Time-domain Fitting Research Articles

Time-domain fitting and director reorientation of proton NMR s=ectra of molecules dissolved in liquid-crystal solvents

Time-domain fitting and director reorientation of proton NMR s=ectra of molecules dissolved in liquid-crystal solvents

Improved algorithm for noniterative time-domain model fitting to exponentially damped magnetic resonance signals

This communication is concerned with a new method of fitting a physical model function to a magnetic resonance signal, directly in the time domain. Our primary aim is analysis of the signal in quantitative terms, i.e., describing the signal in terms of physically meaningful parameters with their statistical errors. Before explaining the new method we make some remarks about the place of time-domain model fitting in spectral analysis. The notion of quantitative description just defined goes beyond constructing a spectrum from the available time-domain data. This judgment is supported by the observation that recently proposed methods for constructing a spectrum (l-3), although very useful for many purposes, have not provided values of the physical parameters involved. In fact, if the latter are wanted afterward, it is then still necessary to fit an appropriate model function to the spectrum (4, 5). Furthermore, in addition to the fitting, the degree to which the spectrum constructed approximates the “ideal” spectrum is to be assessed. On the basis of these considerations, we advocate the use of timedomain model fitting, if quantitative description of a signal is needed. On the other hand, if the primary need is to have a spectrum, one may well use methods such as proposed in Refs. (Z-3), some of which require significantly less computer time. If the signal decays exponentially, which is not uncommon in magnetic resonance, an additional advantage of remaining in the time domain emerges, namely that the fitting procedure can be made noniterative. To the best of our knowledge noniterative fitting procedures are not yet available for the frequency domain. A method of noniterative fitting has recently been devised by Kumaresan and Tufts (6) and was subsequently applied to magnetic resonance (7, 8) under the name LPSVD; see also (9) for a related method. An error analysis was given in (6, IO). We are here concerned with an alternative noniterative model fitting procedure, devised by Kung et al. (II) using the so-called state space formalism. The method can handle considerably more data points than LPSVD because polynomial rooting is avoided. At the same time, the residue of the fit is usually better than that of LPSVD. We shall indicate that the basic idea can be explained with elementary matrix algebra, without invoking the state space formalism. In addition, a formula for efficient computer implementation is given.

The two-dimensional frequency-domain analysis of nuclear spin relaxation data. Simultaneous measurements of spin-spin and spin-lattice relaxation times

While most NMR pulse sequences designed to measure nuclear spin relaxation times yield true two-dimensional time-domain data, it is standard practice to Fourier transform these in only the t 2 dimension and to extract the relaxation time from a time-domain fit in the t 1 dimension. As an alternative to this procedure, the consequences of a full two-dimensional frequency-domain analysis are examined. It is shown that exponential time-domain relaxation data produce excellent Lorentzian lines in the frequency domain. Accurate relaxation times may be determined from the full width at half height using LW = 1 π · T n , where n = 1 or 2. An advantage of the frequency-domain fit is that it requires only two parameters instead of three, as often used in conventional analyses. In addition, new pulse sequences which allow the simultaneous determination of both T 1 and T 2 in a single 2D experiment are demonstrated for single-line spectra. In this case, only three parameters are necessary to determine both relaxation times, as compared with five when conventional procedures are employed.

The evaluation in time domain of mass transfer parameters from chromatographic peaks

Impulse response experiments have been carried out in packed beds in order to determine parameters as interstitial velocities, dispersion coefficients, internal diffusion coefficients and film transfer coefficients. The transfer function of the Kubín-Kucera model is solved in the Fourier domain. Two types of beds are considered, viz. long columns (about 6 m), packed with small particles and a short tube (1 m), packed with particles with a diameter in the order of the tube diameter. Moment analysis, Fourier analysis and fitting in the time domain are applied. It is shown that for Gaussian curves Fourier analysis has no advantage with respect to the method of moments. Moreover, for asymmetric curves a combination of Fourier analysis and time domain fitting is to be preferred. This is exemplified with a system where internal diffusion is the most important process. By applying Fourier analysis and fitting in the time domain, the interstitial velocity, axial dispersion coefficient and internal diffusion coefficient could be determined from a single experimental run with an accuracy better than 10 percent. For the experiments, performed in this work, the contribution of the film resistance compared with internal resistance proved to be negligible. The values obtained for the internal tortuosity factor and for the Péclet number agree with literature.

Chromatographic studies of CO and CO2 adsorption on activated alumina by weighted fourier analysis.

Fourier analysis for the parameter estimation in adsorption chromatography is, in practice, by no means equivalent to time-domain fitting, though Parseval''s theorem guarantees such relation. To reduce the effect of the tail of the response curves, the RTD curves are weighted with an exponential function, and then Fourier transformed. Intraparticle diffusivities, adsorption rate and equilibrium constants are successfully determined by this method, and comparisons are made between weighted and un-weighted results.

Mixing in trickle flow through packed beds

Axial mixing and mean residence times of the liquid phase in two-phase trickle flow in packings have been measured in terms of the dispersion model. The systems studied were 1 4 in. Raschig rings in a 2-in. diameter bed and 1 in. Lessing rings and 2 in. Raschig rings in a 2 ft diameter bed. The results were analysed by both moments and time-domain fit methods. The moments method was found to be unsuitable. The Peclet numbers obtained by time-domain fitting were reproducible and agreed with published figures obtained by frequency-domain methods. They were of the order of 1.1 to 10 times as large as those generally reported previously. The results are correlated by Pe 1 = 1.00(Re 1) 0.70Ga −0.32 The dispersion model is probably adequate for most applications, as it describes the RTD up to twice the mean residence time, which is approximately 90% of the flow elements. It underestimates the tailing response beyond twice the mean residence time.

New generator design isolates double frequency vibration : Power Plant Engineering, Volume 44, No. 12

This paper is about modeling of a two-pole cage induction machine equipped with a embedded force actuator for active control of lateral rotor vibrations. A fourth-order complex-valued linear differential-equation is derived to describe how the control voltage affects the rotor vibration. The model is identified using complex exponent sweeps since the model has explicit time-dependence. Validation measurements are done with white-noise signals. Time-domain fit with the presented techniques is better than with traditional ones.