Introduction to Covariance NMR

Resolution and sensitivity are two key factors in NMR spectroscopy. In the case of 2D NMR, the resolution of the direct dimension (f2) depends, among other things, on the number of acquired complex points, whilst the resolution of the indirect dimension (f1) is directly proportional to the number of increments (or number of acquired FIDs). In general, it could be said that the resolution along the direct dimension comes for free in the sense that increasing the number of data points does not augment the acquisition time of the experiment significantly. However, increasing the number of t1 data points (increments) has a direct impact in the length of the experiment as can be seen from the Total acquisition time for a 2D NMR spectrum:

T = n*N1*Tav

Where n is the number of scans per t1 increment, N1 is the number of T1 increments and Tav is the average length of one scan. This usually means the resolution of the indirect dimension f1 is kept lower than that of f2.

For example, let’s consider a COSY spectrum (magnitude mode) of Strychnine which has been acquired with 1024 data points along the direct dimension and with 128 t1 increments. It’s clearly appreciated that resolution along F2 is much higher than along F1. For example, doublets corresponding to protons H20a and H23 are resolved in F2 but not in F1.

How could this be improved? We could try to extrapolate the FID (somehow) along the columns (F1) to a higher number of points (e.g. 1024). A well known and very simple technique is simply to add zeros, a process called zero-filling, which basically is equivalent to a kind of interpolation in the frequency domain. For example, in this particular case we could try to extrapolate the FID (along t1) from 128 to 1024 points in order to match the number of points along f2. The figure below shows the results:



It can be observed now that the resolution along f1 is slightly higher than in the previous case but we still cannot distinguish the inner structure of the multiplets (e.g. H20a and H23). Zero Filling is certainly a good technique to improve resolution but, of course, it cannot invent new information from where it does not exist. In this case we have zero filled from 128 to 1024 data points (e.g. 4 fold). In theory, zero filling by at least a factor of two is highly recommended because it enforces causality but beyond that, the gain in resolution is purely cosmetic. We might get more data points per hertz, but no new information is achieved as it’s shown in the figure above.

Is there a better way to extrapolate the FID? Yes, and the answer is very evident: Linear Prediction. In a few words, Forward Linear Prediction uses the information contained in the acquired FID to predict new data points so that we are artificially extending the FID in a more natural way than with Zero Filling. Of course, we cannot create new information with this process but the resulting spectrum will look better. This is illustrated in the next figure:



In this case, we have extended the t1-FID from 128 to 512 data points and then zero-filled up to 1024. Now we can see that the f1-lines are narrower but the couplings cannot be resolved yet.

In recent years there has been great interest in the development of new methods for the time-efficient processing of 2D (nD in general) NMR data. One of the more exciting methods is the so-called Covariance NMR, a technique developed by Brüschweiler at al. In fact, there are several types of Covariance NMR: Direct, Indirect Covariance NMR (there is a third method, Unsymmetrical Indirect Covariance which can be considered as a subtype of Indirect Covariance NMR). In this post I will cover only the first type, Direct Covariance NMR leaving, the other 2 types for future posts.

Before going any further with Covariance NMR, let’s see the results of applying this technique to the same spectrum. This is what we get:

At first glance, this result looks like a kind of magic: Now the splitting corresponding to H20a and H23 are clearly resolved in both dimensions. Actually, the resolution along F1 is virtually analogous to that of the F2 dimension. How did we arrive to such a good result? From an intuitive stand point, what we have attained here was a transfer of the resolution of F2 to the F1 dimension. In other words, we have applied a mathematical process which takes advantage of the higher spectral resolution in the F2 dimension to transfer it to the F1 one.

Mathematically, direct Covariance NMR is extremely straightforward as defined by the following equation:

C = (F'F) (1)

Where C is the symmetric covariance matrix, F is the real part of the regular 2D FT spectrum and F' its transpose (NOTE: direct covariance NMR can also be applied in the mixed frequency-time domain, i.e., when the spectrum has been transformed along F2. In this case, a second FT will not be required – nor apodization nor phase correction along the indirect dimension).

In order to approximate the intensities of the covariance spectrum to those of the idealized 2D FT spectrum, the square root of C should be taken. Root squaring may also suppress false correlations which may be present in F'F due to resonance overlaps. Taking the square root of C matrix is in practice done using standard linear algebra methods (in short, diagonalizing the matrix and then reconstructing C^(1/2) using eigenvectors and the square roots of eigenvalues).

So direct Covariance NMR allows us to produce a 2D spectrum in which the resolution in both dimensions is determined by the resolution of the spectrum along the direct dimension. I have created a stacked plot representation with the 1D projections obtained from the different processing methods described in this post (Zero-Filling, Linear Prediction and Direct Covariance NMR). I think that this plot shows the power of the Covariance NMR method.



There is nothing magic about Covariance NMR. It’s a process with a strong mathematical background based on the well-known Parserval’s theorem. An elaborate discussion of this method is beyond the scope of this post (at least for the time being), but it has been nicely described in the following articles by Brüschweiler et al.

[1] R. Bruschweiler and F. Zhang, J. Chem. Phys., 120, 5253 (2004)
[2] F. Zhang and R. Bruschweiler, Chem Phys Chem, 5, 794 (2004)
[3] R Bruschweiler, J. Chem Phys., 121, 409 (2004).
[4] N. Trbovic, S. Smirnov, F. Zhang, and R. Bruschweiler, J. Magn Reson, 171, 277 (2004).
[5] F. Zhang, N. Trbovic, J Wang, and R. Bruschweiler, J. Magn. Reson., 174, 219, 2005).
[6] Y CHen, F. Zhang, W. Bermel and R Bruschweiler, JACS, 128, 15564 (2006)
[7] F Zhang and R Bruschweiler, Angew Chem., Intl. Ed., Engl., 46, 2639 (2007)
[8] Y. Chen, F. Zhang, D. Snyder, Z Gan, L. Bruschweiler-Li, and R. Bruschweiler, J. Biomol. NMR, 38, 73 (2007)

If you still feel skeptical about Covariance NMR, just try it out. We have recently implemented this technique in Mnova. This version is still in alpha stage but if you want to try it, just send me an email (my email is carlos followed by ‘at’ and mestrelab.com) and I will give you a link to download this alpha version of Mnova.

BTW, I’m proud to say that MestReC was the first NMR software package (apart from the in-house algorithms developed by Brüschweiler group) which offered Covariance NMR processing (here is the proof). Now we have ported and greatly enhanced MestReC old implementation in Mnova. Now Covariance NMR is not only much faster than it was in MestReC; it’s also easier to use and more versatile. I hope you will enjoy it!.

Before I finish, I’d like to acknowledge Dr. Fengli Zhang for his support while implementing Covariance NMR into MestReC.

MestReJ: A free tool for the prediction of vicinal proton-proton 3J(HH) coupling constants

Scalar coupling constants are sensitive to the geometrical features of a molecule and therefore, their magnitude provides a direct insight into the geometry and electronic structure of a molecule. For example, the Karplus equation [J. Chem. Phys., 30, 11 (1959), J. Am. Chem. Soc., 85, 2870 (1963)] describes the relationship between the 3J coupling constant and the dihedral angle between vicinal hydrogens.

After the pioneering work of Karplus, several other generalized Karplus equations have been proposed for the mutual dependence of J and the dihedral angle. Among these, Haasnoot-de Leeuw-Altona (HLA) are by far the most widely used. Applications including other generalized Karplus equations are scarce which hinder their general use for the common organic chemist. Such is the case of the more recent and precise Díez-Altona-Donders (DAD) equations, developed by Altona’s group.

A few years ago, we developed an easy to use J pocket calculator MestReJ which you can now download directly from the link below and use for free with no strings attached.

Download MestReJ at the Mestrelab Research Chemistry Software Product Page

MestReJ is a very easy little application to work with: it uses a Newman projection of the fragment under observation and a plot of the J values against the torsion angle HCC’H’. It implements the two kinds of generalized Karplus equations developed by the Altona’s group: the classical Haasnoot-de Leeuw-Altona equations and the more recent and precise Díez-Altona-Donders equations. The Colucci-Jungk-Gandour, the Barfield-Smith and the Karplus equations are also implemented in the program. For further information, see this article:

A Graphical Tool for the Prediction of Vicinal Proton-Proton ³J_HH Coupling Constants

Navarro-Vazquez, A.; Cobas, J. C.; Sardina, F. J.; Casanueva, J.; Diez, E.
J. Chem. Inf. Comput. Sci.; 2004; 44(5); 1680-1685. DOI: 10.1021/ci049913t

I hope you will find this application useful in your research

A very graphical display of the value of promotional literature

Is promotional literature a valuable investment for a company like ours and an appreciated resource by our customers, or just a way of killing trees for no gain?. In the photo below you can see the take of some conference attendees on this question, which is taking Mestrelab to reevaluate whether we should be making this effort to produce the literature

BayDOSY: What’s under the Hood

I have already posted about our efforts towards the evaluation of diffusion NMR experiments by means of a new Bayesian-based approach, the so-called bayDOSY.
Recently Stan Sykora has given a talk at GIDRM conference covering some background of this technique for NMR DOSY analysis, including its basic math principles and advantages over other techniques. Also, the limitations of this new data evaluation scheme are noted, as well as potential extensions designed to address some of these limitations.
This talk is available in PDF format from the web page below:

http://www.ebyte.it/stan/Talk_GIDRM_2008.html

As I have already offered, should you be interested in testing bayDOSY, just drop me a line and I will give you a special version of Mnova

2D Phase Correction

‘Understanding NMR Spectroscopy’ by James Keeler is one of my favorites NMR books which I highly recommend to anyone seriously interested in NMR spectroscopy. All NMR concepts, ranging from quantum mechanics to product operator formalism and data processing are very elegantly and clearly exposed.
These days I’m working on several points regarding 2D phase correction and while consulting this book, I found a phrase which immediately caught my attention. In page 240, you can read this:

(…) However, phasing a two-dimensional spectrum is not quite so straightforward as phasing in one dimension, as it is not feasible to recompute the whole spectrum after each trial phase correction is applied – to do so would simply be too time consuming. (…)

Indeed this is true, to the best of my knowledge, with most existing NMR software packages but not with Mnova or iNMR, in which the recommended way to iteratively adjust the phase of a 2D spectrum is by real time operations (drag & drop) on the full 2D hypercomplex matrix, in just the same way as you do it with 1D spectra. In fact, I believe that this way is much more convenient than the traditional, old fashioned way of extracting some selected traces (cross-sections) parallel to one of the dimensions, calculating the phase on them, and then applying the correction to the full 2D spectrum.
Obviously, real time 2D phase correction was not possible at early times of 2D NMR development (eighties), when computers were not fast enough to make this task possible, so that the 1D cross-sections approach was the only feasible way. However, since the late nineties, advances in computer technology made real time processing of 2D NMR possible, not only regarding phase correction but any other iterative process such as weighting. For example, it is now possible to iteratively adjust the weighting functions in a 2D spectrum and see, in real time, the results in the processed full 2D spectrum.

SPECTRa

During the flight to Stockholm (where I’m having a great time btw), I read with great interest this article:

SPECTRa: The Deposition and Validation of Primary Chemistry Research Data in Digital Repositories
DOI: 10.1021/ci7004737

In my opinion, the project presented in this article is a great initiative which I can only applaud and support. There is, however, a point which I would like to comment on because it’s not fully clear to me which regards to the way in which NMR spectra are stored in the repository.

The authors have decided to use JCAMP as the format for file input to their repositories. They do not specify which actual data is being saved in these JCAMP files, the processed spectrum or the FID (or both). I hope that they are saving the original FID and not only the processed spectrum, otherwise data preservation will be broken. I think this is a very important point which deserves some further clarification. This is how I see it:

The most important piece of information in an NMR experiment is, for sure, the acquired FID, not the processed spectrum. A chemist could have processed an FID to produce a spectrum in such a way that some spectral features are lost. For example, he/she could have applied a very large line broadening function which will make the analysis of the finer structure of some multiplets impossible. If the original FID has not been kept, the option to re-process the spectrum to calculate those lost couplings will not be viable (in some cases, Inverse Fourier Transform and/or some resolution enhancements procedures could help, but only in a very limited extent). In fact, there are many processing operations which could alter, irreversibly, either the qualitative or quantitative information present in the NMR experiment.

In short, I’m strongly convinced that any system aimed at preserving all information contained in NMR experiments should keep the original acquired data points, the FID. This is something I have learnt during the many years of development of both MestReC and Mnova: Mnova keeps, in addition to the processed spectrum, all the original files as they were acquired in the spectrometer. And I know that iNMR does the same thing, though in a different way (Mnova packs all the files within a single binary file, whereas iNMR keeps all the files separately and a processing log file so that the processed spectrum does not need to be saved. From my point of view, both approaches are equivalent and perfectly valid).

Bayesian DOSY: a New Approach to Diffusion Data Processing

Yesterday I blogged about basic concepts on DOSY NMR. From an experimental stand point, the pulse sequences are not very complex, being the Pulse Field Gradient Stimulated Echo (PFGSE) experiment proposed by Tanner one of the basic pulse sequences used to determine diffusion by NMR. This pulse sequence can be considered as a building block for a number of extended pulse sequences designed to minimize some sources of artefacts caused by thermal convection currents in the sample, background gradients, radiation damping, zero order coherences in strongly coupled spin systems, etc. Other effects such as J-modulation and Cross-Relaxation have also been considered.

The tricky part comes in the mathematical DOSY transformation. As I mentioned earlier, there exists different approaches, each of them with their own strengths and weaknesses. Today I would like to introduce a new method for DOSY transformation based on Bayesian Theory which has been implemented in our software Mnova as a result of our collaboration with Stan Sykora. We call this new algorithm BDT (Bayesian DOSY Transform)

A formal description of the Bayesian theory and its particular implementation for DOSY transformation is beyond this blog entry, but we are currently writing an article which will give all the details of the method as implemented in Mnova.
In short, this Bayesian approach assigns an a-priori probability (this is the key word in Bayesian context) to the elements in a space defined by the entities to be estimated. In this context, an entity is the pair (f,d) where f is frequency and d is a mono-diffusion coefficient. Actually, it should be (f,d,w) where w is a weight (intensity), but this weight is factored out by finding its optimal value (since the dependence is linear, this can be done explicitly). So we are actually finding the a-posteriori probability for the sentence there is a component - no matter how intense - at f that has such and such d. Next, a final normalization process modifies the probability still further (composite probability) by looking at the intensity in the spectrum at location f (this would be real probability if the spectrum were normalized to 1).

Stan Sykora will be presenting the mathematical background and physical insights necessary to understand this new method as well as some real life examples processed with Mnova in the GIDRM conference to be held in Bressanone (Brixen) next September.

So how does the algorithm perform? We first tested the algorithm by simulating with Matlab a diffusion experiment with 2 synthetic peaks with frequencies of 100 and 200 Hz and diffusion coefficients of 1 and 0.01 (dimensionless).


Application of BDT yields the following DOSY spectrum:



The synthetic DOSY spectrum was changed in order to include artificial noise and different degrees of peak overlap and diffusion distances. We’ve found the performance of the algorithm to be excellent in all the cases analyzed.

Next we implemented the algorithm in Mnova and we applied it with a sample of an aqueous solution of potassium N-methyl-N-oleoltaurate (a surfactant) with TSP at 23 C (The original Varian FID file has been obtained from the VARIAN NMR USER GROUP LIBRARY which was submitted by Brian Antalek as a sample for this DECRA algorithm). This is the original raw data after automatic Fourier Transform, phase and baseline correction in Mnova:


After applying BDT, we obtain a DOSY spectrum in which the three components are clearly well resolved in the diffusion dimension and the absolute values are in accordance with the expected values


Below you can see the DOSY spectrum of a mixture of Caffeine + 2-EthoxyEthanol + Water acquired in a Bruker instrument by my friend Andy Soper.

Main properties of the algorithm

Compared to other approaches, we believe that this Bayesian method implemented in Mnova appears extremely promising. It automatically avoids having exact, unnatural zeros anywhere in the resulting DOSY map since every point of the 2D map has a well defined value of statistical congruence with the data. Moreover, the BDT maps show ‘normal’ line widths in the f-direction, correctly positioned and resolved peaks in the d-direction and quantitatively correct horizontal and vertical projections – a combination difficult to achieve by any other means.
The approach can be easily extended to non-exponential cases arising from overlapping lines and, like all Bayesian methods, incorporate additional information available from other sources (a-priori knowledge). Likewise, it is possible to place a statistical premium on alignment of spectral peaks along horizontal lines in the [f,d] plot.

Do you want to try it out yourself?

The algorithm is readily available in the alpha stage in Mnova. From this post, I would like to offer this special version to anyone interested in trying the BDT algorithm with his own data sets. I will very much appreciate any feedback from you.
Just write to me at the email address below and I will give you the instructions on how to get the software.

DOSY NMR

NMR diffusion experiments provide a way to separate the different compounds in a mixture based on the differing translation diffusion coefficients (and therefore differences in the size and shape of the molecule, as well as physical properties of the surrounding environment such as viscosity, temperature, etc) of each chemical species in solution. In a certain way, it can be regarded as a special chromatographic method for physical component separation, but unlike those techniques, it does not require any particular sample preparation or chromatographic method optimization and maintains the innate chemical environment of the sample during analysis.

The measurement of diffusion is carried out by observing the attenuation of the NMR signals during a pulsed field gradient experiment. The degree of attenuation is a function of the magnetic gradient pulse amplitude (G) and occurs at a rate proportional to the diffusion coefficient (D) of the molecule. Assuming that a line at a given (fixed) chemical shift f belongs to a single sample component A with a diffusion constant D_A, we have

(1) S(f,z) = S_A(f) exp(-D_AZ)

where SA(f) is the spectral intensity of component A in zero gradient (‘normal’ spectrum of A), DA is its diffusion coefficient and Z encodes de different gradient amplitudes used in the experiment.

Depending on the type of experiment, there are various formulae for Z in terms of the amplitude G of the applied gradient and one or more timing parameters such as Δ (time between two pulse gradients, related to echo time) and δ (gradient pulse width). In the original Tanner-Stejskal method using two rectangular gradient pulses, for example,

(2) Z =γ²G²δ²(Δ-δ/3)

Eq. (2) holds strictly for simple PFG-NMR experiments, and is modified slightly to accommodate more complicated pulse sequences.

In practice, a series of NMR diffusion spectra are acquired as a function of the gradient strength. For example, the figure below shows the results of a series of 1H-NMR diffusion experiments for a mixture containing caffeine, 2-Ethoxyethanol and water.



It can be observed that the intensities of the resonances follow an exponential decay. The slope of this decay is proportional to the diffusion coefficient according to equation (1). All signals corresponding to the same molecular species will decay at the same rate. For example, peaks corresponding to water decay faster than the peaks of caffeine and 2-Ethoxyethanol

The DOSY transformation

As far as data processing of raw PFG-NMR spectra is concerned, the goal is to transform the NxM data matrix S into an NxR matrix (2D DOSY spectrum) as follows:



The horizontal axis of the DOSY map D is identical to that of S and encodes the chemical shift of the nucleus observed (general 1H). The vertical dimension, however, encodes the diffusion constant D. This is termed Diffusion Ordered Spectroscopy (DOSY) NMR. In the ideal case of non-overlapping component lines and no chemical exchange, the 2D peaks align themselves along horizontal lines, each corresponding to one sample component (molecule).

The horizontal cut along such a line should show that component’s ‘normal’ spectrum. Vertical cuts show the diffusion peaks at positions defining the corresponding diffusion constants.

The mapping S=>D will be henceforth called the DOSY transformation. This transformation is, unfortunately, far from straightforward. Practical implementations include mono and biexponential fitting, Maximum Entropy, and multivariate methods such as DECRA.

Recently, Mathias Nilsson and Gareth A. Morris have proposed the so-called ‘Speedy Component Resolution’ (Anal. Chem., 2008, 80, 3777–3782) as an improved variation of the Component Resolved (CORE) method (J. Phys. Chem, 1996, 100, 8180). This is a multivariate-based method and the examples used in the article show an excellent performance of the algorithm.

Following a different approach, we have recently developed a brand new method for DOSY processing which has been included in Mnova. I will leave the details (and how to get the program to try it out) for the next post. In the meantime, should you be interested, just drop me a line. 

Sensitivity Enhancement for free?

In my previous post I discussed about NMR FID/spectra having two channels, the Real and Imaginary parts and that in general, only the Real part is displayed. In fact, we could use the term Stereo-FID in the same fashion as we use Stereo-Sound (remember that NMR spectra span audio frequencies).
If we think about these stereo signals as coming from two receiver coils in quadrature and assuming simultaneous detection, we could formulate the following scenario:

It’s a common practise to acquire several (hundred or thousand) FIDs which are then added together (signal averaging). Because NMR responses build up in proportion to the number of signals recorded, N, whilst the noise varies randomly from one measurement to the next and thus, adds up more slowly, as sqrt(N), there is an overall improvement in sensitivity of sqrt(N).
If the noise in the Real and Imaginary channels were totally independent, it should be possible to add to the Real channel the Imaginary channel (after a 90º phase shift to make it phase coherent with regard to the Real part) so that we could achieve a further improvement of sensitive of sqrt(2)!. Would this be possible? This will be the subject of this post.

Of course, this ‘trick’ would only work if the noise in the two channels were totally independent. In the NMR field it is assumed that the experimental noise is stationary and white (i.e. the correlation between two consecutive points is zero). But what about the correlation between the noise in the two different channels? Are they correlated? This is very easy to analyze experimentally with Mnova by writing a very simple script which calculates the Correlation Coefficient (Pearson). For example, consider the spectrum below. We could use this script to calculate the Pearson Correlation Coefficient using the points between 5000 & 10000 which is a signal free region


As shown in the figure below, the correlation coefficient between the noise in both channels is almost zero. You can repeat this operation with different spectra and you will arrive to similar results

So it appears as if the noise in both channels is statistically uncorrelated, something which should be intuitively expected as the 2 acquisition channels are orthogonal. Does this mean that the noise in both channels is independent?
We can make a very simple experiment: We could phase a spectrum in order to make the real part perfectly in phase and then change the phase of the imaginary part in such a way that the peaks in both channels become perfectly phase-coherent. Next, both channels can be added so that the signals will increase 2 fold. On the other hand, if the noise in both channels were phase-incoherent, it will increase more slowly and therefore the overall S/N should increase by sqrt(2).

We can carry out this experiment very easily by applying a 90º phase shift into the imaginary channel and then summing up both channels. This is done with the following script:


Before applying this script, we need to calculate the SNR of the original spectrum. We can use the central peak of the Chloroform multiplet as a reference peak to estimate the SNR as depicted in the figure below:

Next, we can apply the script above (sumReImPhased) to apply a 90º phase shift to the imaginary part followed by the addition of the imaginary channel to the real one. If the SNR of this new spectrum is calculated we get the following result:

We can appreciate that the Chloroform peak is now twice the height, but the standard deviation has also increased two-fold, so the SNR remains constant. What a disappointment!

Conclusions:

We have first found out that the noise in the Real and Imaginary channels is statistically uncorrelated. However, this does not mean that they are independent. Just as sin(x) and cos(x) functions are orthogonal "uncorrelated", but not independent (sin^2(x) = 1 – cos^2(x)).
And we have shown that noise in the Re and Im channels is certainly not independent: the SNR does not improve at all.
The essence is that if the spectrometer has just one coil (which is, to the best of my knowledge, always the case in spectroscopy), then the noise in the two orthogonal channels would not be independent and thus the sqrt(2) sensitivity enhancement is not possible. It will be necessary to have two separate coils (and two receivers) in quadrature to have uncorrelated real and imaginary noise. Why is this not possible? I don’t really know, most likely because of lack of space …

References:

Experimental Noise in Data Acquisition and Evaluation III. Exponential Multiplication, Discrete Sampling, and Truncation Effects in FT Spectroscopy. Dr. S. Sýkora.

Real NMR

We often forget that NMR spectra are complex entities. By complex I don’t mean complicated or difficult to understand (which is, unfortunately, a very common thing too), but by the numbers in the complex space, that is, formed by real and imaginary numbers. One reason that can justify why NMR spectra are barely considered as formed by complex numbers is that NMR software, in general, only displays the real part. However, the imaginary part is present in the background (unless discarded e.g. to save memory) and it’s actually very important. The imaginary part makes it possible to correct the phase of the spectra in such a way that we can get nice absorptive lines (e.g. Lorentzian / Gaussian lines) which provide higher resolution than out-of-phase spectra (e.g. combination of absorptive and dispersive lines. See figure below).




NMR spectra, being of complex nature, can also be displayed in magnitude mode which makes the spectra phase insensitive. This can be advantageous in those cases in which phase correction is difficult, but at the expense of poorer resolution (though in the case of 13C NMR, magnitude mode can facilitate automatic processing, as I have blogged before). In addition, phase information can be very useful in some cases (e.g. DEPT, NOESY, edited NMR experiments, etc).


Acquisition of NMR spectra as complex numbers is also important (in addition to making phase correction possible) to distinguish positive from negative frequencies (quadrature detection. See this link for more information).

Mnova does not have a built-in function to show the imaginary part of the spectrum, but in the case you really need to see it, I will offer here two solutions:

1. One obvious way is by realizing that that the Re and Im channels differ by a phase shift of 90º. Thus, if we apply a (zero order) phase correction of +/-90º, the real part displayed in the software becomes analogous or equivalent to the imaginary counterpart.

2. There is a second alternative: it’s possible to write a very simple script which swaps the real with the imaginary part of the spectrum. This is the script:

1. function swapRI()
2. 
   
3. {
4. 
   
5. var w = new DocumentWindow(mainWindow.activeWindow());
6. 
   
7. //get active spectrum
8. 
   
9. var spectrum = nmr.activeSpectrum();
10. 
    
11. if (!spectrum.isValid() || spectrum.isReal)
12. 
    
13. return;
14. 
    
15. if (spectrum.dimCount > 1)
16. 
    
17. return; //1D only
18. 
    
19. var dCount = spectrum.dimCount;
20. 
    
21. var npts = spectrum.count(1);
22. 
    
23. //To print the value of one of the spectral points
24. 
    
25. for (var i=0; i<npts; i++)
    
26. 
    
27. {
28. 
    
29. var tmp = spectrum.real(i);
30. 
    
31. spectrum.setReal(i, spectrum.imag(i));
32. 
    
33. spectrum.setImag(i, tmp);
34. 
    
35. }
36. 
    
37. spectrum.update();
38. 
    
39. w.update();
40. 
    
41. }
42. 
    
43. }

Having said that NMR spectra are of complex nature and why this is important, then, how is it possible, from a hardware point of view, to get real and imaginary numbers? After all, NMR detectors are just measuring ‘real’ values (i.e. induced oscillating voltage in a resonant RF coil). When we say Re and Im numbers in this context, we could have said X and Y values, or u and v values, etc. What matters here is that we are using 2 (orthogonal) detectors to measure (generally simultaneously) the magnetization along X and Y axis. We know that the Re and Im components in the complex space are related by a phase angle of 90º, analogous to the X and Y axis. Thus, the values recorded by the detector placed along the X axis can be considered as the real part of the spectrum (or the cosine component), whereas the values measured in the detector placed in the Y axis will correspond to the imaginary part (or sine component) of the spectrum.

What I would like to highlight here is that the Re and Im, in principle, could appear as 2 INDEPENDENT physical measures, even though they are (generally) recorded simultaneously. In other words, the Re and Im channels are UNCORRELATED. However, this is not completely true. I have already mentioned that the dispersive and absorptive lines (corresponding to Real and Im components of a phased spectrum) are related by a phase shift of 90º. More strictly speaking, the Re and Im components are connected by the well known Kramers–Kronig relation being the Hilbert Transform the mathematical bridge connecting both components. This transform makes it possible to obtain the imaginary part from the real one and the other way round. How about the noise? Is the noise in the imaginary and real parts uncorrelated? In that case, theoretically, it should be possible to achieve a further sqrt(2) gain in sensitivity when quadrature detection is used. This issue will be the subject of my next post.

References
Quadrature versus linear detection, and where do complex MR data come from
Nothing is better