if user i (i = 1, 2, . . . ,K) is perfectly paired, and the proof is
completed.
Remark 1: It has been revealed in both [14] and [15] that the
quasi-degradation probability is high for i.i.d Rayleigh distributed
channels, especially in a heterogenous environment.
Moreover, by adopting the proposed user pairing algorithms,
this probability can be further increased. In other words,
most of the users can be paired perfectly after performing
the proposed user pairing algorithm. Therefore, the overall
diversity achieved by the proposed beamforming algorithms
is close to N −K +2, which can be validated in our simulation
results. On the other hand, the overall diversity is related to the
number of user pairs having quasi-degraded channels. That
means, to obtain a good diversity performance, an efficient
user pairing algorithm should maximize the number of user
pairs having quasi-degraded channels, which is also the goal
of the proposed PBPA/IBPA.
To derive the total power consumption for the proposed
beamforming algorithms (PH-NOMA in combination with
PBPA/IBPA) and conventional ZFBF, we first define some
user sets. Note that the total user set is