Frequency domain Lorentzian lineshape function

The Lorentzian lineshape is obtained for liquid samples under ideal high-resolution NMR conditions10 and is readily derived from the Bloch equations.31 The normalized Lorentzian lineshape function in the frequency domain is given by... 

The definition of the convolution product is quite clear like the one of the Fourier transforms, it has a given mathematical expression. An important property of convolution is that the product of two functions corresponds to the Fourier transform of the convolution product of their Fourier transforms. In the context of high-resolution FT-NMR, a typical example is the signal of a given spin coupled to a spin one half. In the time domain, the relaxation gives rise to an exponential decay multiplied by a cosine function under the influence of the coupling. In the frequency domain, the first corresponds to a Lorentzian lineshape while the second corresponds to a doublet of delta functions. The spectrum of such a spin has a lineshape which is the result of the convolution product of the Lorentzian with the doublet of delta functions. In contrast, the word deconvolution is not always used with equal clarity. Sometimes it is meant as the strict reverse process of convolution, in which case it corresponds to a division in the reciprocal domain, but it is often used more loosely to mean simplification. This lack of clarity is due to the diversity of solutions offered to the problem of deconvolution, depending on the function to be deconvoluted, the quality one wishes to obtain, and other parameters. 

As usual there is a reciprocal relationship between the time domain and the frequency domain. The more rapid is the damping (decay) of the signal (i.e., larger a and shorter lifetime t = 1/a), the wider the Lorentzian and dispersion lineshape functions become in the spectral domain (Figure 4). 

A spectrum may be analyzed either in the frequency domain (i.e., the Eourier transform spectrum) or in the time domain (i.e., the raw signal). The simplest methods in both approaches attempt to fit model functions to the different resonances in the spectrum. Typically, the models might be Lorentzian or Gaussian lineshapes in the frequency domain or exponentially decaying sinusoids in the time domain. 

Secondly, it has not always been possible to use corresponding functions in the analyses in the two domains. For some of the functions used to represent lineshape components in the frequency domain, such as the Gaussian and Lorentzian, the corresponding FID function is known analytically, and can therefore be used in fitting FIDs. This is not the case for some functions that have been used to represent lineshape components believed to be intermediate in character between a Gaussian and a Lorentzian. This can make comparison of the results problematic. However, since none of these functions has any theoretical basis, there is no reason to prefer functions that are tractable in the frequency domain over those that are tractable in the time domain. 

The effect of 1-D spin dynamics on the ESR line was investigated in various 1-D paramagnetic systems in the 1970s [20-23]. Basically, if the cutoff frequency is less than the linewidth, (Oc < A co, the divergence of f(cu) is felt on the line, which no longer has the Lorentzian shape derived from motional narrowing. The lineshape F(o>) can be analyzed either directly in the frequency domain or in the time domain by considering the total spin time correlation function, G(/) = (S (t)S (O)), with S = Sx-Sx, which is the Fourier transform of F((o) ... 

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