If I Had a Bell

Weighting functions are so easy in practice and so (apparently) difficult in theory, that I wonder if we can just ignore the latter and pretend it doesn't exist. It's a matter of language. Instead of moving towards simplification, we keep creating more names, more languages, more ways to measure things. If you have learned NMR from the literature, you'll discover a different language in software, and can't find the vocabulary for the translation. The same happens when you want to write your own article: how you describe your NMR processing? It's the inverse translation. Last but not least, different softwares have different names and units for the weighting functions. Don't be mislead by the names, what matters is the shape of the window function. Ideally, it should be reproduced into your manual. Given that the shape often depends on a user-settable parameter, they can't illustrate all the possible cases. You can, however, ask the program to show the plot of your weighting function and/or the weighted FID. This is the best vocabulary. If the function grows (moving from left to right), then it can enhance the resolution.
Apart from this effect of visual translation, it's neither necessary nor advisable to observe the effect of the weighting function on the FID, when you can directly observe the effect on the spectrum in frequency domain. The FID is dominated by a few intense signals, often coming from the solvents, that are of little importance to you. In frequency domain you can monitor the effect of weighting on the important signals only.
Remember that weighting means selective attenuation, but also alteration and degradation, which you must be ready to accept: it's a part of the game. It improves the appearance of some signals and not of others. I go to the frequency domain, zoom into the region that I want to improve, and monitor the effect of varying the parameters (just like phase correction).
In some cases there's no parameter to adjust. Typical is the case of the squared cosine bell. It has a shape similar to that of a gaussian bell and it means that you could equivalent similar results using either. It's easy to calculate which value you need for the sigma, in case you choose the gaussian function, but it's even easier to use a cosine bell, with no parameter at all to set!
blue: weighted with a squared cosine bell
red: weighted with a gaussian bell (line broadening = 0.93324 / acquisition time).
The example contains four lessons:

1. non branded weighting functions can be better than the branded ones
2. anybody can be creative with them
3. there's no optimal weight and, even when it exists, a small variation corresponds to another weight that's almost as good, for all practical purposes.
4. If your software provides a limited number of functions, it doesn't mean that your possibilities are limited.