Data Analysis Time-domain

The use of nonlinear least-squares analysis is ubiquitous in the analySK of fluorescence data, pairicululy time-domain and fiequency-domain data. A usefiil introduction to the principles of least-squares analysis is found in the compact but informative book by Bevington (1969). The applications of these concepts to diverse types of fluorescence data can be found in edited volumes (Brand and Johnson, 1992 Johnson and Brand, 1994). For more basic information about statistics and spectroscopy, one can examine several introductoiy texts (Ihylor, 1982 Mark and Workman, 1991). 

In the analysis of vibration data there is often the need to transform the data from the time domain to the frequency domain or, in other words, to obtain a spectrum analysis of the vibration. The original and inexpensive system to obtain this analysis is the tuneable swept-filter analyzer. Because of inherent limitations of this system, this process, despite the use of automated sweep, is time-consuming when analyzing low frequencies. When the spectra data needs to be digitized for computer inputing, there are further limitations in capability of tuneable filter-analysis systems. 

The process of vibration analysis requires gathering complex machine data and deciphering it. As opposed to the simple theoretical vibration curves shown in Figures 43.1 and 43.2, the profile for a piece of equipment is extremely complex. This is tme because there are usually many sources of vibration. Each source generates its own curve, but these are essentially added together and displayed as a composite profile. These profiles can be displayed in two formats time-domain and frequency-domain. 

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. 

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. 

Comparative analysis directly compares two or more data sets in order to detect changes in the operating condition of mechanical or process systems. This type of analysis is limited to the direct comparison of the time-domain or frequency-domain signature generated by a machine. The method does not determine the actual dynamics of the system. Typically, the following data are used for this purpose (1) baseline data, (2) known machine condition, or (3) industrial reference data. 

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. 

Time domain FLIM Theory, instrumentation, and data analysis... 

Recently, a method used for the analysis of frequency-domain data has been proposed for the analysis of time-domain images. The AB-plot or phasor plot provides a useful graphical representation of lifetime data that can be used for the segmentation of the images prior to data fitting [47, 48], With this method, data fitting may be avoided in many instances. 

Barber, P. R., Ameer-Beg, S. M., Gilbey, J. D., Edens, R. J., Ezike, I. and Vojnovic, B. (2005). Global and pixel kinetic data analysis for FRET detection by multi-photon time-domain FLIM. In Multiphoton Microscopy in the Biomedical Sciences V.Vol. 5700. SPIE, San Jose, CA, USA, pp. 171-81. 

ESE envelope modulation. In the context of the present paper the nuclear modulation effect in ESE is of particular interest110, mi. Rowan et al.1 1) have shown that the amplitude of the two- and three-pulse echoes1081 does not always decay smoothly as a function of the pulse time interval r. Instead, an oscillation in the envelope of the echo associated with the hf frequencies of nuclei near the unpaired electron is observed. In systems with a large number of interacting nuclei the analysis of this modulated envelope by computer simulation has proved to be difficult in the time domain. However, it has been shown by Mims1121 that the Fourier transform of the modulation data of a three-pulse echo into the frequency domain yields a spectrum similar to that of an ENDOR spectrum. Merks and de Beer1131 have demonstrated that the display in the frequency domain has many advantages over the parameter estimation procedure in the time domain. 

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. 

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. 

Fig. 5. Discriminant PLS analysis of NMR time domain CPMG data acquired at 100 MHz versus mealiness in Cox s apples having three degrees of mealiness and labelled (Coxa, Coxb and Coxc), acquired in the data for Ref. 34.

The terms cepstrum and cepstral come from inverting the first half of the words spectrum and spectral they were coined because often in cepstral analysis one treats data in the frequency domain as though it were in the time domain, and vice versa. The value of cepstral analysis comes from the observation that the logarithm of the power spectrum of a signal consisting of two echoes has an additive periodic component due to the presence of the two echoes, and therefore the Fourier transform of the logarithm of the power spectrum exhibits a peak at the time interval between them. The... 

A most convenient way to solve the differential equations describing a mass transport problem is the Laplace transform method. Applications of this method to many different cases can be found in several modern and classical textbooks [21—23, 53, 73]. In addition, the fact that electrochemical relationships in the so-called Laplace domain are much simpler than in the original time domain has been employed as an expedient for the analysis of experimental data or even as the basic principle for a new technique. The latter aspect, especially, will be explained in the present section. 

---