Nyquist Theorem and Resolution in Multidimensional NMR

Two relations determine main limitations of multidiemnional NMR.

Linewidths are inversely proportional to maximum evolution time in each dimension:

lw α 1/tmax

And the spectral width is limited by time delay between evolution time points:

sw = 1/Δt

Keeping constant number of time points (i.e. constant experiment time) you cannot improve resolution AND spectral range.

But I want both ...

Above limitations are strictly connected with conventional point distribution. Equispaced points are comfortable for FT processing, but you have to care of above sw/lw dependence.

Maybe I should try non-conventional sampling?

Like this ...

But Fourier can't transform that!

No, you are wrong. Fourier transform is just integral, and may be calculated using any data. All you have to do is to use REALLY multidimensional transform, not a sequence of 1D FTs :

Instead use one, but really multidimensional procedure. For each frequency point of spectrum integrate product of your signal and appropriate period function. For example, if you want to transform signal ,
take PAIRS of frequencies (ω1,ω2) and calculate sum over a evolution time space.

Where w(t1,t2) are weighting terms, if you want to weight your signal.

How about quadrature ...?

In standard 1D Fourier Transform, quadrature is needed if you want to get correct frequency values AND signs. Elegant description of this procedure is given by complex notation:

In case of multidimensional FT high-order commutative equivalents of complex numbers should be used.In general, they are called Clifford Algebras.

I am not familiar with Clifford Algebras...

It doesn't matter. In fact, calculating real part of the Multidimensional FT is just adding four transforms of four signal modulations :

These four modulations are the same that you have to measure in conventional experiment. You can also process echo-antiecho modulated signals by combining them to achieve above form.

Have you ever tried to use it or is it just a theory?

Yes, and we achieved great impovement in resolution. That's one of examples, fragment of 3D HNCA of Ubiquitin :

Resolution for free? I am a bit suspicious ...

Of course nothing is for free. Random sampling generates so called "sampling noise" even for perfect signals:

But this noise is in fact part of a peak. Its shape is connected with frequency and intensity of peak, just like (in conventional case) "sinc" function is a result of Fourier Transform of unweighted FID. So, shape of noise can be easily calculated and substracted from spectrum. You just need a list of frequencies and linewidths of strongest peaks. They can be picked from noisy spectrum which you want to clean. Cleaning can reveal very small peaks possibly hidden under noise. We have prepared special cleaning program called Handy which is part of MFT package.

How about other sampling schemes?

Presented algorithm is general way of processing sparse time domain data. If you use radially sampled time domain data as an input you will get spectra (nad artifacts) of the same shape as in well known Projection Reconstruction method!

We have also tried spiral sampling of time domain. Artifacts in this case are much wider spread (in fact they are placed on spiral also in frequency domain)than in radial case.However, we proved that sampling artifacts have the smallest magnitude in the case of random sampling.

OK, I am still a bit suspicious, but I want to try this method...

You can get all needed programs for free, just click "Get MFT package" on the right side of this page, and register. Actual version gives output in both Sparky and NMRPipe format. It takes raw Varian FID or spectra processed in NMRPipe in direst dimension as an input. All details for execution of MFT programs are given in included README file. Details for MFT transform can be found in our works:

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