# Weighting in 1D Spectroscopy

The appearance of a 1D spectrum can often be enhanced by proper weighting. iNMR introduces a simplified and effective way for achieving this result, which will certainly satisfy both the expert and the beginner. Follow these steps:

- Fourier transform and phase correct the spectrum.
- Choose the command: ‘Process/Visual Weighting’. The corresponding dialog will appear.
- Specify what you are aiming for: Sensitivity (positive line broadening) or Resolution? (negative line broadening).
- Move slowly the upper thumb (exponential). You will see the spectrum change.
- Continue moving the thumb to the right and back to the left. Observe the differences and choose the point where the spectrum looks better for your purposes.
- You can stop here and dismiss the dialog, or you can try further optimization with the second slider (gaussian); (the gaussian always increase sensitivity at the expense of resolution; the sensitivity/resolution switch has no effect on the gaussian slider).
- You can continue to change alternatively the two weights until optimization is complete.

This is just the easiest way to apply weighting functions. There is another, faster solution, which you would prefer only if you already know in advance the coefficients of the weight to apply. At the bottom of the FT dialog you can select the functions and specify the values. The interactive weighting lets you visually follow what happens, so you don't need to know the numerical values to apply. Just act on the slider and stop when the spectrum looks good.

# Let's Get Lost

The exponential-gaussian transformation is a simple graphic process that does not require any mathematical background. What you have, at any stage of the transformation, is still your authentic NMR spectrum. The optimal result is the one that looks the best to your eyes. If you really want to know what's happening, here is the theory.

When you multiply two functions in time domain, after FT you have the “convolution” of their respective FT. The convolution has simultaneously the shapes of the two ancestor functions. In your case the ancestor functions are the FID and the weighting function, while the convolution corresponds to the weighted time domain spectrum.
The line-width of the convoluted signal is the sum of the natural linewidth and the linewidth specific of the weighting function. This is the reason why iNMR asks the user for the linewidth **W**, shows only W and never shows any other parameter. The reader can, however, use the following formulae to calculate the hidden parameters λ and σ. In the following ω is the angular frequency ( ω = 2 π ν ) and W is the full linewidth at half height, measured in Hz.

### exponential function

f(t) = exp( - λ t ).

F(ω) = λ / (λ^{2} + ω^{2}).

λ = π W.

### gaussian function

f(t) = exp( - σ^{2} t^{2} / 2 ).

F(ω) = √(2 π) exp[-ω^{2} / (2 σ^{2} )] / σ.

σ = π W / √ log_{e}2 = 1.2 π W.

# Summary

When the user chooses “sensitivity” the sign of the exponential is negative and the line broadening is positive. When the user chooses “resolution” the sign of the exponential is positive and the line broadening is negative. The gaussian coefficient is squared, therefore it makes no difference if it is positive or negative.