# Non-uniform sampling (NUS) in 2D-NMR

The NMR technicians where I work have recently started replacing the standard 2D NMR experiments (COSY, HSQC) with NUS variants (25% sampling). For the uninitiated, the rationale behind the technique is summarised by Claridge (who does a better job of explaining than I would):

The classical sampling of 2D (and more generally multi-dimensional) NMR experiments requires the uniform sampling of data in the indirect dimension(s) that allows for the processing of the data by the discrete Fourier transform. This means a sequential, stepwise increment of the t1 period of a 2D data set is made to the limit t1 which dictates the resolution in this dimension. The number of such t1 increments employed ultimately defines the total duration of the experiment. The method of non-uniform sampling (NUS) seeks to reduce the number of data points collected in the indirect dimension(s) and so reduces the total experiment time.

Claridge, T.D.W. High-Resolution NMR Techniques in Organic Chemistry, 2016

My experience of these experiments has been overwhelmingly positive so far: shorter experiment times and greater resolution, even when acquiring spectra of small quantities (0.1 - 0.5 mg) of large molecules (ca. 1000 Da).

The challenge is knowing what to do when the spectrum doesn't come out looking quite right. With 'traditional' experiments, running more scans often fixes a multitude of sins (up to a point), but with random point sampling I can't see this holding up so well.

The processing of data is also causing some confusion - TopSpin and MNova give (often very) different results, despite both having NUS processing capability. This processing issue leads me to suspect that many of the issues I'm having with spectra from NUS pulse sequences are actually due to the processing (which isn't terribly well covered in the TopSpin manual).

My question, therefore, is what considerations should be taken into account with NUS spectra: are there instances where its better to take the hit and run a long HSQC? What do I need to be careful of when processing data to ensure spectra come out looking okay?

• I never used NUS myself, but from what I heard the "CS" mode for processing in Topspin gives much better results than the "MDD" mode, but also takes longer. – Mad Scientist Jul 21 '18 at 14:10
• If you suspect the NUS itself is the problem (rather than the processing), one can usually control the percentage of points sampled during NUS (e.g. collect 50% of the points). If the problems disappear as you increase the number of points collected (eventually reaching 100% or uniform sampling) then you have your answer. – Bryan Hanson Jul 22 '18 at 0:27

I think a really full answer will require delving into the theory of NUS, which (just like the Fourier transform) goes way beyond NMR as a field: the idea of undersampling a signal and then reconstructing it is of great interest in a variety of engineering areas. I can't give that level of detail, so I don't pretend that this is really an answer, but maybe having some of this written down will help.

I'm really not an expert in this, but I can backup Mad Scientist's comment that for 2D data, TopSpin's reconstruction mode (the MDD_mod processing parameter) generally should be set to cs and not mdd. I suspect this may require a license from Bruker, so it might be best to check with the relevant NMR manager. Mnova, on the other hand, uses a different reconstruction algorithm entirely, so it's not overly surprising that there might be some differences in performance.

The amount of undersampling you want to do will also affect the quality of your results. I think a good rule of thumb is to make sure the ratio of points measured in $$t_1$$ to number of peaks in each column of the spectrum is somewhere around 2 to 5. The reasoning is simple: if you need to detect $$k$$ peaks in the frequency domain, then you need to measure some multiple of $$k$$ signals in the time domain. In other words, the more data you want to pick up, the more signals you need to measure. For a HSQC where each column has very few frequencies, you can get away with very sparse sampling, and 25% is likely to be fine (as long as your TD1 is not too small). For other spectra like COSY, TOCSY, HMBC where there is less empty space in the spectrum, this may not work quite as well.

As far as I'm aware, increasing the number of scans should have a similar effect for both uniform and non-uniform sampling. The main changes are the increase in S/N, as well as possibly better artefact suppression achieved via phase cycling. However, with modern gradient-selected experiments (especially the standard experiments like HSQC, COSY, etc.) you should not really need extremely long phase cycles to achieve good artefact suppression. In my experience the difference in spectral quality between NS=2 and greater is already pretty minimal. Bumping up the number of scans is mainly useful for increasing S/N. [Obviously, if the reason why the NUS reconstruction is failing is because of too poor S/N, then increasing the number of scans will help.]

Lastly, this is a trivial point, but I'd also suggest double-checking the other processing parameters (window functions, zero-filling etc.) after the reconstruction is done. It's quite easy to miss something out (for example, changing QSINE to SINE) and inadvertently get ugly peak shapes. I learnt (and am still learning) it the hard way.

Having said all this, what should you do if you find your NUS data is not good enough? I don't think there are really many answers to this, although someone more knowledgeable will be able to make better suggestions. One is to play around with the reconstruction algorithm, but unless you are very well-versed with the theory, this is essentially limited to trial-and-error with a dropdown box. The other is to just reacquire the spectrum with less undersampling, especially if you suspect you may have pushed it a bit too much.

• I am curious as to why non-uniform sampling rate enhance would resolution? Isn't the basic purpose of any sampling process to represent a continuous signal as accurately as possible? I can understand the logic the NUS can save time but resolution enhancement is something interesting. I didn't study the theory of NUS. – M. Farooq Jan 20 at 3:24
• @M.Farooq It's not explicit, but (at least in the field of NMR) when people talk about using NUS to gain resolution in nD NMR ($n \geq 2$), they usually mean given the same experimental time. Thus, while under uniform sampling one may measure 32 increments in one spectral dimension, NUS allows one to measure 32 points in the same amount of time and then reconstruct 128 points from them, for example. [Of course, one could always try to reconstruct 128 points from the uniformly sampled 32 points to get the higher resolution; so actually it's a more subtle question.... – orthocresol Jan 25 at 20:02
• ... of whether the reconstruction of undersampled NUS data, is better than the reconstruction of a smaller amount of uniformly sampled data. That's right about where I'll draw the line and say I don't know enough to make a conclusion, although the NUS people will always tell you yes, it is better :-) and I am inclined to believe them.] – orthocresol Jan 25 at 20:04