Frequency axis

Figure 11 Single detector signal provided by the inspection of inner circonferential notch tube sample, and corresponding scalogram. Time axis (in s) and frequency axis (in Hz) have been scaled according to the speed of evolution of the detector in ihe tube (500 mm/s).

Most response functions of poiymers obey a time-temperature or frequency-temperature superjDosition [43, 44], A change in temperature is equivaient to a shift of tire iogaritiimic frequency axis ... 

We can sample the energy density of radiation p(v, T) within a chamber at a fixed temperature T (essentially an oven or furnace) by opening a tiny transparent window in the chamber wall so as to let a little radiation out. The amount of radiation sampled must be very small so as not to disturb the equilibrium condition inside the chamber. When this is done at many different frequencies v, the blackbody spectrum is obtained. When the temperature is changed, the area under the spechal curve is greater or smaller and the curve is displaced on the frequency axis but its shape remains essentially the same. The chamber is called a blackbody because, from the point of view of an observer within the chamber, radiation lost through the aperture to the universe is perfectly absorbed the probability of a photon finding its way from the universe back through the aperture into the chamber is zero. 

The key dimension m NMR is the frequency axis All of the spectra we have seen so far are ID spectra because they have only one frequency axis In 2D NMR a stan dard pulse sequence adds a second frequency axis Only pulsed FT NMR spectrometers are capable of carrying out 2D experiments... 

The reset time, which is nser-adjnstable, can range from 0.05 seconds to 80 minutes or more, depending on controller design. The reset time constant, when converted to frequency 1/2(Tr) Hz (where Tr is the reset time in seconds), determines the frequency where the reset and proportional response characteristics of the controller merge (see Fig. 8-64Z ). Tuning the reset adjustment on the controller moves the reset frequency to the left or right along the frequency axis and thereby affec ts the reset ac tion of the controller. 

Equation 1.11 can be used to relate the dynamic behavior at one temperature Tj to that at another, T2, as illustrated in Figure 1.6. When the temperature is raised to T2, the curves are displaced laterally by the distance, log aj, on the logarithmic frequency axis, where log Oj now reflects the change in characteristic response frequency of molecular segments when the temperature is changed from Ti to T2. Thus, log aj is given by... 

To ensure that no information is lost on Fio)) as the dilation is discretized, the scale factors 2 " for m g Z must cover the whole frequency axis. This can be accomplished by requiring the wavelets to satisfy the following... 

Two-dimensional NMR A strange concept, when we consider that all the spectra we have previously dealt with were of course plotted in two dimensions, the two axes (dimensions) being a frequency axis (horizontal, expressed in ppm rather than in Hz for reasons we have already discussed) and an intensity axis. To understand the basic idea of two-dimensional NMR (2D NMR) we should first remind ourselves that while the spectrum we see and use is plotted as a... 

Generally, the rheology of polymer melts depends strongly on the temperature at which the measurement is carried out. It is well known that for thermorheological simplicity, isotherms of storage modulus (G (co)), loss modulus (G"(complex viscosity (r (co)) can be superimposed by horizontal shifts along the frequency axis ... 

A theoretical COSY spectrum for two coupled methine protons (AX system) is shown in Figure 9.44, the chemical shifts of the two protons being 8A and 8X ppm. The theoretical 1 -D proton spectrum is shown alongside each frequency axis, Ft and F2, the two scales being equal. Two types of... 

Fig. 5.4 From top down the measured absorption profiles of methane, methyl chloride, and ethylene obtained using a WGM locked to the laser. In each case, the top trace shows the amplified variation in dip depth and the bottom trace is the transmission profile of the gas in a 16 cm absorption cell. The frequency axis shows the tuning range. Reprinted from Ref. 4 with permission. 2008 Optical Society of America...

The subscript ref refers to the values at the reference temperature. If we plot the measurement frequency multiplied by the constant aT all the data superimpose on the frequency axis. The correction for the moduli is to multiply by the function... 

As the electrochemical reaction is confined to the boundaries of the thin film, the voltammetric response exhibits a quasireversible maximum. The position of the quasireversible maximum on the log frequency axis depends on the kinetics of the overall reaction at the thin-film electrode, i.e., reflecting the coupled electron-ion transfer (4.3). Analyzing the evolution of the quasireversible maximum measured with different redox probes and various transferring ions, it has been demonstrated... 

Ti oc The observed NMR signal generally decays exponentially, which supports the conclusion that there is rapid material exchange between surface and bulk phases of liquid in the pore. The shift on the frequency axis associated with the freezing event is consistent with the blocking of rapid exchange between bulk and surface phases i.e., elimination of the bulk phase. A similar power law is found for tetradecane in Vycor above the freezing temperature. 

In this expression e(/ ) is the dielectric constant —it is a function of frequency —along the imaginary frequency axis / it is measurable as the dissipative part of the spectrum of dielectric constant for any material. The latter is an experimentally determined function of frequency for each of the three components, and the complicated expression in Equation (64) is integrated over all frequencies. 

Figure 3. Comparison of the experimental and theoretical Na2 - Na(3s) + Na(3d) yields as a function of w2. In the calculation, an intermediate v = 33, J = 31, 33 resonance is used and to is fixed at 17,720.7 cm"1. The intensities of the two laser fields are 7(

A cut along the pump laser frequency axis for a nonresonant dump laser frequency yields REMPI spectra. The same cut for a resonant dump laser frequency show a splitting of the 2 < - 1 transition that can be explained in the intuitive picture of a dynamic Stark effect. (For a closer discussion of this point, see Refs. 6 and 30). A cut along the dump laser frequency... 

All experiments described previously deal with one-dimensional 13C NMR spectra Spectral parameters such ai chemical shifts, coupling constants and relaxation times are evaluated from spectra with one frequency axis as one dimension. A second dimension comes into the NMR experiment when two spectral parameters, such as chemical shifts and coupling constants or chemical shifts of different nuclei, are represented in an area limited by two frequency axes. These experiments are based on pulse sequences facilitating multiplicity analysis and partly involving further sensitivity enhancement of the 13C NMR experiment. 

How do we extract the chemical shifts of all nuclei in the sample from the free-induction decay signal The answer is our old friend the Fourier transform. The FID is called a time-domain signal because it is a plot of the oscillating and decaying RF intensity versus time, as shown in Fig. 10.4 (the time axis is conventionally labeled t2, for reasons you will see shortly). Fourier transforming the FID produces afrequency-domain spectrum, a plot of RF intensity versus the frequencies present in the FID signal, with the frequency axis labeled v2 for frequency or F2 for chemical shift, as shown in Fig. 10.1. So the Fourier transform decomposes the FID into its component frequencies, revealing the chemical shifts of the nuclei in the sample. 

Figure 8.2.9 CSI experiment using four samples stacked vertically. The resulting contour plot is a spatial/chemical-shift map, which can be analysed by taking slices along the frequency axis to yield individual spectra for each sample position. Reprinted with permission from Hou, T., Smith, J., MacNamara, E., Macnaughtan, M. and Raftery, D., Anal. Chem., 73, 2541-2546 (2001). Copyright (2001) American Chemical Society...

In the internal representation model the time-frequency plane is divided in cells with a resolution of 20 ms along in the time axis (time index m) and of 0.2 Bark along the frequency axis (frequency index /). A first approach was to use the power ratio between the output y and input x, py px in every (At, A f) cell (m, /) as a correction factor for the noise disturbance L (m, l) in that cell (nomenclature is chosen to be consistent with [Beerends and Stemerdink, 1992]). 

Contrary to the point by point approach the diagonalization method consists of the generation of an entire lineshape function in one step. (13, 14, 57-60) Time-consuming calculations are carried out only once. The resulting set of complex numbers can be used for a simple calculation of the lineshape (absorption and dispersion modes) at any desired point on the frequency axis. Thus, the complex matrix from equation (147) can be diagonalized by a similarity transformation using an co-independent complex matrix W ... 

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