Phase Cycling and Gradient Pulses - The James Keeler Group

step
pulse
phase
phase shift experienced
by transfer with ∆p = +1
equivalent
phase
rx. phase to
select ∆p = –3
difference
1 0 0 0 0 0
2 90 –90 270 270 270 – 270 = 0
3 180 –180 180 180 180 – 180 = 0
4 270 –270 90 90 90 – 90 = 0
Here we see quite different behaviour. The equivalent phases, that is the phase
shifts experienced by the transfer with ∆p = 1, match exactly the receiver
phases determined for ∆p = –3, thus the phases in the "difference" column are
all zero. We conclude that the four step cycle selects transfers both with ∆p =
–3 and +1.
Some more work with tables such as these will reveal that this four step
cycle suppresses contributions from changes in coherence order of –2, –1 and 0.
It selects ∆p = –3 and 1. It also selects changes in coherence order of 5, 9, 13
and so on. This latter sequence is easy to understand. A pathway with ∆p = 1
experiences a phase shift of –90° when the pulse is shifted in phase by 90°; the
equivalent phase is thus 270°. A pathway with ∆p = 5 would experience a
phase shift of –5 × 90° = –450° which corresponds to an equivalent phase of
270°. Thus the phase shifts experienced for ∆p = 1 and 5 are identical and it is
clear that a cycle which selects one will select the other. The same goes for the
series ∆p = 9, 13 ...
The extension to negative values of ∆p is also easy to see. A pathway with
∆p = –3 experiences a phase shift of 270° when the pulse is shifted in phase by
90°. A transfer with ∆p = +1 experiences a phase of –90° which corresponds to
an equivalent phase of 270°. Thus both pathways experience the same phase
shifts and a cycle which selects one will select the other. The pattern is clear,
this four step cycle will select a pathway with ∆p = –3, as it was designed to,
and also it will select any pathway with ∆p = –3 + 4n where n = ±1, ±2, ±3 ...
9.5.2 General Rules
The discussion in the previous section can be generalised in the following way.
Consider a phase cycle in which the phase of a pulse takes N evenly spaced
steps covering the range 0 to 2π radians. The phases, φ k
, are
φ k
= 2πk/N where k = 0, 1, 2 ... (N – 1).
To select a change in coherence order, ∆p, the receiver phase is set to
–∆p × φ k
for each step and the resulting signals are summed. This cycle will, in
addition to selecting the specified change in coherence order, also select
pathways with changes in coherence order (∆ p ± nN) where
n = ±1, ±2 ..
The way in which phase cycling selects a series of values of ∆p which are
related by a harmonic condition is closely related to the phenomenon of aliasing
in Fourier transformation. Indeed, the whole process of phase cycling can be
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