  Non-orthogonal multiple access (NOMA) is considered as one of the predominant
multiple access technique for the next-generation cellular networks. We
consider a 2-user pair downlink NOMA system with imperfect successive
interference cancellation (SIC). We consider bounds on the power allocation
factors and then formulate the power allocation as an optimization problem to
achieve {$\alpha$-Fairness} among the paired users. We show that
{$\alpha$-Fairness} based power allocation factor coincides with lower bound on
power allocation factor in case of perfect SIC and $\alpha > 2$. Further, as
long as the proposed criterion is satisfied, it converges to the upper bound
with increasing imperfection in SIC. Similarly, we show that, for $0<\alpha<1$,
the optimal power allocation factor coincides with the derived lower bound on
power allocation. Based on these observations, we then propose a low complexity
sub-optimal algorithm. Through extensive simulations, we analyse the
performance of the proposed algorithm and compare the performance against the
state-of-the-art algorithms. We show that even though Near-Far based pairing
achieves better fairness than the proposed algorithms, it fails to achieve
rates equivalent to its orthogonal multiple access counterparts with increasing
imperfections in SIC. Further, we show that the proposed optimal and
sub-optimal algorithms achieve significant improvements in terms of fairness as
compared to the state-of-the-art algorithms.
