Sunday, 5 January 2014

Reference Deconvolution


In an idealized situation, according to quantum mechanics theory, NMR transitions in liquid state and excluding dynamic effects such as chemical exchange are of Lorentzian shape [1]. In practice NMR lineshapes are never pure Lorentzians due to a number of reasons [2], ranging from magnetic field inhomogeneity and magnetic field noise to sample temperature gradients, sample spinning  or FID weighting, to cite a few. 
Another important property that might affect significantly the final observed lineshape is the following: Even in molecules of modest size the number of distinct peaks might be thousands times smaller than that of quantum transitions. As a simple example, the number of transitions of a molecule containing 15 would be 245760 whereas only a few hundreds of peaks would be observed in the spectrum. As a result, an NMR peak is actually an envelope of a distribution of myriads of Lorentzians and its shape is dominated by the coupling pattern of the spin system. 
Whilst this kind of line broadening affects signals differently across the spectrum and is very difficult (or even impossible) to resolve by post-processing operations, there are many other distortions that affect all resonances in the spectrum in the same way. These include lineshape distortions caused by poorly shimmed samples and they can be removed by using a post-processing technique known as Reference Deconvolution [3] 

Reference Deconvolution

This technique is used to remove the instrumental lineshape distortion by deconvolving the experimental NMR spectrum using a reference signal, usually one within the same spectrum (which should be an isolated singlet) known to be subject to the identical lineshape distortions. Finally, once the lineshape distortion is removed, the spectrum can be reconvoluted with a known lineshape, typically a Lorentzian, so that the result will be a corrected spectrum in which the instrumental distortion has been replaced by the ideal lineshape. 
Actually, the concept of deconvolution is very simple: If S(f) is the experimental frequency domain spectrum, it can be decomposed into two main components, the ideal spectrum I(f) and the instrumental distortion D(f). In other words, the observed experimental spectrum is the result of a contaminated ideal spectrum. Mathematically, this contamination is expressed with the concept of convolution which is represented by the symbol *

S(f) = I(f) * D(f)

The goal here is to find the function D(f) so that the ideal spectrum I(f) can be recovered:

I(f) = S(f) [*]-1 D(f)

Where [*]-1 denotes deconvolution. That is to say that the ideal spectrum can be recovered by means of a deconvolution which consists basically in reversing the effects of the convolution.  

In practice, this process is more efficiently done in the time domain:

I(t) = S(t) / D(t)

This is possibly by considering the Convolution Theorem which states that point-wise multiplication in one domain (i.e. time domain) is equivalent to convolution in the other FT domain  (e.g. frequency domain). 

The complete process of Reference Deconvolution will be illustrated with an example using Mnova NMR and one 300MHz 1H-NMR spectrum in deuterated acetone (kindly provided by Gareth Morris) in which the homogeneity of the static field was deliberately perturbed. The spectrum corresponds to ODBC (ortho-dichlorobenzene) and has been folded several times in order to optimize digitization:

First, after issuing command Process/Reference deconvolution in Mnova, the User needs to select a well resolved reference signal in the frequency domain spectrum. In order to avoid numerical instabilities this signal should be a singlet. The reason is that if the reference signal has some multiplicity (i.e. a doublet), inverse FT of this reference signal (remember that Reference Deconvolution takes place in the time domain) might result in an FID with zeroes at regular intervals. As this time domain signal will be used in the denominator of the reference deconvolution function, this would result in severe discontinuities.  

In this particular example, as in many others, a convenient reference could be the TMS signal (0 ppm), which in principle should be a singlet (disregarding the 13C and 29Si satellites, more about this in a moment), but as it can be noticed, it shows lineshape errors and spinning sidebands due to a combination of poor shimming and sample spinning. 
In the figure below, the result of selecting the reference signal with Mnova is depicted. 

In the Reference Deconvolution dialog box, there are two check boxes, 29Si and 13C satellites and the explanation is this: The TMS reference signal comes with the presence of small 29Si and 13C satellites flanking the central peak at 3.3 and 59 Hz, respectively. Since the reference line is supposed to be representative of all signals in the spectrum, any fine structure which is unique to the reference should generally be removed before deconvolution is performed. The 13C satellites are not usually a concern as they are quite distant from the main TMS peak, but the 29Si satellites are more problematic owing to their close proximity to the central signal. 
So when, for example, the 29Si satellites option is ticket, the software will automatically synthesize the peaks corresponding to the 29Si satellites which in turn will be part of the reference FID model. 


Once the reference region is selected, the software calculates a reference FID, Sr(t), by zeroing all the spectrum except the selected region followed by an inverse FT (there are some additional processing steps required to avoid the negative effects of the long tails of the imaginary components – The interested reader is referred to the [2-3] for further details). 

Having calculated the reference FID, Sr(t), an ideal reference FID Si(t) can be computed by simply simulating an FID using the set of frequencies and amplitudes for the parent signal (e.g. TMS) and any attendant satellites, and decay rate which depends on the target lineshape, a value that can be specified in the reference deconvolution dialog box (in this example a value of 0.35 Hz has been used). With these two reference FIDs, it is possible to calculate the correction function c(t) by simply taking the (complex) ratio of both:

c(t) = Si(t) / Sr(t)

This correction function c(t) simply represents the (inverse of the) instrumental function responsible of the lineshape distortion. Multiplying the original experimental FID, Se(t), by this function yields a corrected FID, Sc(t) which may then Fourier Transformed to yield a corrected spectrum Fc having the lineshape of the specified ideal lineshape:

Sc(t) = c(t) x Se(t)
Fc = FT[Sc(t)]

Overall, the result of applying reference deconvolution to the ODBC spectrum used in this example is shown in the figure below. 


Reference deconvolution is a powerful processing method to remove some distortions that affect all the peaks in a spectrum in the same way. In practice, this is done by extracting the distorted component from a reference signal and deconvolving the whole imperfect spectrum. 
Present implementation of the algorithm in Mnova 9.0 supports 1D spectra only thus far, but extensions to 2D spectra are planned. 


[1] Why are spectral lines Lorentzian? 

[2] Metz, K. R., Lam, M. M., & Webb, A. G. (2000). Reference deconvolution: A simple and effective method for resolution enhancement in nuclear magnetic resonance spectroscopy. Concepts in Magnetic Resonance, 12(1), 21–42. (link

[3] Morris, G. A., Barjat, H., & Home, T. J. (1997). Reference deconvolution methods. Progress in Nuclear Magnetic Resonance Spectroscopy, 31(2), 197–257. 

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