 James Keeler

James Keeler works on high-resolution nuclear magnetic resonance methods and explains their physical basis through experiments and mathematical models. His university profile identifies this research focus and his authorship of Understanding NMR Spectroscopy.^[ref] The author-hosted Irvine notes give a particularly concrete account of how receiver electronics turn rotating nuclear magnetization into a signed frequency measurement.^[ref]

A single receiver channel proportional to $\cos (\Omega t)$ cannot distinguish offsets $+\Omega$ and $-\Omega$, because cosine is even. Two receiver references separated in phase by $\pi /2$ give quadrature detection: the two outputs represent $\cos (\Omega t)$ and $\sin (\Omega t)$ and combine into the complex number $e^{i\Omega t}$. This determines the sense of rotation in the rotating frame as well as its rate.^[ref]

For an original numerical illustration, offsets of $+100\mathrm{Hz}$ and $-100\mathrm{Hz}$ at $t=1.25\mathrm{ms}$ have phases $+\pi /4$ and $-\pi /4$. Both give the same real channel $1/\sqrt{2}$, but the imaginary channels are respectively $+1/\sqrt{2}$ and $-1/\sqrt{2}$. This is why retaining both receiver channels preserves information lost by a single real measurement.