Sensitivity of Nonuniform Sampling NMR

Many information-rich multidimensional experiments in nuclear magnetic resonance spectroscopy can benefit from a signal-to-noise ratio (SNR) enhancement of up to about 2-fold if a decaying signal in an indirect dimension is sampled with nonconsecutive increments, termed nonuniform sampling (NUS). This work provides formal theoretical results and applications to resolve major questions about the scope of the NUS enhancement. First, we introduce the NUS Sensitivity Theorem in which any decreasing sampling density applied to any exponentially decaying signal always results in higher sensitivity (SNR per square root of measurement time) than uniform sampling (US). Several cases will illustrate this theorem and show that even conservative applications of NUS improve sensitivity by useful amounts. Next, we turn to a serious limitation of uniform sampling: the SNR by US decreases for extending evolution times, and thus total experimental times, beyond 1.26<i>T</i>₂ (<i>T</i>₂ = signal decay constant). Thus, SNR and resolution cannot be simultaneously improved by extending US beyond 1.26<i>T</i>₂. We find that NUS can eliminate this constraint, and we introduce the matched NUS SNR Theorem: an exponential sampling density matched to the signal decay always improves the SNR with additional evolution time. Though proved for a specific case, broader classes of NUS densities also improve SNR with evolution time. Applications of these theoretical results are given for a soluble plant natural product and a solid tripeptide (<i>u</i>-¹³C,¹⁵N-MLF). These formal results clearly demonstrate the inadequacies of applying US to decaying signals in indirect <i>n</i>D-NMR dimensions, supporting a broader adoption of NUS