Improving the Resolution of MPM Recovered Relaxometry Parameters with Proper Time Domain Sampling

The matrix pencil method (MPM) is a powerful tool for processing transient nuclear magnetic resonance (NMR) relaxation signals with promising applications to increasingly complex problems. In the absence of signal noise, the eigenvalues recovered from an MPM treatment of transient relaxometry data reduce to relaxation coefficients that can be used to calculate relaxation time constants for known sampling time ∆t. The MPM eigenvalue and relaxation coefficient equality as well as the resolution of similar eigenvalues and thus relaxation coefficients degrade in the presence of signal noise. The relaxation coefficient ∆t dependence suggests one way to improve MPM resolution by choosing ∆t values such that the differences between all the relaxation coefficient values are maximized. This work develops mathematical machinery to estimate the best ∆t value for sampling damped, transient relaxation signals such that MPM data analysis recovers a maximum number of time constants and amplitudes given inherent signal noise. Analytical and numerical reduced dimension MPM is explained and used to compare computer-generated data with and without added noise as well as treat real measured signals. Finally, the understanding gleaned from this effort is used to predict the best data sampling time to use for non-discrete, distributions of relaxation variables.