Lorentz-Gauss transformation

The weighting functions used to improve line shapes for such absolute-value-mode spectra are sine-bell, sine bell squared, phase-shifted sine-bell, phase-shifted sine-bell squared, and a Lorentz-Gauss transformation function. The effects of various window functions on COSY data (absolute-value mode) are presented in Fig. 3.10. One advantage of multiplying the time domain S(f ) or S(tf) by such functions is to enhance the intensities of the cross-peaks relative to the noncorrelation peaks lying on the diagonal. 

Figure 3.37. The Lorentz-Gauss transformation ( Gaussian multiplication ) can be used to improve resolution, (a) Raw FID and spectrum following Fourier transformation and results after the L-G transformation with (b) lb = -IHz, gb = 0.2 and (c) lb =...

Figure 4b. Aromatic portion of the spectrum of the same sample recorded by disk acquisition and transformation of 128 K data points using a total of 48 K hardware memory. Resolution was fur -ther enhanced by Lorentz-Gauss transformation (6).

Figure 2 Background signal (2) in the NMR spectrum of a multinuclear probe head. (A) ai NMR spectrum of a 0.1 M solution of [Et2AIO(CH2)2NEt2]2- (B) Spectrum obtained with the test-tube filled with solvent only. For resolution enhancement, the same Lorentz-Gauss transformation was used in both cases before Fourier transformation. Reprinted with permission of Wiley-VCH from Benn R and Rufinska A (1986) Angewandte Chemie, International Edition in English 25 861-881.

Fig. 4. Two-dimensional (2D) spectra of cyclo(Pro-Gly), 10 mM in 70/30 volume/volume DMSO/H2O mixture at CLio/27r = 500 MHz and T = 263 K. (A) TCX SY, t = 55 ms. (B) NOESY, Tm = 300 ms. (C) ROESY, = 300 ms, B, = 5 kHz. (D) T-ROESY, Tin = 300 ms, Bi = 10 kHz. Contours are plotted in the exponential mode with the increment of 1.41. Thus, a peak doubles its intensity every two contours. All spectra are recorded with 1024 data points, 8 scans per ti increment, 512 fi increments repetition time was 1.3 s and 90 = 8 ps 512x512 time domain data set was zero filled up to 1024 x 1024 data points, filtered by Lorentz to Gauss transformation in u>2 domain (GB = 0.03 LB = -3) and 80° skewed sin" in u), yielding a 2D Fourier transformation 1024 x 1024 data points real spectrum. (Continued on subsequent pages)...

If we can set Rre to cancel exactly the original decay of the FID then the result of this process is to generate a time-domain function which only has a Gaussian decay. The resulting peak in the spectrum will have a Gaussian lineshape, which is often considered to be superior to the Lorentzian as it is narrower at the base the two lineshapes are compared in Fig. 4.12. This transformation to a Gaussian lineshape is often called the Lorentz-to-Gauss transformation. 

Much of the popularity of these functions probably rests of the fact that there is only one parameter to adjust, rather than two in the case of the Lorentz-to-Gauss transformation. 

Apodization is the process of multiplying the FID prior to Fourier transformation by a mathematical function. The type of mathematical or window function applied depends upon the enhancement required the signal-to-noise ratio in a spectrum can be improved by applying an exponential window function to a noisy FID whilst the resolution can be improved by reducing the signal linewidth using a Lorentz-Gauss function. ID WIN-NMR has a variety of window functions, abbreviated to wdw function, such as exponential (EM), shifted sine-bell (SINE) and sine-bell squared (QSINE). Each window function has its own particular parameters associated with it LB for EM function, SSB for sine functions etc. 

This is incompatible with Maxwell s equations, as shown below by using Gauss s law, Eq. (2.7.16), and the Lorentz force, Eq. (2.7.24). Assume that the two systems S and S move at velocities v and v and relative velocity V= v — v. If we use the Galileian transformation and assume that the charge q and the electric displacement D is the same in the two systems ... 

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