What a fun afternoon :) Every time I think I'm ready to push the SEND
button on this summary, someone else checks in with a great suggestion!
Thanks to all 15+ respondents for your various wisdom on this. What a great
crowd! I do believe I (mostly) get it! I'll only mention a few.
Glenn Facey, your blogs on quadrature detection
<
http://u-of-o-nmr-facility.blogspot.com/2008/08/what-is-quadrature-detection.html>
and phasing
<
http://u-of-o-nmr-facility.blogspot.com/2010/11/phase-of-nmr-spectrum.html>
were very helpful as a starting point, getting me thinking about the
complex FT. Clearly that was what I was missing.
I think the best discussion that I found regarding the complex FT is in Chapter
31 of The Scientist and Engineer's Guide to Signal Digital Processing, 1999
<
https://www.analog.com/media/en/technical-documentation/dsp-book/dsp_book_Ch31.pdf>,
by Steven Smith. I particularly like the ending paragraph of that chapter:
*Look at this situation this way. In spite of its abstract nature, the
complexFourier transform properly describes how physical systems behave.
When werestrict the mathematics to be real numbers, problems arise. In
other words,these problems are not solved by the complex Fourier transform,
they areintroduced by the real Fourier transform. In the world of
mathematics, thecomplex Fourier transform is a greater truth than the real
Fourier transform.This holds great appeal to mathematicians and
academicians, a group thatstrives to expand human knowledge, rather than
simply solving a particularproblem at hand.*
So that is what I was needing from a pure math sense. It's not that we are
getting the standard real and imaginary results of an FT. It's that we
starting with two channels, placing one in a real part and one in an
imaginary part of an input signal and then transforming them *together *to
produce those r an i components of our NMR spectra that we know and love.
That's quite elegant.
Thank you, Kosuke Ohgo, for your link to the chemistry stackExchange
discussion
<
https://chemistry.stackexchange.com/questions/32447/why-does-the-transverse-magnetization-in-fourier-transform-nmr-consist-of-ixiiy>
nicely illustrating the key point of how, mathematically, adding that
phase-shifted quad signal gets us better detection with the reference in
the middle of the spectrum rather than the edge. Very nice!
And final kudos to Rich Shoemaker for pointing out that our AVANCE III uses
digital quadrature detection, taking this all to another level of
sophisticated electronics. This is very nicely described in Chapter 3
<
https://edisciplinas.usp.br/pluginfile.php/6595411/mod_resource/content/1/Practical%20aspects%20of%20High-res%20NMR.pdf>
of Claridge's High-Resolution NMR Techniques in Organic Chemistry
<
https://www.sciencedirect.com/bookseries/tetrahedron-organic-chemistry-series/vol/27/suppl/C>,
which several respondents suggested. (Not 100% sure that chapter copy is
legit, but I read it anyway.)
Many others alluded to these ideas, of course. Thank you all!
My brain is full for the day.
Bob
Received on Thu Feb 16 2023 - 11:45:29 MST