= NMR phase cycling
{c}
<NMR phase cycling> systematically changes <radio frequency> pulse phases and the receiver phase across repeated <nuclear magnetic resonance> acquisitions so that selected <signals> reinforce while unwanted contributions cancel. Pulse phase controls the rotation axis in the <rotating frame>; receiver phase controls how transverse <magnetization> is represented in the recorded complex <signal>. A phase change does not mean changing the resonant nucleus or physically rotating the sample.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[Chapter9, sections9.1–9.2.3, pp.9-1–9-5: selection, pulse phase and receiver phase]{ref}
An original two-acquisition example makes the subtraction explicit. Suppose changing an excitation pulse by $\pi$ reverses the desired contribution $S$ while leaving an unwanted background $B$ unchanged. Before correcting the receiver phases, the two measurements are
$$
s_1=S+B,\qquad s_2=-S+B.
$$
Applying the corresponding receiver sign and averaging gives $(s_1-s_2)/2=S$. With invented values $S=5$ and $B=2$, the recorded values7 and−3 yield5. Adding them without the intended sign would instead retain the background.
The assumptions matter. If the background changes between acquisitions, the residual is $(B_1-B_2)/2$. Nor does a two-step cycle separate contributions that transform in exactly the same way under its phase changes. More elaborate cycles select particular changes in <NMR coherence>, rather than indiscriminately removing every source of error.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[Chapter9, section9.1: multiple possible pathways and systematic selection]{ref}
Phase patterns can also be built into a train of refocusing pulses. <XY8> is an important example for controlling cumulative <NMR pulse errors>. Within-train pulse compensation and across-acquisition selection are related uses of phase control, but the eight pulses of an <XY8> block do not by themselves specify a complete receiver cycle.
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