### Why one of the rf pulses in a multiple-pulse sequence does not need to be phase cycled?

Consider a three-pulse sequence for simplicity. It is easy to show that Pulse-3 does not need to be phase cycled, for example.

The receiver phase φ_{rec} is defined by

φ_{rec} = -φ1*Δp1 - φ2*Δp2 - φ3*Δp3, (**Eq. 1**)

where φi is the phase of Pulse-i and Δpi is the difference between the coherence order p
after Pulse-i and that before Pulse-i. That is

Δp1 = p1 - p0, Δp2 = p2 - p1, and Δp3 = p3 - p2

where p0 = 0 and p3 = -1.

Since Δp1 + Δp2 + Δp3 = -1 whatever the coherence transfer pathway is,
we replace Δp3 in the expression of the receiver phase (**Eq. 1**) by

Δp3 = -1 - Δp1 - Δp2.

As a result, the expression of the receiver phase becomes

φ_{rec} - φ3 = -(φ1 - φ3)*Δp1 - (φ2 - φ3)*Δp2, (**Eq. 2**)

which can be rewritten as

ψ_{rec} = -ψ1*Δp1 - ψ2*Δp2.

The phase of Pulse-3 does not appear in the definition of the receiver phase.

This result is in agreement with the well-known property in phase cycling:

"**additing a constant phase to the receiver phase and to those of all the rf pulses does
not change the NMR signal**".

Application of this property to **Eq. 1** gives **Eq. 2**.