I have a challenge for you. It's a puzzle that I can't find the solution to. Before your challenge, I need to finish the story of my challenge (I already told the beginning and the middle part: see previous posts). To tell my story, however, I need to explain my language first. When I asked for help, on September 9, nobody understood what I really needed, because we speak different languages. While I look at the mathematical description of NMR processing, you think at the software command that performs it. This time I will try to be the clearest I can. The world of NMR is complicated because every program stores data in a different way. There is the sequential storage and the storage in blocks, the little endian and the big endian, the interleaved and the non-interleaved. Add to this that the spectrometer is free, when writing the data on disc, to change the sign to a part of the points. When acquiring a multidimensional spectrum, which is made of many FIDs, it is also free to collect those FIDs in any order (this is the case of Varian's, which has two different ways to acquire a 3D spectra, although one was enough, and none of the two ways is vaguely similar to the Bruker way...). NMRPipe adds a further complication, because when it read Varian data it changes the sign (and the users can't know). There are also different FT algorithms that don't yield the same spectrum (the real part is always the same, but the sign of the imaginary part looks like an arbitrary choice).
I don't want to talk about these dull details: they are hackers' business (and, unfortunately, my business too).
I want to talk, instead, about phase-sensitive detection, because this is where my deepest troubles came from.
Phase-sensitive detection brings so many advantages, in terms of sensitivity and resolution, that it's almost a must in NMR spectroscopy. To accomplish it, the spectrometer measures the magnetization twice (in time or in space). When the spectrum arrives on the disc, two numbers are stored where there used to be one. To accomplish phase-sensitive detection in bidimensional spectroscpy, two FIDs are acquired instead of one. To accomplish phase-detection in 3-D spectroscopy, two bidimensional experiments are performed instead of one. In summary, a 3D FID contains 8 values that correspond to an identical combination of t1, t2 and t3 coordinates (a couple of couples of couples). In practice it can become a mess, but the rules are simple enough. Whatever the number of dimensions, each and every value has a counterpart (think like brother and sister). You don't need to think in terms of rows, planes or cubes. Just remember that each sister has a corresponding brother and vice versa. There is no difference between 2D and 3D spectra. There is an extra operation, however, that is absent in 1D spectroscopy. It's called the Ruben-States protocol (you can add more names to the brand, if you like; I call it, more generically: "shuffling"). How does it work?
We have a white horse and a white mare (brother and sister). We know that every mare has a brother and every horse has a sister. This is also true for the mother of the two white horses. She had a black brother, with a black son and a black daughter. Now, let's swap the black horse with the white mare. The members of the new pairs have the same sex. With this combination you can continue the race towards the frequency domain and win. The trick works in almost all kinds of homo-nuclear spectra. In the 2D case, you have to perform two FTs. The shuffling goes in between. In the 3D case, there are 3 FTs and 2 intermediate shuffles: FT, shuffle, FT, shuffle, FT. Maybe you have heard about alternatives called States-TPPI or the likes. Actually they all are exactly the same thing.
Our trick is not enough when gradient-selection is employed in hetero-nuclear spectroscopy. You hear the names "sensitivity enhanced" or "echo-antiecho" or "Rance-Key". It's easier for me to remember Ray Nance instead, although he had nothing to do with today's subject. Whatever the name, this kind of experiments require an ADDITIONAL step, to be performed before anything else, or at least before the shuffling. Here is what you have to do. Let's say that A is the pair of white horses and B the pair of black ones. Calculate:
S = A + B.
D = A - B.
Now rotate D by 90 degrees. This latter operation has no equivalent in the realm of animals. It would consists into a double transformation: the brother would become female and the sister a... negative male, if such an expression has any sense. Finally, replace A with S and B with the rotated D.
The 3 operations involved (addition, subtraction and phase rotation) all commute with the FT. Knowing this fact, I have always moved Ray Nance from his canonical place (before the first FT), to the moment of the shuffle, and fused the two together. In this way I have eliminated a processing step. The condensed work-flow looks simpler to my eyes, but this is a matter of opinion.
Unfortunately, this was the reason why I could not process Varian 3D spectra: I have tried all the possible sign inversions, but no one worked. Every time the number of peaks was the double (or quadruple) than it should have been. The mirror image was mixed with the regular spectrum.
If, instead, I process the raw data as described in literature, that is before the standard workflow, I get the correct spectrum. Now: I still believe that sum, addition and phase rotation commute with FT. I can apply this property to all the 2D spectra and to all the Bruker 3D spectra I have met so far. The property ceases to work in the case of Varian spectra.
Question: do you know why? I don't.