Summing in quadrature

A must be summed in quadrature with the FWHM keV noise AA in order to obtain the measured energy resolution M. . 

Fig. 8.18 Results of the address level calibration [13]. Left RMS of width of the six address level peaks for all operable ROCs in the detector. Right Separation between the mean of two neighboring peaks. The separation is given in units of sigma, defined by summing in quadrature the RMS widths for adjacent peaks...

A pre-requisite for the successful extraction of key NMR parameters from an experimental spectrum is the way it is processed after acquisition. The success criteria are low noise levels, good resolution and flat baseline. Clearly, there are also experimental expedients that can further these aims, but these are not the subject of this review per se. In choosing window functions prior to FT, the criteria of low noise levels and good resolution run counter to one another and the optimum is just that. Zero filling the free induction decay (FID) to the sum of the number acquired in both the u and v spectra (in quadrature detection) allow the most information to be extracted. 

Summing all errors in quadrature results is a 27 ppm-40 ppm uncertainty. The main sources of uncertainty are therefore statistical, reference wavelengths and dispersion function determination. All major error sources are soft and may be reduced further. Methods of reducing statistical uncertainty by improving spectrometer efficiency are being investigated and improved flux from the EBIT has been achieved in other studies [26],... 

When the light-matter interaction is directly in quadrature form, s is odd and at most only 5(5 — ) different frequencies are incident since each field, but one, appears with both absorptive and emissive components. Thus the absolute phase of only one field survives the sum in Eq. (2.12c). The odd field is the Fourier component at ot)j so that finally = < y(r) - y(r) = 0. That is. 

For induction motors, the total current can be approximated by the vector sum of the torque producing component and the magnetising or flux producing component of motor current. These two current vectors are in quadrature, and as the hoist torque approaches zero, the motor current approaches magnetising current. Electrical RMS calculations for hoist motors must be based on motor current, and so the calculation based on torque current only is not accurate for induction motors, and under-estimates the induction motor current. For greatest accuracy the duty cycle must be calculated with the correct motor characteristics. 

The spot is not a simple circle, so it is not clear how to combine these two values, but adding them in quadrature (rss, or root-of-the-sum-of the-squares") should give us a good estimate - about 100 pm. This is larger than most modern detectors. If we want a smaller spot, we must use a smaller aperture (or move it farther away), or use a shorter focal length lens and work with a faster lens, or a shorter wavelength. ... 

The total uncertainty is the rss (root-of-the-sum-of-the-squares, or addition-in-quadrature ) of these two, or about 0.56 mV. 

To Calculate Noise from a Set o/H.Hf Values The square of the noise in the narrow spectral band around each frequency is the PSD times the spectral bandwidth. Noise adds in quadrature, so the square of the total noise (also known as the variance) is the sum over all frequencies of the PSD times the frequency intervals between the PSD values ... 

In fact, each linear polarizability itself consists of a sum of two temis, one potentially resonant and the other anti-resonant, corresponding to die two doorway events, and D, and the window events, and described above. The hyperpolarizability chosen in equation (B1.3.12) happens to belong to the generator. As noted, such tliree-coloiir generators caimot produce Class I spectroscopies (fiill quadrature with tliree colours is not possible). Only the two-colour generators are able to create the Class I Raman spectroscopies and, in any case, only two colours are nomially used for the Class II Raman spectroscopies as well. 

The AB and AX systems of all 13C —13C bonds appear in one spectrum when the INADEQUATE pulse sequence (Fig. 2.48) is applied. Complete interpretation usually becomes difficult in practice due to signal overlapping, isotope shifts and AB effects (Section 2.9.4). A separation of the individual 13C— 13C two-spin systems by means of a second dimension would be desirable. It is the frequency of the double quantum transfer (d e) in Fig. 2.48 which introduces a second dimension to the INADEQUATE experiment. This double quantum frequency vDQ characterizes each 13CA — I3CX bond, as it depends on the sum of the individual carbon shieldings vA and vx in addition to the frequency v0 of the transmitter pulse located in the center of the spectrum if quadrature detection is applied [69c, 71] ... 

Table 1. Comparison of theoretical and experiment results. Where two errors are listed in the experiment column, the first is the statistical and the second is the systemmatic. The error in the difference column is the quadrature sum of the experimental and theoretical error...

The many existing quadrature formulas differ only in the choice of functions to fit to the data points. Two of the simplest approximations are to fit straight lines between successive points and sum the area under the lines, and to fit parabolas to successive triplets of points and sum the areas under the parabolas. These approximations lead to the quadrature formulas known respectively as the trapezoidal rule and Simpson s rule. We will discuss each in turn. 

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