Full-Matrix Analysis of the Error Propagation in Two-Dimensional Chemical-Exchange and Cross-Relaxation Spectroscopy

Propagation of errors in a multispin system, from the experimental peak volumes into magnetization-exchange (cross relaxation or chemical exchange) rates, was calculated analytically. An explicit expression was derived for the error propagator. The error for each magnetization-exchange rate constant has a minimum, with the position dependent on the structure of the exchange network. Error analysis allows optimization of the mixing time for arbitrarily complex exchange systems. In cross-relaxation spectroscopy, the error analysis provides the upper and the lower boundaries of calculated interproton distances for all spin pairs. These limits differ throughout the molecule for the same distance. The differences are a consequence of the spin diffusion, which the present analysis fully takes into account.