Free induction decay acquisition in NMR

Recently, there was an interesting Q&A exchange on free induction decay acquisition in NMR [Is intensity in NMR spectra acquired at regular time intervals?](https://chemistry.stackexchange.com/questions/162908/is-intensity-in-nmr-spectra-acquired-at-regular-time-intervals) with useful answers. I have not acquired FIDs in NMR or used the raw data but rather dealt with microwave rotational spectra free induction decay data-which is purely real with its own pecularities.


The interesting point for me in NMR-FID was the mathematical notion of a *complex* free induction decay. Although the FID experiment measures "real" values, since there a quadrature detection, and the two coils are orthogonal, it makes sense to *treat* it as a complex number.


My questions are:


(i) What is the criterion of calling one signal as real and the other imaginary from the two orthogonal coils. I mean, both coils are orthogonal to each other. I think this label must be arbitrary to call one real and the other imaginary (i.e., the labels can be exchanged). This website [MRI questions](https://mriquestions.com/real-v-imaginary.html) says the same.


(ii) Which FID is used for displaying the NMR spectrum? Both FID from the two receivers should appear like a free induction curve since are both *real* experimental values. Discrete Fourier transform of each should yield the same frequency information i.e., the same NMR spectrum. There must a be phase difference but frequency information must be the same. Is this correct?





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> (1) What is the criterion of calling one signal as real and the other imaginary from the two orthogonal coils
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Assuming that you have detection axes aligned with the $x$- and $y$- axes ($y$ being offset by $+90^\circ$ from $x$), then for a single spin with precession frequency $\omega$, the signals you get will be:


$$S\_x = \cos(\omega t); \quad S\_y = \sin(\omega t),$$


and then we choose to define the complex signal as


$$S := S\_x + \mathrm{i}S\_y = \exp(\mathrm{i}\omega t).$$


In this case $x$ is the real axis and $y$ is the imaginary axis. Complex FT of this yields a peak at frequency $\omega$, as desired. However, this isn't the only way to do it. If we wanted to swap them around, we chould choose $y$ as the real axis and $x$ as the imaginary axis. However, we would need to be careful to include a minus sign in the definition:


$$S' := S\_y - \mathrm{i}S\_x = -\mathrm{i}\exp(\mathrm{i}\omega t)$$


This is the same signal as before but with an additional global phase factor, which is unimportant. Notice that if we defined $S' := S\_y + \mathrm{i}S\_x$ we would get a peak with negative frequency. In some sense, the imaginary isn't the $x$-axis, it's the *negative* $x$-axis. I think the summary from this is that the imaginary axis should be offset from the real axis by $+90^\circ$ (and not $-90^\circ$).



> 
> (2) Which FID is used for displaying the NMR spectrum?
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Both, actually: we don't FT only the real or imaginary part of the FID. We form a complex FID and perform a complex FT on that, i.e. $\int\_{-\infty}^\infty f(t)\exp(-\mathrm{i}\omega t)\,\mathrm{d}t$ to get a complex spectrum.


Traditionally, only the real part of the spectrum is displayed to the user, though.



> 
> [...] Both FID from the two receivers should appear like a free induction curve since are both real experimental values. Discrete Fourier transform of each should yield the same frequency information i.e., the same NMR spectrum. There must a be phase difference but frequency information must be the same. Is this correct?
> 
> 
> 


Yes, both real and imaginary components of the FID will contain the same frequency information. However, neither component alone is sufficient for providing quadrature detection, because one signal is


$$S\_x = \cos(\omega t) = \frac{\exp(\mathrm{i}\omega t) + \exp(-\mathrm{i}\omega t)}{2}$$


and the other signal is


$$S\_y = \sin(\omega t) = \frac{\exp(\mathrm{i}\omega t) - \exp(-\mathrm{i}\omega t)}{2\mathrm{i}}$$


so if each of them is individually FT'd, you get duplicate peaks at $\pm\omega$, hence the need to combine them. (If your frequencies are only positive, then of course you don't need two components; you just discard the negative frequency parts, or equivalently use a cosine transform instead, i.e. $\int\_{-\infty}^\infty f(t)\cos(\omega t) \,\mathrm{d}t$. But in NMR, the frequencies are always going to have both signs.)


As for the spectra, you can reconstruct the imaginary part of the spectrum from the real part of the spectrum via a Hilbert transform. The resulting formulae relating the two parts are also known as the [Kramers–Kronig relations](https://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations).


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