Fourier domain fitting

In determining the partition coefficient and the solute diffusion coefficient in the stationary polymer phase, either moment analysis or Fourier domain fitting was used. The two techniques are described below. 

Fourier domain fitting. The Fourier transform of the experimental elution curve is calculated. The parameters a and 3 are then determined using a fitting procedure in the Fourier domain that is equivalent to a least-squares criterion in the time domain. With Fourier domain estimation, model parameters are chosen to minimize the difference between the Fourier transforms of experimental and theoretical elution curves. The Fourier transform of a bounded, time varying response curve, f(t), is defined as... 

Procedure. The apparatus and general procedures of the capillary IGC experiment are described elsewhere (1.2.36-). Each measurement was conducted at three different earner gas flow rates (between 2 to 20 cm/s). For each experiment, an estimate of a and B was obtained using moment analysis and used as an initial guess for the Fourier domain fitting. The values of B2 at the three different earner gas flow rates were plotted versus 1/tc. Using equation (31), x2/D was estimated from the slope of B2 versus 1/tc, using a linear least-squares. 

It should be noted that the use of an appropriate analytic function instead of discrete replica may be beneficial as this (1) is less influenced by noise and (2) allows one to reproduce the wings of the resonances, which are neglected in the original CLEAN algorithm due to intensity threshold. On the other hand, if the line widths in the Fourier domain are mainly due to signal truncation, the fitted parameters poorly reflect the tme signal properties, and this may affect the performance of described procedure. 

As mentioned before, the smallest observable frequency (v ,in) in a continuous signal is the reciprocal of the measurement time ( I2T ). Because only those frequencies are considered which exactly fit in the measurement time, all frequencies should be a multiple of namely n/2T with n = -< to -l-oo. As a result the Fourier transform of a continuous signal is discrete in the frequency domain,... 

There is significant debate about the relative merits of frequency and time domain. In principle, they are related via the Fourier transformation and have been experimentally verified to be equivalent [9], For some applications, frequency domain instrumentation is easier to implement since ultrashort light pulses are not required, nor is deconvolution of the instrument response function, however, signal to noise ratio has recently been shown to be theoretically higher for time domain. The key advantage of time domain is that multiple decay components can, at least in principle, be extracted with ease from the decay profile by fitting with a multiexponential function, using relatively simple mathematical methods. 

Fitting model predictions to experimental observations can be performed in the Laplace, Fourier or time domains with optimal parameter choices often being made using weighted residuals techniques. James et al. [71] review and compare least squares, stochastic and hill-climbing methods for evaluating parameters and Froment and Bischoff [16] summarise some of the more common methods and warn that ordinary moments matching-techniques appear to be less reliable than alternative procedures. References 72 and 73 are studies of the errors associated with a selection of parameter extraction routines. 

These maxima in the Fourier transform data, which correspond to the different chromium coordination shells, were isolated using a filter window function. The inverse transform of each peak was generated and fitted using a non-linear least squares program. The amplitude and phase functions were obtained from the theoretical curves reported by Teo and Lee (2 ). The parameters which were refined included a scale factor, the Debye-Waller factor, the interatomic distance, and the threshold energy difference. This process led to refined distances of 1.97(2) and 2.73(2) A which were attributed to Cr-0 and Cr-Cr distances, respectively. Our inability to resolve second nearest neighbor Cr-Cr distances may be a consequence of the limited domain size of the pillars. 

Many methods for the fitting of data obtained from pulsed NMR have been described in the literature. The methods may, for example, be classified as either time domain (TD) or FD methods. Alternatively, they may be described as black-box methods or as interactive methods. An excellent review is given by de Beer and van Ormondt.20 There is now a consensus21 that FD and TD fitting methods are equivalent in terms of y2 parameter estimation if potential artifacts introduced by Fourier transformation are handled properly. TD and FD fitting can truly be equivalent (i.e. same 2 minima) if 2 is determined over the whole data range (a consequence of the power theorem on the Fourier transform) and if the model used to fit the experimental spectrum is correct. Very often, however, the model used is only an approximation. 

The original linear prediction and state-space methods are known in the nuclear magnetic resonance literature as LPSVD and Hankel singular value decomposition (HSVD), respectively, and many variants of them exist. Not only do these methods model the data, but also the fitted model parameters relate directly to actual physical parameters, thus making modelling and quantification a one-step process. The analysis is carried out in the time domain, although it is usually more convenient to display the results in the frequency domain by Fourier transformation of the fitted function. 

The process of going from the time domain spectrum fit) to the frequency domain spectrum F(v) is known as Fourier transformation. In this case the frequency of the line, say 100 MHz, in Figure 3.7(b) is simply the value of v which appears in the equation... 

A computer digitizes the time domain spectrum fit) and carries out the Fourier transformation to give a digitized F( v). Then digital-to-analogue conversion gives the frequency domain spectrum F(v) in the analogue form in which we require it. 

Varian VXR 500 NMR spectrometer operating at a proton frequency of 500 MHz. The spectral width was 6000 Hz with the carrier frequency at the HDO resonance. The solvent resonance was suppressed by presaturation. Each FID was composed of 16 k data points with 80 transients. The delay between successive transients was 2.8 s. The time domain data were processed by zero-filling to 32 k points, exponential multiplication (0.5 Hz) and Fourier transformation. Chemical shifts were referenced to internal sodium 3-(trimethylsilyl)-propionate-2,2,3,3-d4. The Kj values were obtained by nonlinear least square fit of the data to the equation... 

Laplace transformations are mainly used in signal analysis of electrical circuits for mathematical convenience. Differential and integral equations can often be reduced to nonlinear algebraic equations of the complex variable p in the transform domain. Many of the properties of the Fourier transformation can be taken over simply by substituting (ohy p. Particularly useful are the Laplace transforms L for differentiation and for integration. They can be expressed in terms of the transform F] p) of a function fit) by... 

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