ElPulpo: a qNMR friendly resolution enhancement algorithm

Introduction 

Spectroscopic techniques, particularly NMR, frequently encounter the challenge of overlapping peaks. Thus, the need to enhance the resolution of such spectra has been the subject of extensive research for decades. In NMR, the pursuit of higher resolution began with hardware advancements, notably the development of sophisticated shimming systems. These systems were designed to improve magnetic field homogeneity and stability, as well as the stability of the radio frequency (RF) sources used. However, these improvements are inherently limited by the natural linewidths of the resonance peaks of the samples, which are predominantly dictated by relaxation effects and molecular mobility. The latter is often insufficient to fully average out dipolar interactions and other inhomogeneities. 

With the advent of computers in NMR technology, digitized spectra became the norm, and a persistent drive emerged to enhance resolution post-acquisition through computational algorithms. Over the decades, numerous algorithmic strategies have been proposed, yet the consensus in the scientific community suggests that the potential for further enhancements still exists. 

In this context, we introduce “ElPulpo“, a novel resolution enhancement algorithm specifically designed for 1D NMR spectra. ElPulpo distinguishes itself by its computational agility and its streamlined approach, requiring only one key parameter: the ‘Reference Linewidth’. Notably, a good default value of this parameter can be pre-estimated by means of another algorithm that we have used for years and integrated with ElPulpo.  

Very importantly, ElPulpo was meticulously designed with the following requirements in mind: 

• It respects the intrinsic quantitative nature of NMR spectra, upholding its compatibility with quantitative NMR (qNMR) standards. Regardless of how ElPulpo affects the lineshapes of the spectral peaks, it does not alter their integrals —both absolute and relative—affirming its suitability for qNMR analysis.  
• ElPulpo works in the frequency domain, applying a number of evaluation steps. All these steps are rigorously (mathematically) linear and therefore fully compatible with NMR spectra of mixtures, no matter how complicated.
• ElPulpo keeps under control the propagation of experimental noise. In mode 1 (see below) it effectively attenuates noise by approximately 50%. In modes 2 and 3, the noise propagation factor is around 0.8 and 1.2, respectively. Even with the higher modes that most aggressively reduce the linewidths of spectral peaks, the noise propagation factor is kept below approximately 3.5.
• ElPulpo operates both on the real and imaginary parts of the spectrum and, by its mathematical nature, maintains the Hilbert-transform relationship between them. In practice, this means that it is fully compatible with spectral phasing practices, both manual and automatic. Spectra can be phased or re-phased either before or after the application of ElPulpo, a property that is often very handy. 

This document will not delve into the mathematical intricacies of ElPulpo; instead, it will focus on its practical application, demonstrating its effectiveness through various examples.  

ElPulpo 

ElPulpo is available through the Mnova command Processing/More Processing…/ElPulpo Resolution Enhancement and requires two fundamental parameters: 

Resolution Mode: This parameter balances resolution enhancement against noise effects in the algorithm’s operation. Lower modes (e.g., 1, 2) prioritize signal integrity, improving resolution conservatively while minimizing noise propagation. They are preferred when maintaining a high signal-to-noise ratio (SNR) is important. 

Higher modes (e.g. 4, 5) lean towards more aggressive resolution enhancement, suited for scenarios where resolution is paramount, albeit with decreased noise tolerance. This trade-off allows users to adapt the algorithm’s performance to specific analytical needs. The default value is mode 2. 

The figure below illustrates the impact of employing two distinct modes. On the left side, the original, unprocessed spectrum is displayed (cyan curve), overlaid with the spectrum processed by ElPulpo in mode 2 (red line). On the right side, the same spectrum is presented, this time processed using mode 5. It’s important to note that ElPulpo is engineered to preserve peak integrals; as it narrows line widths for enhanced resolution, it correspondingly increases peak heights to maintain integral constancy.  

Reference Line Width: This parameter is pivotal for the algorithm’s performance. The ideal value for the Reference Line Width should match the actual line width of the peaks in the spectrum. However, given the variability in peak widths across a spectrum, the algorithm requires a compromise value. The software facilitates this by offering an automatic setting for this parameter. It calculates the average line width of all peaks in the spectrum and then adjusts this average by a factor, currently set to 0.8.  

It is important to understand the behavior of the algorithm in response to adjustments in the Reference Line Width (refLW): When refLW is small, its influence on the peak diminishes, tending towards zero, minimally affecting the peak. Conversely, increasing refLW initially leads to peak narrowing, but exceeding a certain point introduces negative side lobes. 

Optimal peak refinement occurs when refLW matches the original line width (originalLW) of the peaks, avoiding negative lobes. If refLW is larger than originalLW, further narrowing of the central peak is accompanied by the emergence of negative lobes. On the other hand, when refLW is less than originalLW, narrowing diminishes and no lobes are formed. 

The figure below demonstrates the influence of different Reference Line Width settings in ElPulpo processing. Again, the cyan curve depicts the original spectrum, while the red curve represents the spectrum processed by ElPulpo. On the left, a Reference Line Width of 0.94, approximating the actual linewidth of the spectrum, is used. On the right, a slightly larger Reference Line Width of 1.14 Hz is applied. This adjustment leads to some excursions of the peaks into the negative domain, a direct consequence of the increased line width parameter. 

Why ‘El Pulpo’? The Story Behind the Name: 

The reason why we have called the algorithm ElPulpo is that we imagine it as an octopus (but one with 13 tentacles) that crawls above the spectrum and, at each point, spreads its tentacles far enough (compared to the reference linewidths) to 13 different locations in the spectrum. It friendly ‘palpates’ the spectrum around those points and, using a separate multi-point interpolation algorithm, assesses the value of the complex spectrum at those pre-selected complex points. After that, using a composite Savitsky-Golay convolution filter, it uses the 13 values to extract a desired, theoretically determined combination of the original spectrum (derivative 0) with its higher derivatives (from 0 to M, where M is the ‘mode’ number), and then uses the computed value to replace the one of the point at which it has started  

The spreading of the octopus tentacles before getting a feel of the spectrum is essential to make the method (and especially its noise propagation factor) little sensitive to spectrum digitization that may range from quite poor (common for narrow peaks in NMR spectroscopy) to very dense, even magnitudes higher than necessary.