Where X and Y are the real and imaginary FIDs coming out of the ADC. Thus, we can speak of the "receiver phase" as the point of view from which we view the FID signal in the rotating frame of reference. If the receiver phase is shifted by 90° (from x' to y' axis), we mean that the "real" channel has been shifted counterclockwise by 90° from the x' axis to the y' axis, and the "imaginary" channel has been shifted counterclockwise by 90° from the y' axis to the —x' axis. With this 90° shift, for example, a coherence of Iy would be observed as Ix, and in a four-scan phase cycle with A$r = 90°, we would observe a four-scan sequence of coherences Ix, Iy, —Ix, —Iy as 4Ix in the sum-to-memory. To keep track of these phase shifts, a shorthand notation is often used where 0 stands for a 0° phase shift (real part of receiver on the x' axis), 1 stands for a 90° phase shift (receiver on the y' axis), 2 stands for a 180° phase shift (receiver on the — x' axis), and 3 stands for a 270° phase shift (receiver on the — y' axis). In this notation, the four-scan receiver phase cycle with A$r = 90° would be written: 0 12 3.

In general, the receiver phase should follow the phase shifts that result from a shift in the phase of a pulse

For a series of pulses (a pulse sequence), we can select the change in coherence order Ap resulting from each of the pulses if we phase cycle all of the pulses and then calculate the effect of the desired coherence pathway on the final phase. If we diagram the coherence pathway, we can note the change in coherence order Ap caused by each pulse and then calculate the receiver phase change necessary to make the desired combination of Ap's add together at the receiver while all other pathways cancel:

where A$p is the phase increment for each pulse and the sum is taken over all pulses that are phase cycled. The phase cycling of pulses must be done independently for this selection to work; which means that if the first pulse has a phase shift of 180° (x, — x or 0 2) and the second pulse has a phase shift of 90° (x, y, —x, — y or 0 1 2 3), it would be necessary to have eight scans in our phase cycle:

Pulse 1: 00002222

Pulse 2: 01230123

How selective can we be in terms of Ap? If we cycle the phase of a pulse by 360°/N, then with N scans in the phase cycle we are effectively putting up a "mask" that permits every Nth value of Ap to get through the mask. For example, if N = 2 (pulse phase x, — x or 0 2), and we set the receiver phase to A$r = —Ap x A$p = —(—1) (180°) = —180° = 180°, we would permit Ap = — 1 as planned, but we would also allow Ap = —3, —5, —7 and so on as well as Ap = 1, 3, 5, and so on. That is, the "holes" in our mask are spaced every second value of Ap. To block Ap = 1 and permit Ap = — 1, we would need a four-scan phase cycle (N = 4, AOp = 90o) and the "mask" would have holes every fourth value of Ap, allowing Ap = -5, -1,3, and so on. This is a much more effective phase cycle, but it takes twice as many scans to complete it:

Pulse: Receiver:

0 0

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