Line Gaussian

Matched filter The multiplication of the free induction decay with a sensitivity enhancement function that matches exactly the decay of the raw signal. This results in enhancement of resolution, but broadens the Lorentzian line by a factor of 2 and a Gaussian line by a factor of 2.5. 

Reproduced from ref. 48, with permission.) (a) Experimental spectrum at 120 K, (b) simulation with parameters of Table 4.15 and constant 4.1 G Gaussian line widths, (c) simulation with constant 4.1 G line widths, wg = 0.0049. (Reproduced with permission from ref. 48, copyright (1997) American Chemical Society.)... 

A Gaussian line and its first derivative are shown in Figures 5.5 and 5.6. Comparison with Figures 5.1 and 5.2 shows that the Gaussian line is somewhat fatter near the middle but lacks the broad wings of the Lorentzian line. 

In the practice of solid-state bioEPR, a Lorentzian line shape will be observed at relatively high temperatures and its width as a function of temperature can be used to deduce relaxation rates, while a Gaussian line will be observed at relatively low temperatures and its linewidth contains information on the distributed nature of the system. What exactly is high and low temperature, of course, depends on the system for the example of low-spin cytochrome a in Figure 4.2, a Lorentzian line will be observed at T = 80°C, and a Gaussian line will be found at T 20°C, while at T 50°C a mixture (a convolution) of the two distributions will be detected. 

Two samples of the same phosphor crystal have quite different thicknesses, so that one of them has a peak optical density of 3 at a frequency of vo. while the other one has a peak optical density of 0.2 at vq. Assume a half width at half maximum of Av = IGHz and a peak wavelength of 600 nm, and draw the absorption spectra (optical density versus frequency) for both samples. Then show the absorbance and transmittance spectra that you expect to obtain for both samples and compare them with the corresponding absorption spectra. (To be more precise, you can suppose that both bands have a Lorentzian profile, and use expression (1.8), or a Gaussian line shape, and then use expression (1.9).)... 

G Using the expression for the second moment of Gaussian lines, we can write ... 

The composite filter 7(g)) may either be the true inverse filter, truncated for oo large if necessary, or any of the variations described in Section IV. In their original work, Rendina and Larson chose 7(g)) = (co)/t(co), where //(co) is a Gaussian line-broadening function that limits the ultimate resolution obtainable but yields a manageable 7(g)). For their studies Rendina and Larson used Ns = 4. 

The truth of this is readily seen in the following hypothetical example. Assume that a spectrum of Gaussian lines is to be scanned at a rate such that the maximum Fourier component is 100 Hz. We might then establish the electronic bandpass such that a 100-Hz component is attenuated less than 3 dB. Without noise we would sample at 200 Hz. However, significant noise signals exist out to at least six times the passband frequency of 100 Hz, which means that we must sample at 1200 Hz to avoid aliasing the noise as we deconvolve. This is an extremely conservative approach, and one might well sample less frequently without difficulty. 

The net result of these considerations is that, for a spectrum of Gaussian lines, one should sample 10 times per resolution element ( FWHM of isolated lines) and that the scan rate should be adjusted to yield a scan rate of 10 times constants to scan one resolution element (Blass, 1976a). 

We have carried out simulations using polynomial least-squares filters of the type described by Savitzky and Golay (1964) to determine the impact of such smoothing on apparent resolution. For quadratic filters, a filter length of one-fourth of the linewidth (at FWHM) does not seriously degrade the apparent resolution of two Gaussian lines in very close proximity. 

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