useful when doing waterfall analysis. Even though the actual location of each frequency component may be slightly out of position, the profile is more visible when closely packed into a waterfall display. However, it is not normally used for single-channel FFT analysis. 

Fig. 8 Increase of wavelength (filled triangles) and amplitude (open circles) of the wrinkles with increasing plasma dose (oxygen plasma exposure time multiplied by plasma power). The values were evaluated by quantitative FFT-analysis applied to AFM height-images...

FIGURE 1 TEM (A) and HRTEM (B) images of CeOj nanopolyhedra. TEM (C) and HRTEM (D) images of CeOj nanorods, inset is a fast Fourier transform (FFT) analysis. TEM (E) and HRTEM (F) images of Ce02 nanocubes, inset is a fast Fourier transform (FFT) analysis. Reprinted with permission from Mai et al. (2005). Copyright 2005 American Chemical Society. 

Power consumption or torque fluctuations are influenced by granule properties (PSD, shape index, and apparent density) and the granulation time. Fluctuation of torque/power consumption and intensity of spectrum obtained by FFT analysis can be used for end-point determination. 

Fig. 2.39 (a) Cross sectional analysis of AFM data incl. 1D-FFT analysis along the selected line, (b) the 2D-FFT shows the typical periodicity in 2D the R value agrees well with the value obtained from the ID analysis in panel (a)... 

Fig. 3.52 Left (a) Schematic of intermittent contact mode AFM and phase imaging right (b) intermittent contact AFM phase image of a 30 nm thin block copolymer films on silicon [(poly(isoprene)-b-poly(ferrocenyl dimethylsilane), 29 kg/mol/15 kg/mol], which displays a in-plane worm-like surface pattern of poly(ferrocenyl dimethylsilane) in a matrix of poly (isoprene). From the 2D FFT analysis (inset) an average repeat period of 33 nm was estimated. Reprinted with permission from [116]. Copyright 2000. American Chemical Society...

Figure 5. NeNePo spectrum of Ag4 recorded in a one-color experiment (385 nm) at 20 K ion temperature. Trace A (insert) shows the uncorrected, mass-selected AgJ yield. Trace B is a composite of two measurements, and the signal has been corrected for the FFT analysis. (Figure taken from Ref. 212).

Figure 9. FFT analysis of the sum of sine wave perturbation left side, no optimization right side, optimization of phases, (a) Perturbation voltage in the time domain, (b) Perturbation voltage in the frequency domain, (c) Complex plane plots of simulated impedance spectra with 5% noise added to the current response. Solid lines show response without noise.

Table 8.3 shows the usual octave and one-third octave bands. As the name suggests, a one-third octave band instrument makes three measurements in each octave as opposed to the single measurement of the octave band instrument. A narrow band instrument, on the other hand, uses DSP to implement analysis FFT, and in the current state of the art, FFT analysis allows the analyzed frequency range to be sliced up into a large number of smaller intervals, limited in number only by the measured time interval length and the available computer power. 

AFM micrographs show (a) the S-layer recrystallized in vitro at pH = 3 in the presence of an APTES-Si substrate, (b) height profile based on (a), (c) zoomed image for lattice symmetry estimation, (d) FFT analysis based on high-quality image (b) showing the p4 symmetry (see black arrows). 

The optimum bubble conditions can be confirmed by a spectral analysis of the sound pressure. A secondary sound field is emitted by the oscillating bubbles that can be analyzed by the model and measured inside the reactor. Because of the nonlinear bubble motion under a sinusoidal driving pressure, a FFT-analysis of a hydrophone signal can be used to identify the dominant bubble behavior inside a reactor. Having a short depth of penetration, these signals can be used to map active zones. As a measure for developed and active cavitation the measured ratio... 

Figure 7 shows the time domain data for PE of 5 mm thickness, which is typical of the waveforms obtained from the polymers examined. The data were sampled at the rate of 2ns/word, which is fast enough to memorize the waveform in the MHz range. Similar waveform data were obtained for the PE sample with a thickness of 2 mm. Then the power spectra for these time domain data were calculated using FFT analysis, and those for PE of 2 mm and 5 mm thicknesses re shown in Fig. 8, revealing broad frequency components up to 12MHz and the maximum amplitude at different positions. 

Figure 16 shows the temperature and frequency dependence of ultrasonic absorption, which were obtained by FFT analysis of the ultrasonic signals observed at various temperatures. It is obvious that the absorption reaches a maximum around the ferroelectric phase transition of about 66 °C. Based on... 

The choice of frequencies that are included in the multisine stimulus waveform is an interesting area for discussion. In the case where all frequencies are stimulated in the applied multisine waveform, the FFT analysis is not able to differentiate between signals that are the response of the cell to the same frequency in the stimulus waveform, or signals that are simply harmonics of the stimulus frequencies due to the nonlinearity of the cell, and this can lead to very poor results. Of course it is... 

Much improved measurement can be obtained by reducing the number of stimulus frequencies in the multisine waveform. Some commercially available analyzers have either a preprogrammed list of frequencies or allow the multisine frequencies to be selected by the user. In this case, many frequencies in the FFT analysis band are deliberately not stimulated. The applied multisine frequencies are usually chosen so that each stimulus frequency is not coincident with the main harmonics of lower frequencies. In this case, the main components of harmonic distortion from each frequency in the stimulus waveform do not interfere with other stimulus frequencies (though of course it is not possible to avoid all significant harmonic frequencies). Without doubt, the impedance results from this approach are much improved compared to those obtained by the method where all frequencies are stimulated. 

Great care must also be exercised in the acquisition of data that is to be used for FFT analysis. In order to avoid problems with aliasing of frequencies (see section 3.1.3.4 for a discussion of this problem), it is essential to apply rigorous analog and digital filtering techniques, and the measurement sample rate must be sufficiently high (at least two times the maximum frequency of interest) otherwise out of band... 

Figure 3. Original signal prepared, (a) Signal waveform, (b) FFT analysis spectrum.

The main advantages of hydrophones are (1) the relative ease of use, (2) the fact that FFT analysis can be carried out, a feature which gives information on the type of cavitation (transient vs stable, access to chaos via a cascade of period doubling in the acoustic emissions from pulsating bubbles), (3) the topology of the reactors, (4) the shape of a pulsed signal in pulsed ultrasound, etc. ... 

Fast Fourier transformation (FFT) analysis of the normalized resistance signals for upper and lower sensors during multicyclic loading test performed on laminated glass composite specimens (Fig. 14.9) is given in Fig. 14.10(a) and (b), respectively. 

The FFT analysis allows a more detailed quantitative analysis of the effect of the lacquer system on the surface of the car body sheet (Fig. 8a). The wavelength-dependent contribution values are the highest for surface AR. They increase with increasing profile wavelength. The curve of surface PP is very similar. For surface BC, the values are typically one third as large below a wavelength of about 0.4 mm above which they match the values of curve AR. Surface TC contains additional lower amplitude contributions up to a wavelength of about 6 mm. 

For an identical set of static conditions to those described in Section 4.2 a series of frequency domain estimation experiments was performed. Typical results (again for an eccentricity ratio of 0.4) are shown in Fig. 5. These results show the evolution (with Increasing frequency) of the estimates of the four parameter states. The levels of forcing for these experiments was chosen to provide peak displacement amplitudes corresponding to some 30 per cent of clearance. Predictions of the various frequency response functions using both the estimated coefficients and FFT analysis are shown in Fig. 6. 

Transform (FFT) analysis or by autoregressive modern techniques. However, the processing of HRV and their analysis in frequency domain are not straight forward. The RR series must be first submitted to preprocessing procedures to produce a series of equidistantly sampled data suitable for spectral analysis. There are various methods to quantily the HRV and it can be derived from either heart period or heart rate. These signal have the same informative content, but the results obtained from each have shown considerable discrepancies, in as much as the relationship between them is non linear [1]. 

In the context of STFT, windowing is the process whereby shorter portions of a sampled sound are detached for FFT analysis (Figure 3.4). The windowing process must be sequential, but the windows may overlap (Figure 3.5). The effectiveness of the STFT process depends... 

Figure 3.4 Windowing is a process in STFT whereby shorter portions of the sound are detached for FFT analysis...

---