Frequency domain performance

Frequency domain performance, involving comparison of cumulative frequency distributions of the observed data and model predictions. In many situations, considering the various sources of error discussed earlier, it may... 

Frequency domain performance has been analyzed with goodness-of-fit tests such as the Chi-square, Kolmogorov-Smirnov, and Wilcoxon Rank Sum tests. The studies by Young and Alward (14) and Hartigan et. al. (J 3) demonstrate the use of these tests for pesticide runoff and large-scale river basin modeling efforts, respectively, in conjunction with the paired-data tests. James and Burges ( 1 6 ) discuss the use of the above statistics and some additional tests in both the calibration and verification phases of model validation. They also discuss methods of data analysis for detection of errors this last topic needs additional research in order to consider uncertainties in the data which provide both the model input and the output to which model predictions are compared. 

In order to analyze the vibrations of a single molecule, many molecular dynamics steps must be performed. The data are then Fourier-transformed into the frequency domain to yield a vibrational spectrum. A given peak can be selected and transformed back to the time domain. This results in computing the vibra-... 

T. Analysis of the signal has primarily been performed in the time-domain although some applications are beginning to appear using frequency-domain techniques. The main features of the time-domain signal that are used for analysis are ... 

As mentioned earlier, we acquire data in the time domain but to make sense of it, we need to view it in the frequency domain. This is where the Fourier transformation comes in. There is not too much to do here - there are no parameters to change, ft is a necessary step but the automatic routines will perform this for you with no input. 

At present, two main streams of techniques exist for the measurement of fluorescence lifetimes, time domain based methods, and frequency domain methods. In the frequency domain, the fluorescence lifetime is derived from the phase shift and demodulation of the fluorescent light with respect to the phase and the modulation depth of a modulated excitation source. Measurements in the time domain are generally performed by recording the fluorescence intensity decay after exciting the specimen with a short excitation pulse. 

The least-squares method is also widely applied to curve fitting in phase-modulation fluorometry the main difference with data analysis in pulse fluorometry is that no deconvolution is required curve fitting is indeed performed in the frequency domain, i.e. directly using the variations of the phase shift and the modulation ratio M as functions of the modulation frequency. Phase data and modulation data can be analyzed separately or simultaneously. In the latter case the reduced chi squared is given by... 

Prior to describing the possible applications of laser-diode fluorometry, it is important to understand the two methods now used to measure fluorescence lifetimes these being the time-domain (Tl)/4 5 24 and frequency-domain (FD) or phase-modulation methods.(25) In TD fluorometry, the sample is excited by a pulse of light followed by measurement of the time-dependent intensity. In FD fluorometry, the sample is excited with amplitude-modulated light. The lifetime can be found from the phase angle delay and demodulation of the emission relative to the modulated incident light. We do not wish to fuel the debate of TD versus FD methods, but it is clear that phase and modulation measurements can be performed with simple and low cost instrumentation, and can provide excellent accuracy with short data acquisition times. 

Frequency domain techniques offer advantages over time domain techniques for real-time applications. In the frequency domain the measurements are performed within limited frequency bandwidths. The noise in limited bandwidths is reduced, in most cases substantially. Figures 9.7 illustrates this concept. In Figure 9.7a the... 

This comparison between time and frequency domain measurements is performed at submegahertz frequencies in order to avoid the issue of deconvolution of time domain signals. At megahertz frequencies time domain measurements encounter an additional limitation, these signals must be deconvoluted to isolate the sensor response from the instrument response. The need for deconvolutions adds extra software and computation time, which limits the versatility of time domain techniques for real-time applications. No deconvolutions are necessary in the frequency domain as shown below. 

At the present time, two methods are in common use for the determination of time-resolved anisotropy parameters—the single-photon counting or pulse method 55-56 and the frequency-domain or phase fluorometric methods. 57 59) These are described elsewhere in this series. Recently, both of these techniques have undergone considerable development, and there are a number of commercially available instruments which include analysis software. The question of which technique would be better for the study of membranes is therefore difficult to answer. Certainly, however, the multifrequency phase instruments are now fully comparable with the time-domain instruments, a situation which was not the case only a few years ago. Time-resolved measurements are generally rather more difficult to perform and may take considerably longer than the steady-state anisotropy measurements, and this should be borne in mind when samples are unstable or if information of kinetics is required. It is therefore important to evaluate the need to take such measurements in studies of membranes. Steady-state instruments are of course much less expensive, and considerable information can be extracted, although polarization optics are not usually supplied as standard. 

A new value of frequency is specified and the calculations repeated. Table 12.3 gives a FORTRAN program that performs alt these calculations, The initial part of the program solves for all the steadystate compositions and flow rates, given feed composition and feed flow rate and the desired bottoms and distillate compositions, by converging on the correct value of vapor boilup Vg. Next the coeflicients for the linearized equations arc calculated. Then the stepping technique is used to calculate the intermediate g s and the final P(j transfer functions in the frequency domain. 

The Fourier transform, in essence, decomposes a function into sinus functions of different frequency that sum to the original function. It is often useful to think of functions and their transforms as occupying two domains. These domains are often referred to as real and Fourier space, which are in most physics applications time and frequency. Operations performed in one domain have corresponding operations in the other. Moving between domains allows for operations to be performed where they are easiest or most advantageous [127]. 

Most NMR spectra consist of a number of signals and their time-domain spectra appear as a superposition of a number of traces of the type shown in Figure 5.3. Such spectra are quite uninterpretable by inspection, but Fourier transformation converts them into ordinary frequency-domain spectra. The time-scale of the FID experiment is of the order of seconds during which the magnetisation may be sampled many thousands of time. Data sampling is accomplished by a dedicated computer that is also used to perform the Fourier transformation. 

Analysis This pull-down menu is only available for frequency domain data (spectra) and allows a few simple analytical tasks to be performed such as peak picking, calibration, integration or simple spectral analysis. 

Processing in general transforms an original time domain signal s(t) with the aid of some processing function(s) into a manipulated frequency domain signal S"(f). Manipulations can be performed either in the time domain ( s(t) => s (t) ) prior to the Fourier transformation, or in the frequency domain (S (f) S"(f)) after the Fourier... 

NMR data i,s usually processed using one or more processing functions, some of which are applied in the time domain, others in the frequency domain. Each processing function in the time domain f(t) also has its counterpart F(f) in the frequency domain and forms a Fourier pair. In principle the same effect in the final spectrum S"(f) may be obtained with a given processing function, applied either in the time or the frequency domain as long as a few important rules are followed when performing the... 

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