Exponential multiplication

Recent reports 54 seem to indicate that the resolution of the notoriously difficult solid-state spectra of coals may be enhanced by such techniques as double exponential multiplication and convolution difference. Differential relaxation behaviour as discussed in connection with intermolecular effects in carbohydrates and low temperature methods may further improve identification. 

Most simply this can be done by multiplying each data point in the FID by an exponential decay term that starts at unity but decays to a negligible value at its end. Such an exponential multiplication (EM) is a simple and effective way to increase the signal-to-noise ratio at the expense of added line broadening (LB). The LB term in this function can be altered... 

Figure 1.36 Various selected apodization window functions (a) an unweighted FID (b) linear apodization (c) increasing exponential multiplication (d) trapezoidal multiplication (e) decreasing exponential multiplication (f) convolution differ-...

The process of exponential multiplication just described produces a rapid decay of the FID and the production of broad lines suppressing the decay of the FID gives narrow lines and better resolution, with increased noise level. An alternative approach to resolution enhancement is to reduce the intensity of the earlier part of the FID. Ideally, we should use a function that reduces the early part of the FID, to give sharper lines, as well as reduces the tail of the FID, to give a better signal-to-noise ratio. 

The tail of the FID contains very little information rather, most of the relevant information is in the initial large-volume portion of the FID envelope. The loss of the tail of the FID should not, therefore, significantly affect the quality of the data. Another manipulation to compensate for the lost information is exponential multiplication, in which the FID is multiplied by a negative exponential factor. 

Apodization (exponential multiplication) is used to improve the signal-to-noise ratio, and it does not affect the chemical shifts of the NMR signals. 

Figure 5 shows the original FID and the result when this is multiplied by mathematical functions either exponential multiplication (EM) or shaped sine bell (SSB, a sine function). 

This has the effect of smoothing the FID away to zero, thus yielding lovely peaks. We call this exponential multiplication for obvious reasons ... 

It is also possible to play other mathematical tricks with the FID. For example, we may want to make our signals appear sharper so we can see small couplings. In this case, we want our FID to continue for longer (an infinite FID has infinitely thin lines when Fourier transformed). To do this we use Gaussian multiplication . This works exactly the same way as exponential multiplication but uses a different mathematical function (Figure 4.2). 

Exponential multiplication The application of a mathematical function to an FID which has the effect of smoothing the peak shape. Signal/noise may be improved at the expense of resolution. 

Fig. 2.15. Signal noise enhancement by exponential multiplication at the cost of resolution sample hexamethylphosphoramide (70%) in tetradeuLeriomethanol (30%) at 30 C and 15.08 MHz coupled ...

FIGURE 1. ID-INADEQUATE spectrum of (SieMeii)2SiMe2 (4) 500 mg silane in 2 ml CgDg, 30 mg Cr(acac)3 added T = 333 K 6400 FIDs accumulated gentle exponential multiplication insert shows the couplings at Si4 in an expanded scale. Reproduced by permission of Elsevier Science from Reference 11... 

Bruker uses the command EM (exponential multiplication) to implement the exponential window function, so a typical processing sequence on the Bruker is EM followed by FT or simply EE (EF = EM + FT). Varian uses the general command wft (weighted Fourier transform) and allows you to set any of a number of weighting functions (lb for exponential multiplication, sb for sine bell, gf for Gaussian function, etc.). Executing wft applies the window function to the FID and then transforms it. 

Processing Exponential multiplication with line broadening of 5-8 Hz gives acceptable line widths for lignin signals... 

Answer The data were Mated with an exponential multiplication [before Fourier transformation to improve the signal-to-noise ratio in the frequency spectrum ( 1.3.4). The line broadening parameter used was 1 Hz, forcing a rapid decay of the FID. This gives a greater. linewidth than is imposed by the field homogeneity and relaxation rates,... 

In 2D spectroscopy the Gaussian function is often preferred over exponential multiplication because little broadening at the base of the resonance is induced by this filter, and significant sensitivity improvement can still be obtained. 

Varian VXR 500 NMR spectrometer operating at a proton frequency of 500 MHz. The spectral width was 6000 Hz with the carrier frequency at the HDO resonance. The solvent resonance was suppressed by presaturation. Each FID was composed of 16 k data points with 80 transients. The delay between successive transients was 2.8 s. The time domain data were processed by zero-filling to 32 k points, exponential multiplication (0.5 Hz) and Fourier transformation. Chemical shifts were referenced to internal sodium 3-(trimethylsilyl)-propionate-2,2,3,3-d4. The Kj values were obtained by nonlinear least square fit of the data to the equation... 

The multiplication here is a good approximation to exponential multiplication (for A = 2) and to Gaussian multiplication (for A 5= 3). For zero shift, the first data point will be nulled and hence dispersive contributions to the FID are eliminated. An advantage of this function over the commonly used real EM or GM is the fact that the sine functions decay to zero, eliminating truncation oscillations. 

Stillinger, F. H., Exponential multiplicity of inherent structures. Phys. Rev. E 59,48 (1999). Stillinger, F. H., Inherent structures enumeration for low-density materials. Phys Rev. E, 63,... 

As suggested above, the noise amplitude in a spectrum can be attenuated by de-emphasising the latter part of the FID, and the most common procedure for achieving this is to multiply the raw data by a decaying exponential function (Fig. 3.34a). This process is therefore also referred to as exponential multiplication. Because this forces the tail of the FID towards zero, it is also suitable for the aptodisation of truncated data sets. However, it also... 

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