A reader asked me: "What is the shape of a 2-D peak?" and my answer was: "Exponential! (in time domain)". This is all you need to know, I think. The picture should help, as always. If you are reading this blog, it means that you have something like a computer. Even the iPhone is a computer (you can run programs on it) and can display my blog. I suspect, however, that you have a better computer, like I do.
You can use your computer to run simulations and more: you can ask it to fit a model spectrum to an experimental spectrum. This is what I have done. The black line is a (fragment of a projection of a) DQF-COSY. The red line is a model generated by the computer. All I said was:
1) the signal has an exponential decay;
2) it's an anti-phase doublet;
3) here is an experimental spectrum and here are its processing parameters;
4) please fit the model to my spectrum!
The goal was a better estimate of the coupling constant. The distance between the two peaks is 7.3 Hz, while the model contains a J = 8.88 Hz. My impression is that the model is more accurate by 1.6 Hz.
The calculation involved is very little (the whole process was faster than the blinking of an eye). The job of programming was more time-consuming. If you don't like spending time, find someone else that can do the job for you. They ask 0.99$ (0.79€) for an application (for the iPhone). Calculating the Js with more accuracy is certainly worth the price.
There is no secret, no new theory, should you need more explanations please ask.
If you want to know more, I can try to explain what was behind the question of the reader. It's not necessary, it's an un-necessary complication, but it's also well-know theory. If you submit to FT a function (e.g.: an exponential decay) you get a different function (in the example, a Lorentzian). They form a "Fourier pair". Being that most 2-D spectra are weighted with a squared cosine-bell my friend probably wanted to know which other function it makes a pair with.
Now you can realize that it is more a theoretical question than a practical one: if the final goal is to simulate an NMR spectrum, it's enough to know that the signal decays exponentially (and everybody knows it).
The program shown by the picture can simulate a multiplet (with 3 different J values). The multiplicities that can be chosen from the menus are:
but it's trivial to extent the menu and handle quintets and so on. Even without further modifications, you can already simulate a quartet of triplets of doublets."R" is the inverse of the transversal relaxation time. This kind of program can't simulate second-order spectra, but nothing prevents you from writing a program that simulates second-order spectra in time domain.
What happens when the user pushes the "Fit" button? The computer runs a Levenberg-Marquardt optimization, based on first derivatives. How have I found the derivatives without knowing the function that describes a peak? You know: life is simpler when you have a computer...