Time-domain data

The Fourier analyzer is a digital deviee based on the eonversion of time-domain data to a frequeney domain by the use of the fast Fourier transform. The fast Fourier transform (FFT) analyzers employ a minieomputer to solve a set of simultaneous equations by matrix methods. 

Vibration data that is plotted as amplitude versus time is referred to as a time-domain data profile. Some simple examples are shown in Figures 43.1 and 43.2. An example of the complexity of this type of data for an actual piece of industrial machinery is shown in Figure 43.3. 

Time-domain plots must be used for all linear and reciprocating motion machinery. They are useful in the overall analysis of machine-trains to study changes in operating conditions. However, time-domain data are difficult to use. Because all the vibration data in this type of plot are added together to represent the total displacement at any given time, it is difficult to directly see the contribution of any particular vibration source. 

Frequency-domain data are obtained by converting time-domain data using a mathematical technique referred to as Fast Fourier Transform (FFT). FFT allows each vibration component of a complex machine-train spectrum to be shown as a discrete frequency peak. The frequency-domain amplitude can be the displacement per unit time related to a particular frequency, which is plotted as the Y-axis against frequency as the X-axis. This is opposed to time-domain spectrums that sum the velocities of all frequencies and plot the sum as the Y-axis against time... 

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. 

Time-domain data are presented with amplitude as the vertical axis and elapsed time as the horizontal axis. Time-domain profiles are the sum of all vibration components (i.e., frequencies, impacts, and other transients) that are present in the machine-train and its installed system. Time traces include all frequency components, but the individual components are more difficult to isolate than with frequency-domain data. 

With time-domain data, the analyst must manually separate the individual frequencies and events that are contained in the complex waveform. This effort is complicated tremendously by the superposition of multiple frequencies. Note that, rather than overlaying each of the discrete frequencies as illustrated theoretically in Figure 43.18(a), actual time-domain data represents the sum of these frequencies as was illustrated in Figure 43.17. 

For routine monitoring of machine vibration, however, this approach is not cost effective. The time required to manually isolate each of the frequency components and transient events contained in the waveform is prohibitive. However, time-domain data has a definite use in a total plant predictive maintenance or reliability improvement program. 

The frequency-domain format eliminates the manual effort required to isolate the components that make up a time trace. Frequency-domain techniques convert time-domain data into discrete frequency components using a mathematical process called Fast Fourier Transform (FFT). Simply stated, FFT mathematically converts a time-based trace into a series of discrete frequency components (see Figure 43.19). In a frequency-domain plot, the X-axis is frequency and the Y-axis is the amplitude of displacement, velocity, or acceleration. 

Most predictive-maintenance programs rely almost exclusively on frequency-domain vibration data. The microprocessor-based analyzers gather time-domain data and automatically convert it using Fast Fourier Transform (FFT) to frequency-domain data. A frequency-domain signature shows the machine s individual frequency components, or peaks. 

While frequency-domain data analysis is much easier to learn than time-domain data analysis, it does not provide the ability to isolate and identify all incipient problems within the machine or its installed system. Because of this. 

If the vibration analyzer permits acquisition of time-domain data, additional time-waveform data should be obtained from the intermediate guide as well as the inlet and discharge valves. The intermediate guide is located... 

The evolution period tl is systematically incremented in a 2D-experiment and the signals are recorded in the form of a time domain data matrix S(tl,t2). Typically, this matrix in our experiments has the dimensions of 512 points in tl and 1024 in t2. The frequency domain spectrum F(o l, o 2) is derived from this data by successive Fourier transformation with respect to t2 and tl. 

Fig. 3.—A Portion of the Aliphatic Region of the Proton-decoupled, C-N.m.r. Spectra of Native Glycophorin A (in HjO at 30°) and Fully Reductively [ C]Methylated Glycophorins A, A , and A (in H2O at 30°), at 22.5 MHz. [Taken from Ref. 56. Time-domain data were accumulated in 8192 addresses, with a recycle time of 1 s (except for A, where 2 s was used). A digital broadening of 2.8 Hz was applied (A) 1.9 mM virgin glycophorin A, at pH 6.5, after 50,000 accumulations (B) 1.6 mM fully reductively [ C]methylated glycophorin A , at pH 8.5, after 12,015 accumulations (C) 1.6 mM fully reductively [ C]methylated glycophorin A, at pH 7.3, after 14,208 accumulations (D) 1.5 mM fully reductively [ C]methylated glycophorin A °, at pH 7.2, after 12,815 accumulations.]...

In pulse NMR we measure in the time domain i.e., the variation of signal amplitude with time (FID) is recorded. These time-domain data are then subjected to Fourier transformation to convert them into the frequency domain. 

TDj, ti time domain data points TD, time domain pairs of data points SI2 and SIi, total data points in Fj and F, domains, respectively Hz/PTj and Hz/PT], digital resolution in F and F, (real) domains NS, number of acquired transients DS, number of dummy transients Tr, recycle time. 

At the end of the 2D experiment, we will have acquired a set of N FIDs composed of quadrature data points, with N /2 points from channel A and points from channel B, acquired with sequential (alternate) sampling. How the data are processed is critical for a successful outcome. The data processing involves (a) dc (direct current) correction (performed automatically by the instrument software), (b) apodization (window multiplication) of the <2 time-domain data, (c) Fourier transformation and phase correction, (d) window multiplication of the t domain data and phase correction (unless it is a magnitude or a power-mode spectrum, in which case phase correction is not required), (e) complex Fourier transformation in Fu (f) coaddition of real and imaginary data (if phase-sensitive representation is required) to give a magnitude (M) or a power-mode (P) spectrum. Additional steps may be tilting, symmetrization, and calculation of projections. A schematic representation of the steps involved is presented in Fig. 3.5. 

The next step after apodization of the t time-domain data is Fourier transformation and phase correction. As a result of the Fourier transformations of the t2 time domain, a number of different spectra are generated. Each spectrum corresponds to the behavior of the nuclear spins during the corresponding evolution period, with one spectrum resulting from each t value. A set of spectra is thus obtained, with the rows of the matrix now containing Areal and A imaginary data points. These real and imagi-... 

Fourier transformation A mathematical operation by which the FIDs are converted from time-domain data to the equivalent frequency-domain spectrum. 

An important technical development of the PFG and STD experiments was introduced at the beginning of the 1990s the Diffusion Ordered Spectroscopy, that is DOSY.69 70 It provides a convenient way of displaying the molecular self-diffusion information in a bi-dimensional array, with the NMR spectrum in one dimension and the self-diffusion coefficient in the other. While the chemical-shift information is obtained by Fast Fourier Transformation (FFT) of the time domain data, the diffusion information is obtained by an Inverse Laplace Transformation (ILT) of the signal decay data. The goal of DOSY experiment is to separate species spectroscopically (not physically) present in a mixture of compounds for this reason, DOSY is also known as "NMR chromatography."... 

Scalar coupled experiments COSY and TOCSY The correlated spectroscopy (COSY) experiment is one of the most simple 2D-NMR pulse sequences in terms of the number of RF pulses it requires [32]. The basic sequence consists of a 90-C-90-acquire. The sequence starts with an excitation pulse followed by an evolution period and then an additional 90° pulse prior to acquisition. Once the time domain data are Fourier transformed, the data appear as a diagonal in... 

The applicability of the ESE envelope modulation technique has been extended by two recent publications115,1161. Merks and de Beer1151 introduced a two-dimensional Fourier transform technique which is able to circumvent blind spots in the one-dimensional Fourier transformed display of ESE envelope modulation spectra, whereas van Ormondt and Nederveen1161 could enhance the resolution of ESE spectroscopy by applying the maximum entropy method for the spectral analysis of the time domain data. 

Instead of converting the step or pulse responses of a system into frequency response curves, it is fairly easy to use classical least-squares methods to solve for the best values of parameters of a model that fit the time-domain data. 

Processing of time domain data may cause artefacts in the frequency domain. One example for these distortions are truncations at the beginning or at the end of the FID which could lead to severe baseline artefacts which can be reduced by an appropriate filter. Undesired resonances leading to broad lines in the final spectra can be more easily eliminated in time domain by truncating the first few data points. Furthermore, the model functions in time domain are mathematically simpler to handle than the frequency domain analogues, which leads to a reduction of computation time. The advantage of the frequency domain analysis is that the quantification process can be directly interpreted visually. 

Figure 1 shows the noise level obtained with the maximum usable gain of 70 dB. Figure 2 is a F.F.T. of the time domain data of the previous figure. This shows the frequency distribution of the background noise. 

Fig. 4. Two-dimensional (2D) spectra of cyclo(Pro-Gly), 10 mM in 70/30 volume/volume DMSO/H2O mixture at CLio/27r = 500 MHz and T = 263 K. (A) TCX SY, t = 55 ms. (B) NOESY, Tm = 300 ms. (C) ROESY, = 300 ms, B, = 5 kHz. (D) T-ROESY, Tin = 300 ms, Bi = 10 kHz. Contours are plotted in the exponential mode with the increment of 1.41. Thus, a peak doubles its intensity every two contours. All spectra are recorded with 1024 data points, 8 scans per ti increment, 512 fi increments repetition time was 1.3 s and 90 = 8 ps 512x512 time domain data set was zero filled up to 1024 x 1024 data points, filtered by Lorentz to Gauss transformation in u>2 domain (GB = 0.03 LB = -3) and 80° skewed sin" in u), yielding a 2D Fourier transformation 1024 x 1024 data points real spectrum. (Continued on subsequent pages)...

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. 

Fourier transform for different, chemically very similar halomethanes and a mixture thereof. The time-domain data in Figure 7.11 can be directly interpreted as an observation of molecular motion in real time, made possible by the compressed ultrashort pulses in the microscope. From the presence of different oscillatory patterns and beatings, it already becomes clear that the different molecules can be discriminated with high resolution. Correspondingly, the Fourier spectra in Figure 7.11 show markedly different vibrational resonances, which can also be discriminated in the ternary mixture of all components. 

Both Fourier transform commands perform a Fast Fourier Transform (FFT) on the FID. If a baseline correction has not yet been performed on the raw data, a message box will appear which provides the option for performing a baseline correction (see section 5.3.3) on the time domain data (FID) prior to Fourier transformation. 

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